{"id":18145,"date":"2022-04-04T21:24:37","date_gmt":"2022-04-04T21:24:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18145"},"modified":"2022-08-31T17:00:32","modified_gmt":"2022-08-31T17:00:32","slug":"cr-1-operations-with-real-numbers-and-order-of-operations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-1-operations-with-real-numbers-and-order-of-operations\/","title":{"raw":"CR.1: Operations with Real Numbers and Order of Operations","rendered":"CR.1: Operations with Real Numbers and Order of Operations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Model addition of integers with color counters<\/li>\r\n \t<li>Model integer subtraction with color counters<\/li>\r\n \t<li>Multiply integers with different signs<\/li>\r\n \t<li>Multiply integers with the same sign<\/li>\r\n \t<li>Multiply an integer by -1<\/li>\r\n \t<li>Divide integers with the same sign<\/li>\r\n \t<li>Divide integers with different signs<\/li>\r\n \t<li>Divide integers by -1<\/li>\r\n \t<li>Use correct notation to indicate absolute value<\/li>\r\n \t<li>Simplify expressions that contain\u00a0absolute value<\/li>\r\n \t<li>Evaluate expressions that contain absolute value<\/li>\r\n \t<li>Use the order of operations to simplify expressions that involve integer multiplication, division, addition, and subtraction<\/li>\r\n \t<li>Evaluate integer expressions using the order of operations<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Modeling Addition of Integers<\/h2>\r\nWe will first discuss arithmetic operations with integers.\r\n\r\nMost students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.\r\n\r\nThe brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of [latex]2 - 1[\/latex]. Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.\r\n\r\nAddition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.\r\n\r\nWe will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220151\/CNX_BMath_Figure_03_02_024_img.png\" alt=\"This figure has a blue circle labeled positive and a red circle labeled negative.\" \/>\r\nIf we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in the figure below.\r\n\r\nA blue counter represents [latex]+1[\/latex]. A red counter represents [latex]-1[\/latex]. Together they add to zero.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220152\/CNX_BMath_Figure_03_02_001.png\" alt=\"This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.\" \/>\r\nDoing the Manipulative Mathematics activity \"Addition of signed Numbers\" will help you develop a better understanding of adding integers.\r\nWe will model four addition facts using the numbers [latex]5,-5\\text{ and }3,-3[\/latex].\r\n\r\n[latex]5+3 - 5+\\left(-3\\right)-5+35+\\left(-3\\right)[\/latex]\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]5+3[\/latex].\r\n\r\nSolution:\r\n<table id=\"eip-id1168469436825\" class=\"unnumbered unstyled\" summary=\"This figure has five rows. The first row states interpret the expression and has 5 plus 3 means the sum of 5 and 3. The second row states to model the first number and start with 5 positives. It has 5 blue circles. The third row states model the second number and add 3 positives. It has 5 blue circles followed by 3 blue circles. The third row states count the total number of counters. It is followed by 8 blue circles. The fifth row states the sum of 5 and 3 is 8 and is followed by the expression 5 plus 3 equals 8.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]5+3[\/latex] means the sum of [latex]5[\/latex] and [latex]3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220153\/CNX_BMath_Figure_03_02_025_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the second number. Add [latex]3[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220154\/CNX_BMath_Figure_03_02_025_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the total number of counters.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220155\/CNX_BMath_Figure_03_02_025_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The sum of [latex]5[\/latex] and [latex]3[\/latex] is [latex]8[\/latex].<\/td>\r\n<td>[latex]5+3=8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression.\r\n\r\n[latex]2+4[\/latex]\r\n[reveal-answer q=\"510181\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"510181\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220157\/CNX_BMath_Figure_03_02_005_img.png\" alt=\"This figure has six pink circles in a row, representing positive counters. The first two circles are separated from the following four circles.\" \/>\r\n\r\n[latex]6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression.\r\n\r\n[latex]2+5[\/latex]\r\n[reveal-answer q=\"565122\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"565122\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220158\/CNX_BMath_Figure_03_02_006_img.png\" alt=\"This figure has seven pink circles in a row, representing positive counters. Two circles are separated from the following 5 circles.\" \/>\r\n\r\n[latex]7[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]-5+\\left(-3\\right)[\/latex].\r\n[reveal-answer q=\"78976\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"78976\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466019221\" class=\"unnumbered unstyled\" summary=\"This figure has five rows. The first row states to interpret the expression and is followed by negative 5 plus negative 3 means the sum of negative 5 and negative 3. The second row states to model the first number and start with 5 negatives followed by 5 red circles. The third row states model the second number and add 3 negatives followed by a row with 5 red circles and 3 red circles. The fourth row states count the total number of counters followed by 8 red circles. The fifth row states the sum of negative 5 and negative 3 is negative 8, and is followed by the expression negative 5 plus negative 3 equals negative 8.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]-5+\\left(-3\\right)[\/latex] means the sum of [latex]-5[\/latex] and [latex]-3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220159\/CNX_BMath_Figure_03_02_026_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the second number. Add [latex]3[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220201\/CNX_BMath_Figure_03_02_026_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the total number of counters.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220202\/CNX_BMath_Figure_03_02_026_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The sum of [latex]\u22125[\/latex] and [latex]\u22123[\/latex] is [latex]\u22128[\/latex].<\/td>\r\n<td>[latex]-5+-3=-8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression.\r\n\r\n[latex]-2+\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"818374\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"818374\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220204\/CNX_BMath_Figure_03_02_010_img.png\" alt=\"This figure shows a row of 6 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 4 circles.\" \/>\r\n\r\n[latex]\u22126[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression.\r\n\r\n[latex]-2+\\left(-5\\right)[\/latex]\r\n[reveal-answer q=\"875730\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875730\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220204\/CNX_BMath_Figure_03_02_011_img.png\" alt=\"This figure shows a row of 7 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 5 circles.\" \/>\r\n\r\n[latex]\u22127[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe first and second examples\u00a0are very similar. The first example adds [latex]5[\/latex] positives and [latex]3[\/latex] positives\u2014both positives. The second example adds [latex]5[\/latex] negatives and [latex]3[\/latex] negatives\u2014both negatives. In each case, we got a result of [latex]\\text{8-either}8[\/latex] positives or [latex]8[\/latex] negatives. When the signs are the same, the counters are all the same color.\r\nNow let\u2019s see what happens when the signs are different.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]-5+3[\/latex].\r\n[reveal-answer q=\"229832\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"229832\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468228028\" class=\"unnumbered unstyled\" summary=\"This figure has six rows. The first row states interpret the expression and is followed by negative 5 plus 3 means the sum of negative 5 and 3. The second row states model the first number and start with 5 negatives. It is followed by 5 red circles. The third row states model the second number and add 3 positives. It is followed by 5 red circles and 3 blue circles. The fourth row states remove any neutral pairs. It is followed by a row of 5 red circles and under that a row of 3 blue circles. The first three red and blue circles are grouped together. The fifth row states count the result and has 2 red circles. The sixth row states the sum of negative 5 and 3 is negative 2. It is followed by the expressions negative 5 plus 3 equals negative 2\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]-5+3[\/latex] means the sum of [latex]-5[\/latex] and [latex]3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220206\/CNX_BMath_Figure_03_02_027_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the second number. Add [latex]3[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220207\/CNX_BMath_Figure_03_02_027_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove any neutral pairs.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220209\/CNX_BMath_Figure_03_02_027_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the result.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220210\/CNX_BMath_Figure_03_02_027_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The sum of [latex]\u22125[\/latex] and [latex]3[\/latex] is [latex]\u22122[\/latex].<\/td>\r\n<td>[latex]-5+3=-2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNotice that there were more negatives than positives, so the result is negative.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression, and then simplify:\r\n[latex]2+\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"637985\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"637985\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220213\/CNX_BMath_Figure_03_02_016_img.png\" alt=\"This figure shows two rows of counter circles. The first row has 2 dark pink circles, representing negative counters. The second row has 4 light pink circles, representing positive counters.\" \/>\r\n\r\n[latex]\u22122[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression, and then simplify:\r\n[latex]2+\\left(-5\\right)[\/latex]\r\n[reveal-answer q=\"955006\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"955006\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220215\/CNX_BMath_Figure_03_02_023_img.png\" alt=\"This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 5 dark pink circles, representing negative counters.\" \/>\r\n\r\n[latex]\u22123[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]5+\\left(-3\\right)[\/latex].\r\n[reveal-answer q=\"862636\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"862636\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468738226\" class=\"unnumbered unstyled\" summary=\"This figure has six rows. The first row states interpret the expression and is followed by 5 plus negative 3 means the sum of 5 and negative 3. The second row states model the first number and start with 5 positives. It is followed by 5 blue circles. The third row states model the second number and add 3 negatives. It is followed by 5 blue circles and 3 red circles. The fourth row states remove any neutral pairs. It is followed by a row of 5 blue circles and under that a row of 3 red circles. The first three red and blue circles are grouped together. The fifth row states count the result and has 2 blue circles. The sixth row states the sum of 5 and negative 3 is 2. It is followed by the expression five plus negative 3 equals 2.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]5+\\left(-3\\right)[\/latex] means the sum of [latex]5[\/latex] and [latex]-3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220216\/CNX_BMath_Figure_03_02_028_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the second number. Add [latex]3[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220218\/CNX_BMath_Figure_03_02_028_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove any neutral pairs.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220219\/CNX_BMath_Figure_03_02_028_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the result.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220220\/CNX_BMath_Figure_03_02_028_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The sum of [latex]5[\/latex] and [latex]\u22123[\/latex] is [latex]2[\/latex].<\/td>\r\n<td>[latex]5+\\left(-3\\right)=2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression, and then simplify:\r\n\r\n[latex]2+\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"512582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"512582\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220221\/CNX_BMath_Figure_03_02_022_img.png\" alt=\"This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 4 dark pink circles, representing negative counters.\" \/>\r\n\r\n[latex]\u22122[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression:\r\n[latex]\\left(-2\\right)+5[\/latex]\r\n[reveal-answer q=\"556256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"556256\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220222\/CNX_BMath_Figure_03_02_017_img.png\" alt=\"This figure shows two rows of counters. The first row shows 2 dark pink circles, representing negative counters. The second row has 5 light pink circles, representing negative counters. The neutral pairs of one positive and one negative counter are circled leaving three positive counters.\" \/>\r\n\r\n[latex]3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel each addition.\r\n<ol>\r\n \t<li>[latex]4 + 2[\/latex]<\/li>\r\n \t<li>[latex]\u22123 + 6[\/latex]<\/li>\r\n \t<li>[latex]4 + (\u22125)[\/latex]<\/li>\r\n \t<li>[latex]-2 + (\u22123)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"480477\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"480477\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468294641\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]4[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220223\/CNX_BMath_Figure_03_02_035_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add two positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220224\/CNX_BMath_Figure_03_02_035_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many do you have?<\/td>\r\n<td>[latex]6[\/latex]. [latex]4+2=6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468679001\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-3+6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]3[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220225\/CNX_BMath_Figure_03_02_036_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]6[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220226\/CNX_BMath_Figure_03_02_036_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove neutral pairs.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220227\/CNX_BMath_Figure_03_02_036_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220228\/CNX_BMath_Figure_03_02_036_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3[\/latex]. [latex]-3+6=3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467662609\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4+\\left(-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]4[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220229\/CNX_BMath_Figure_03_02_037_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]5[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220230\/CNX_BMath_Figure_03_02_037_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove neutral pairs.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220231\/CNX_BMath_Figure_03_02_037_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220232\/CNX_BMath_Figure_03_02_037_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-1[\/latex]. [latex]4+\\left(-5\\right)=-1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468388113\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>4.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-2+\\left(-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]2[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220233\/CNX_BMath_Figure_03_02_038_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]3[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220234\/CNX_BMath_Figure_03_02_038_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many do you have?<\/td>\r\n<td>[latex]-5[\/latex]. [latex]-2+\\left(-3\\right)=-5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel each addition.\r\n\r\n1. [latex]3 + 4[\/latex]\r\n2. [latex]\u22121 + 4[\/latex]\r\n3. [latex]4 + (\u22126)[\/latex]\r\n4. [latex]\u22122 + (\u22122)[\/latex]\r\n[reveal-answer q=\"767954\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"767954\"]\r\n\r\n1.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220235\/CNX_BMath_Figure_03_02_039_img.png\" alt=\".\" \/>\r\n\r\n2.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220236\/CNX_BMath_Figure_03_02_040_img.png\" alt=\".\" \/>\r\n\r\n3.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220237\/CNX_BMath_Figure_03_02_041_img.png\" alt=\".\" \/>\r\n\r\n4.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220238\/CNX_BMath_Figure_03_02_042_img.png\" alt=\".\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n1. [latex]5 + 1[\/latex]\r\n2. [latex]\u22123 + 7[\/latex]\r\n3. [latex]2 + (\u22128)[\/latex]\r\n4. [latex]\u22123 + (\u22124)[\/latex]\r\n[reveal-answer q=\"388859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"388859\"]\r\n\r\n1.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220239\/CNX_BMath_Figure_03_02_043_img.png\" alt=\".\" \/>\r\n\r\n2.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220241\/CNX_BMath_Figure_03_02_044_img.png\" alt=\".\" \/>\r\n\r\n3.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220242\/CNX_BMath_Figure_03_02_045_img.png\" alt=\".\" \/>\r\n\r\n4.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220243\/CNX_BMath_Figure_03_02_046_img.png\" alt=\".\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we present more examples of using color counters to model addition of integers.\r\n\r\nhttps:\/\/youtu.be\/beI5bQB9a7Y\r\n<h2>Modeling Subtraction of Integers<\/h2>\r\nYou learn as a child how to subtract numbers through everyday experiences. For example, if you have 10 animal cookies and eat 6 of them, you will have 4 animal cookies left.\r\n\r\nReal-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.\r\nWe will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.\r\n\r\nPerhaps when you were younger, you read [latex]5 - 3[\/latex] as <em>five take away three<\/em>. When we use counters, we can think of subtraction the same way.\r\nWe will model four subtraction scenarios\u00a0using the numbers [latex]5[\/latex] and [latex]3[\/latex].\r\n<ul>\r\n \t<li>[latex]5 - 3[\/latex]<\/li>\r\n \t<li>[latex]- 5-\\left(-3\\right)[\/latex]<\/li>\r\n \t<li>[latex]-5 - 3[\/latex]<\/li>\r\n \t<li>[latex]5-\\left(-3\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]5 - 3[\/latex].\r\n\r\nSolution:\r\n<table id=\"eip-id1168469608926\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus 3 means 5 take away 3. The second row states model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so take away 3 positives. It is followed by 5 blue circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 blue circles that are labeled two positives. The fifth row states the difference of 5 and 3 is 2. It is followed by the expression 5 minus 3 equals 2.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]5 - 3[\/latex] means [latex]5[\/latex] take away [latex]3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220309\/CNX_BMath_Figure_03_03_027_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away the second number. So take away [latex]3[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220311\/CNX_BMath_Figure_03_03_027_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the counters that are left.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220313\/CNX_BMath_Figure_03_03_027_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]5 - 3=2[\/latex] .\r\n\r\nThe difference between [latex]5[\/latex] and [latex]3[\/latex] is [latex]2[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression:\r\n\r\n[latex]6 - 4[\/latex]\r\n[reveal-answer q=\"49141\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"49141\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220315\/CNX_BMath_Figure_03_03_003_img.png\" alt=\"This figure shows a row of 6 light pink circles, representing positive counters. The first four are circled.\" \/>\r\n\r\n[latex]2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression:\r\n\r\n[latex]7 - 4[\/latex]\r\n[reveal-answer q=\"400845\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"400845\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220318\/CNX_BMath_Figure_03_03_004_img.png\" alt=\"This figure shows a row of 7 light pink circles representing positive counters. The first four counters are circled.\" \/>\r\n\r\n[latex]3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn teh previous example we subtracted [latex]3[\/latex] positives from positive [latex]5[\/latex]. Now we will subtract\u00a0[latex]3[\/latex] negatives from\u00a0negative [latex]5[\/latex]. Compare the results of this example to the previous example after you read through it.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]-5-\\left(-3\\right)[\/latex]\r\n[reveal-answer q=\"511456\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"511456\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469693755\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so take away 3 negatives. It is followed by 5 red circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 red circles which are labeled two negatives. The fifth row states the difference of negative 5 and negative 3 is negative 2. Next to this is the expression negative 5 minus negative 3 equals negative two.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]-5-\\left(-3\\right)[\/latex] means [latex]-5[\/latex] take away [latex]-3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220319\/CNX_BMath_Figure_03_03_028_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away the second number. So take away [latex]3[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220320\/CNX_BMath_Figure_03_03_028_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the number of counters that are left.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220321\/CNX_BMath_Figure_03_03_028_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5-\\left(-3\\right)=-2[\/latex] .\r\n\r\nThe difference between [latex]-5[\/latex] and [latex]-3[\/latex] is [latex]-2[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression:\r\n\r\n[latex]-6-\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"720789\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"720789\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220324\/CNX_BMath_Figure_03_03_008_img.png\" alt=\"This figure shows a row of 6 dark pink circles, representing negative counters. The last four counters are circled.\" \/>\r\n\r\n[latex]\u22122[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression:\r\n\r\n[latex]-7-\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"188555\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"188555\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220325\/CNX_BMath_Figure_03_03_009_img.png\" alt=\"This figure is a row of 7 dark pink circles representing negative counters. The first four counters are circled.\" \/>\r\n\r\n[latex]\u22123[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that the previous two examples\u00a0are very much alike.\r\n<ul id=\"fs-id801766\">\r\n \t<li>First, we subtracted [latex]3[\/latex] positives from [latex]5[\/latex] positives to get [latex]2[\/latex] positives.<\/li>\r\n \t<li>Then we subtracted [latex]3[\/latex] negatives from [latex]5[\/latex] negatives to get [latex]2[\/latex] negatives.<\/li>\r\n<\/ul>\r\nEach example used counters of only one color, and the \"take away\" model of subtraction was easy to apply.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220326\/CNX_BMath_Figure_03_03_040_img.png\" alt=\"This figure has a row of 5 blue circles. The first three are circled. Above the row is 5 minus 3 equals 2. Next to this is a row of 5 red circles. The first three are circled. Above the row is negative 5 minus negative 3 equals negative 2.\" \/>\r\nNow let\u2019s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]-5 - 3[\/latex].\r\n[reveal-answer q=\"662198\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"662198\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468477542\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by negative 5 minus 3 means negative 5 take away 3. The second row states to model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so we need to take away 3 positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The fourth row states there are no positives to take away so add neutral pairs until you have 3 positives. The fifth row states now take away three positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The blue shaded circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 red circles. The seventh row states the difference of negative five and three is negative 8. It is followed by the expression negative 5 minus 3 equals negative 8.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]-5 - 3[\/latex] means [latex]-5[\/latex] take away [latex]3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220328\/CNX_BMath_Figure_03_03_029_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away the second number.\r\n\r\nSo we need to take away [latex]3[\/latex] positives.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>But there are no positives to take away.\r\n\r\nAdd neutral pairs until you have [latex]3[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220329\/CNX_BMath_Figure_03_03_029_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now take away [latex]3[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220330\/CNX_BMath_Figure_03_03_029_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of counters that are left.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220332\/CNX_BMath_Figure_03_03_029_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5 - 3=-8[\/latex] .\r\n\r\nThe difference of [latex]-5[\/latex] and [latex]3[\/latex] is [latex]-8[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression:\r\n\r\n[latex]-6 - 4[\/latex]\r\n[reveal-answer q=\"647548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"647548\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220334\/CNX_BMath_Figure_03_03_012_img.png\" alt=\"This figure shows a row of 10 dark pink circles, representing negative counters. The first six counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters. These four positive counters are circled.\" \/>\r\n\r\n[latex]\u221210[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression:\r\n\r\n[latex]-7 - 4[\/latex]\r\n[reveal-answer q=\"391259\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"391259\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220336\/CNX_BMath_Figure_03_03_013_img.png\" alt=\"This figure shows a row of 11 dark pink circles, representing negative counters. The first seven counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters. These four positive counters are circled.\" \/>\r\n\r\n[latex]\u221211[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow we will subtract a negative number from a positive number. Think of this as taking away the negative.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]5-\\left(-3\\right)[\/latex].\r\n[reveal-answer q=\"943073\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"943073\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467234371\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states to model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so we need to take away 3 negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The fourth row states there are no negatives to take away so add neutral pairs until you have 3 negatives. The fifth row states now take away three negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The red circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 blue circles. The seventh row states the difference of 5 and negative 3 is eight, beside it the expression 5 minus negative 3 equals eight.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]5-\\left(-3\\right)[\/latex] means [latex]5[\/latex] take away [latex]-3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220337\/CNX_BMath_Figure_03_03_030_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away the second number, so take away [latex]3[\/latex] negatives.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>But there are no negatives to take away.\r\n\r\nAdd neutral pairs until you have [latex]3[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220338\/CNX_BMath_Figure_03_03_030_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away [latex]3[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220339\/CNX_BMath_Figure_03_03_030_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of counters that are left.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220341\/CNX_BMath_Figure_03_03_030_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The difference of [latex]5[\/latex] and [latex]-3[\/latex] is [latex]8[\/latex] .\r\n\r\n[latex]5-\\left(-3\\right)=8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression:\r\n\r\n[latex]6-\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"459252\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"459252\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220342\/CNX_BMath_Figure_03_03_016_img.png\" alt=\"This figure shows a row of 10 light pink circles, representing positive counters. The first six counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.\" \/>\r\n\r\n[latex]10[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression:\r\n[latex]7-\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"632721\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"632721\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220343\/CNX_BMath_Figure_03_03_017_img.png\" alt=\"This figure shows a row of 11 light pink circles, representing positive counters. The first seven counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.\" \/>\r\n\r\n[latex]11[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNow we will do an example that summarizes the situations above, with different numbers. Recall the different scenarios:\r\n<ul>\r\n \t<li>subtracting a positive number from a positive number<\/li>\r\n \t<li>subtracting a positive number from a negative number<\/li>\r\n \t<li>subtracting a negative number from a positive number<\/li>\r\n \t<li>subtracting a negative number from a negative number<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel each subtraction.\r\n\r\n[latex]8 \u2212 2[\/latex]\r\n[latex]\u22125 \u2212 4[\/latex]\r\n[latex]6 \u2212 (\u22126)[\/latex]\r\n[latex]\u22128 \u2212 (\u22123)[\/latex]\r\n[reveal-answer q=\"214895\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"214895\"]\r\n<table id=\"eip-id1168468308805\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8 - 2[\/latex]\r\n\r\nThis means [latex]8[\/latex] take away [latex]2[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]8[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220345\/CNX_BMath_Figure_03_03_041_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]2[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220346\/CNX_BMath_Figure_03_03_041_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8 - 2=6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466328467\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5 - 4[\/latex]\r\n\r\nThis means [latex]-5[\/latex] take away [latex]4[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]5[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220347\/CNX_BMath_Figure_03_03_042_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>You need to take away [latex]4[\/latex] positives.\r\n\r\nAdd [latex]4[\/latex] neutral pairs to get [latex]4[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220349\/CNX_BMath_Figure_03_03_042_img-02.png\" alt=\".\" \/>\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220350\/CNX_BMath_Figure_03_03_042_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]4[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220352\/CNX_BMath_Figure_03_03_042_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220354\/CNX_BMath_Figure_03_03_042_img-05.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5 - 4=-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469609690\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]6-\\left(-6\\right)[\/latex]\r\n\r\nThis means [latex]6[\/latex] take away [latex]-6[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]6[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220355\/CNX_BMath_Figure_03_03_043_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]6[\/latex] neutrals to get [latex]6[\/latex] negatives to take away.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220357\/CNX_BMath_Figure_03_03_043_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove [latex]6[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220359\/CNX_BMath_Figure_03_03_043_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220401\/CNX_BMath_Figure_03_03_043_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]6-\\left(-6\\right)=12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468534936\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>4.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-8-\\left(-3\\right)[\/latex]\r\n\r\nThis means [latex]-8[\/latex] take away [latex]-3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]8[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220403\/CNX_BMath_Figure_03_03_044_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]3[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220404\/CNX_BMath_Figure_03_03_044_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220405\/CNX_BMath_Figure_03_03_044_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-8-\\left(-3\\right)=-5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel each subtraction.\r\n\r\n1. [latex]7 - (-8)[\/latex]\r\n2. [latex]-7 - (-2)[\/latex]\r\n3. [latex]4 - 1[\/latex]\r\n4. [latex]-6 - 8[\/latex]\r\n[reveal-answer q=\"324876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324876\"]\r\n\r\n1.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220406\/CNX_BMath_Figure_03_03_045_img.png\" alt=\".\" \/>\r\n\r\n2.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220408\/CNX_BMath_Figure_03_03_046_img.png\" alt=\".\" \/>\r\n\r\n3.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220409\/CNX_BMath_Figure_03_03_047_img.png\" alt=\".\" \/>\r\n\r\n4.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220410\/CNX_BMath_Figure_03_03_048_img.png\" alt=\".\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel each subtraction.\r\n\r\n1. [latex]4 - (-6)[\/latex]\r\n2. [latex]-8 - (-1)[\/latex]\r\n3. [latex]7 - 3[\/latex]\r\n4. [latex]-4 - 2[\/latex]\r\n[reveal-answer q=\"677150\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"677150\"]\r\n\r\n1.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220411\/CNX_BMath_Figure_03_03_049_img.png\" alt=\".\" \/>\r\n\r\n2.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220412\/CNX_BMath_Figure_03_03_050_img.png\" alt=\".\" \/>\r\n\r\n3.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220413\/CNX_BMath_Figure_03_03_051_img.png\" alt=\".\" \/>\r\n\r\n4.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220414\/CNX_BMath_Figure_03_03_052_img.png\" alt=\".\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nEach of the examples so far have been carefully constructed so that the sign of the answer matched the sign of the first number in the expression. \u00a0For example, in\u00a0\u00a0[latex]\u22125 \u2212 4[\/latex], the result is [latex]-9[\/latex], which is the same sign as [latex]-5[\/latex]. Now we will see subtraction where the sign of the result is different from the starting number.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel each subtraction expression:\r\n<ol id=\"eip-id1168468264062\" class=\"circled\">\r\n \t<li>[latex]2 - 8[\/latex]<\/li>\r\n \t<li>[latex]-3-\\left(-8\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"485889\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"485889\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466285116\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has six rows. The first row states we start with 2 positives. It is followed by 2 blue circles. The second row states we need to take away 8 positives but we have only 2. The third row states add neutral pairs until there are 8 positives to take away. It is followed by a row of 8 blue circles. The first two are separated from the last 6. Under the last six there is a row of 6 red circles. The fourth row states then take away the 8 positives. It is followed by a row of 8 blue circles. They are circled. The fifth row states find the number of counters that are left. There are six negatives. It is followed by 6 red circles labeled six negatives. The sixth row has 2 minus 8 equals negative 6.\">\r\n<tbody>\r\n<tr>\r\n<td>1.\r\n\r\nWe start with [latex]2[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220415\/CNX_BMath_Figure_03_03_031_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We need to take away [latex]8[\/latex] positives, but we have only [latex]2[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add neutral pairs until there are [latex]8[\/latex] positives to take away.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220416\/CNX_BMath_Figure_03_03_031_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away [latex]8[\/latex]\u00a0positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220418\/CNX_BMath_Figure_03_03_031_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the number of counters that are left.\r\n\r\nThere are [latex]6[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220419\/CNX_BMath_Figure_03_03_031_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2 - 8=-6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468388588\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has six rows. The first row states we start with 3 negatives. It is followed by 3 red circles. The second row states we need to take away 8 negatives but we have only 3. The third row states add neutral pairs until there are 8 negatives to take away. It is followed by a row of 8 red circles. The first three are separated from the last 5. Under the last 5 there is a row of 5 blue circles. The fourth row states then take away the 8 negatives. It is followed by a row of 8 red circles. They are circled. The fifth row states find the number of counters that are left. There are five positives. It is followed by 5 blue circles. The sixth row has negative 3 minus negative 8 equals 5.\">\r\n<tbody>\r\n<tr>\r\n<td>2.\r\n\r\nWe start with [latex]3[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220422\/CNX_BMath_Figure_03_03_032_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We need to take away [latex]8[\/latex] negatives, but we have only [latex]3[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add neutral pairs until there are [latex]8[\/latex] negatives to take away.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220423\/CNX_BMath_Figure_03_03_032_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away the [latex]8[\/latex] negatives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220425\/CNX_BMath_Figure_03_03_032_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the number of counters that are left.\r\n\r\nThere are [latex]5[\/latex] positives.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220426\/CNX_BMath_Figure_03_03_032_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-3-\\left(-8\\right)=5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel each subtraction expression.\r\n<ol id=\"eip-id1168466205125\" class=\"circled\">\r\n \t<li>[latex]7 - 9[\/latex]<\/li>\r\n \t<li>[latex]-5-(-9)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"678756\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"678756\"]\r\n\r\n1.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220427\/CNX_BMath_Figure_03_03_020_img.png\" alt=\"This figure shows a row of 9 light pink circles, representing positive counters. The first seven are separated from the last two. The entire row is circled. Below the last two light pink circles is a row of two dark pink circles, representing negative counters.\" \/>\r\n\r\n[latex]\u22122[\/latex]\r\n\r\n2.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220428\/CNX_BMath_Figure_03_03_021_img.png\" alt=\"This figure shows a row of 9 dark pink circles, representing negative counters. The first five are separated from the last four. The entire row is circled. Below the last four dark pink circles is a row of four light pink circles, representing positive counters.\" \/>\r\n\r\n[latex]4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel each subtraction expression.\r\n<ol id=\"eip-id1168468470083\" class=\"circled\">\r\n \t<li>[latex]4 - 7[\/latex]<\/li>\r\n \t<li>[latex]-7-\\left(-10\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"164098\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"164098\"]\r\n\r\n1.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220429\/CNX_BMath_Figure_03_03_022_img.png\" alt=\"This figure shows a row of seven light pink circles, representing positive counters. The first four are separated from the last three. The entire row is circled. Below the last three light pink circles is a row of three dark pink circles, representing negative counters.\" \/>\r\n\r\n[latex]\u22123[\/latex]\r\n\r\n2.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220430\/CNX_BMath_Figure_03_03_023_img.png\" alt=\"This figure shows a row of 10 dark pink circles, representing negative counters. The first seven are separated from the last three. The entire row is circled. Below the last three dark pink circles is a row of three light pink circles, representing positive counters.\" \/>\r\n\r\n[latex]3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen you subtract two integers, there are two possibilities, either the result will have a different sign from the starting number, or it will have the same sign.\r\n\r\nWatch the video below to see more examples of modeling integer subtraction with color counters.\r\n\r\nhttps:\/\/youtu.be\/ZjiL7GfxgiI\r\n<h2>Modeling Multiplication of Integers<\/h2>\r\nSince multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.\r\nWe remember that [latex]a\\cdot b[\/latex] means add [latex]a,b[\/latex] times. Here, we are using the model shown in the graphic below\u00a0just to help us discover the pattern.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220519\/CNX_BMath_Figure_03_04_001.png\" alt=\"This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.\" \/>\r\nNow consider what it means to multiply [latex]5[\/latex] by [latex]-3[\/latex]. It means subtract [latex]5,3[\/latex] times. Looking at subtraction as <em>taking away<\/em>, it means to take away [latex]5,3[\/latex] times. But there is nothing to take away, so we start by adding neutral pairs as shown in the graphic below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220522\/CNX_BMath_Figure_03_04_004-1.png\" alt=\"This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives. Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.\" \/>\r\nIn both cases, we started with [latex]\\mathbf{\\text{15}}[\/latex] neutral pairs. In the case on the left, we took away [latex]\\mathbf{\\text{5}},\\mathbf{\\text{3}}[\/latex] times and the result was [latex]-\\mathbf{\\text{15}}[\/latex]. To multiply [latex]\\left(-5\\right)\\left(-3\\right)[\/latex], we took away [latex]-\\mathbf{\\text{5}},\\mathbf{\\text{3}}[\/latex] times and the result was [latex]\\mathbf{\\text{15}}[\/latex]. So we found that\r\n\r\n[latex]\\begin{array}{ccc}5\\cdot 3=15\\hfill &amp; &amp; -5\\left(3\\right)=-15\\hfill \\\\ 5\\left(-3\\right)=-15\\hfill &amp; &amp; \\left(-5\\right)\\left(-3\\right)=15\\hfill \\end{array}[\/latex]\r\n\r\nNotice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplication of Signed Numbers<\/h3>\r\nThe sign of the product of two numbers depends on their signs.\r\n<table id=\"fs-id1385641\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"This is a table with two columns. The first column states same signs. The second column states product. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under product, it states negative.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Same signs<\/th>\r\n<th>Product<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>\u2022Two positives\r\n\r\n\u2022Two negatives<\/td>\r\n<td>Positive\r\n\r\nPositive<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"fs-id2295230\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"This is a table with two columns. The first column states same signs. The second column states product. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under product, it states negative.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Different signs<\/th>\r\n<th>Product<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>\u2022Positive \u2022 negative\r\n\r\n\u2022Negative \u2022 positive<\/td>\r\n<td>Negative\r\n\r\nNegative<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply each of the following:\r\n<ol>\r\n \t<li>[latex]-9\\cdot 3[\/latex]<\/li>\r\n \t<li>[latex]-2\\left(-5\\right)[\/latex]<\/li>\r\n \t<li>[latex]4\\left(-8\\right)[\/latex]<\/li>\r\n \t<li>[latex]7\\cdot 6[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1168469461529\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-9\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply, noting that the signs are different and so the product is negative.<\/td>\r\n<td>[latex]-27[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467249422\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-2\\left(-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply, noting that the signs are the same and so the product is positive.<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466068278\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4\\left(-8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply, noting that the signs are different and so the product is negative.<\/td>\r\n<td>[latex]-32[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466077286\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>4.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]7\\cdot 6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are the same, so the product is positive.<\/td>\r\n<td>[latex]42[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145306&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n[ohm_question]145306[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video for more examples of how to multiply integers with different signs, and the same sign.\r\n\r\nhttps:\/\/youtu.be\/QY-Za42DItw\r\n\r\nWhen we multiply a number by [latex]1[\/latex], the result is the same number. What happens when we multiply a number by [latex]-1?[\/latex] Let\u2019s multiply a positive number and then a negative number by [latex]-1[\/latex] to see what we get.\r\n\r\n[latex]\\begin{array}{ccc}\\hfill -1\\cdot 4\\hfill &amp; &amp; \\hfill -1\\left(-3\\right)\\hfill \\\\ \\hfill -4\\hfill &amp; &amp; \\hfill 3\\hfill \\\\ \\hfill -4\\text{ is the opposite of }\\mathbf{\\text{4}}\\hfill &amp; &amp; \\hfill \\mathbf{\\text{3}}\\text{ is the opposite of }-3\\hfill \\end{array}[\/latex]\r\nEach time we multiply a number by [latex]-1[\/latex], we get its opposite.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplication by [latex]-1[\/latex]<\/h3>\r\nMultiplying a number by [latex]-1[\/latex] gives its opposite.\r\n\r\n[latex]-1a=-a[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply each of the following:\r\n<ol>\r\n \t<li>[latex]-1\\cdot 7[\/latex]<\/li>\r\n \t<li>[latex]-1\\left(-11\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"459967\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"459967\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468284281\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are different, so the product will be negative.<\/td>\r\n<td>[latex]-1\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that [latex]\u22127[\/latex] is the opposite of [latex]7[\/latex].<\/td>\r\n<td>[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469855058\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are the same, so the product will be positive.<\/td>\r\n<td>[latex]-1\\left(-11\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that [latex]11[\/latex] is the opposite of [latex]\u221211[\/latex].<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145319&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n[ohm_question]145319[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Division with Integers<\/h2>\r\nDivision is the inverse operation of multiplication. So, [latex]15\\div 3=5[\/latex] because [latex]5\\cdot 3=15[\/latex] In words, this expression says that [latex]\\mathbf{\\text{15}}[\/latex] can be divided into [latex]\\mathbf{\\text{3}}[\/latex] groups of [latex]\\mathbf{\\text{5}}[\/latex] each because adding five three times gives [latex]\\mathbf{\\text{15}}[\/latex]. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.\r\n\r\n[latex]\\begin{array}{ccccc}5\\cdot 3=15\\text{ so }15\\div 3=5\\hfill &amp; &amp; &amp; &amp; -5\\left(3\\right)=-15\\text{ so }-15\\div 3=-5\\hfill \\\\ \\left(-5\\right)\\left(-3\\right)=15\\text{ so }15\\div \\left(-3\\right)=-5\\hfill &amp; &amp; &amp; &amp; 5\\left(-3\\right)=-15\\text{ so }-15\\div -3=5\\hfill \\end{array}[\/latex]\r\nDivision of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.\r\n<div class=\"textbox shaded\">\r\n<h3>Division of Signed Numbers<\/h3>\r\nThe sign of the quotient of two numbers depends on their signs.\r\n<table id=\"fs-id2593939\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"This is a table with two columns. The first column states same signs. The second column states quotient. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under quotient, it states negative.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Same signs<\/th>\r\n<th>Quotient<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>\u2022Two positives\r\n\r\n\u2022Two negatives<\/td>\r\n<td>Positive\r\n\r\nPositive<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"fs-id2379645\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"This is a table with two columns. The first column states same signs. The second column states quotient. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under quotient, it states negative.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Different signs<\/th>\r\n<th>Quotient<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>\u2022Positive &amp; negative\r\n\r\n\u2022Negative &amp; positive<\/td>\r\n<td>Negative\r\n\r\nNegative<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nRemember, you can always check the answer to a division problem by multiplying.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide each of the following:\r\n<ol>\r\n \t<li>[latex]-27\\div 3[\/latex]<\/li>\r\n \t<li>[latex]-100\\div \\left(-4\\right)[\/latex]<\/li>\r\n<\/ol>\r\nSolution\r\n<table id=\"eip-id1168467157728\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 15.7812px;\">\r\n<td style=\"height: 15.7812px;\">1.<\/td>\r\n<td style=\"height: 15.7812px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]-27\\div 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Divide, noting that the signs are different and so the quotient is negative.<\/td>\r\n<td style=\"height: 15px;\">[latex]-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467157714\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-100\\div \\left(-4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide, noting that the signs are the same and so the quotient is positive.<\/td>\r\n<td>[latex]25[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145323&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n[ohm_question]145323[\/ohm_question]\r\n\r\n<\/div>\r\nJust as we saw with multiplication, when we divide a number by [latex]1[\/latex], the result is the same number. What happens when we divide a number by [latex]-1?[\/latex] Let\u2019s divide a positive number and then a negative number by [latex]-1[\/latex] to see what we get.\r\n\r\n[latex]\\begin{array}{cccc}8\\div \\left(-1\\right)\\hfill &amp; &amp; &amp; -9\\div \\left(-1\\right)\\hfill \\\\ -8\\hfill &amp; &amp; &amp; 9\\hfill \\\\ \\hfill \\text{-8 is the opposite of 8}\\hfill &amp; &amp; &amp; \\hfill \\text{9 is the opposite of -9}\\hfill \\end{array}[\/latex]\r\nWhen we divide a number by [latex]-1[\/latex] we get its opposite.\r\n<div class=\"textbox shaded\">\r\n<h3>Division by [latex]-1[\/latex]<\/h3>\r\nDividing a number by [latex]-1[\/latex] gives its opposite.\r\n\r\n[latex]a\\div \\left(-1\\right)=-a[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide each of the following:\r\n<ol>\r\n \t<li>[latex]16\\div \\left(-1\\right)[\/latex]<\/li>\r\n \t<li>[latex]-20\\div \\left(-1\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"343427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"343427\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466004748\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]16\\div \\left(-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The dividend, [latex]16[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\r\n<td>[latex]-16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that the signs were different, so the result was negative.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469488934\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-20\\div \\left(-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The dividend, [latex]\u201320[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\r\n<td>[latex]20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that the signs were the same, so the quotient was positive.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145326[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video for more examples of how to divide integers with the same and different signs.\r\n\r\nhttps:\/\/youtu.be\/z5ZFiyLi5Y0\r\n<h2>Absolute Value<\/h2>\r\nWe saw that numbers such as [latex]5[\/latex] and [latex]-5[\/latex] are opposites because they are the same distance from [latex]0[\/latex] on the number line. They are both five units from [latex]0[\/latex]. The distance between [latex]0[\/latex] and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative.\r\nThe symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[\/latex] is written as [latex]|5|[\/latex], and the absolute value of [latex]-5[\/latex] is written as [latex]|-5|[\/latex] as shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220135\/CNX_BMath_Figure_03_01_019.png\" alt=\"This figure is a number line. The points negative 5 and 5 are labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units. Also above the number line the distance from 0 to 5 is labeled as 5 units.\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Absolute Value<\/h3>\r\nThe absolute value of a number is its distance from [latex]0[\/latex] on the number line.\r\nThe absolute value of a number [latex]n[\/latex] is written as [latex]|n|[\/latex].\r\n\r\n[latex]|n|\\ge 0\\text{ for all numbers}[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n[latex]|3|[\/latex]\r\n[latex]|-44|[\/latex]\r\n[latex]|0|[\/latex]\r\n\r\nSolution:\r\n<table id=\"eip-id1165118025864\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|3|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex] is [latex]3[\/latex] units from zero.<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1171842979184\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|-44|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u221244[\/latex] is [latex]44[\/latex] units from zero.<\/td>\r\n<td>[latex]44[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1171842583307\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|0|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex] is already at zero.<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144930&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144931&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144932&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144933&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\nIn the video below we show more example of how to find the absolute value of an integer.\r\n\r\nhttps:\/\/youtu.be\/I8bTqGmkqGI\r\n\r\nWe treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEvaluate:\r\n<ol>\r\n \t<li>[latex]|x|\\text{ when }x=-35[\/latex]<\/li>\r\n \t<li>[latex]|\\mathit{\\text{-y}}|\\text{ when }y=-20[\/latex]<\/li>\r\n \t<li>[latex]-|u|\\text{ when }u=12[\/latex]<\/li>\r\n \t<li>[latex]-|p|\\text{ when }p=-14[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"727778\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"727778\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466007339\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has three rows. The first row has the absolute value of x. The second row states to substitute negative 35 for x and has the absolute value symbol with negative 35. The third row states to take the absolute value and has 35.\">\r\n<tbody>\r\n<tr>\r\n<td>1. To find [latex]|x|[\/latex] when [latex]x=-35:[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|x|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{-35}[\/latex] for x.<\/td>\r\n<td>[latex]\\mid\\color{red}{-35}\\mid[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the absolute value.<\/td>\r\n<td>[latex]35[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469585161\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has four rows. The first row has the absolute value of y. The second row states to substitute negative 20 for y and has inside of the absolute value symbol a negative sign with negative 20 inside of parentheses. The third row states to simplify and has 20 inside of the absolute value symbol. The fourth row states to take the absolute value and has 20.\">\r\n<tbody>\r\n<tr>\r\n<td>2. To find [latex]|-y|[\/latex] when [latex]y=-20:[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|-y|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{-20}[\/latex] for y.<\/td>\r\n<td>[latex]\\mid-(\\color{red}{-20})\\mid[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]|20|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the absolute value.<\/td>\r\n<td>[latex]20[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466911241\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has three rows. The first row has a negative sign in front of the absolute value of u. The second row states to substitute 12 for u and has a negative sign in front of the absolute value symbol with 12 on the inside. The third row states to take the absolute value and has negative 12.\">\r\n<tbody>\r\n<tr>\r\n<td>3. To find [latex]-|u|[\/latex] when [latex]u=12:[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-|u|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{12}[\/latex] for u.<\/td>\r\n<td>[latex]-\\mid\\color{red}{12}\\mid[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the absolute value.<\/td>\r\n<td>[latex]-12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468381460\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has three rows. The first row has a negative sign in front of the absolute value of p. The second row states to substitute negative 14 for p and has a negative sign in front of the absolute value of negative 14. The third row states to take the absolute value and has negative 14.\">\r\n<tbody>\r\n<tr>\r\n<td>4. To find [latex]-|p|[\/latex] when [latex]p=-14:[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-|p|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{-14}[\/latex] for p.<\/td>\r\n<td>[latex]-\\mid\\color{red}{-14}\\mid[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the absolute value.<\/td>\r\n<td>[latex]-14[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNotice that the result is negative only when there is a negative sign outside the absolute value symbol.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144934&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFill in [latex]\\text{&lt;},\\text{&gt;},\\text{or}=[\/latex] for each of the following:\r\n<ol>\r\n \t<li>[latex]|-5|___-|-5|[\/latex]<\/li>\r\n \t<li>[latex]8___-|-8|[\/latex]<\/li>\r\n \t<li>[latex]-9___-|-9|[\/latex]<\/li>\r\n \t<li>[latex]-|-7|___ - 7[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"622738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"622738\"]\r\n\r\nSolution:\r\nTo compare two expressions, simplify each one first. Then compare.\r\n<table id=\"eip-id1166567974475\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|-5|___-|-5|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]5___ - 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Order.<\/td>\r\n<td>[latex]5&gt;-5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1166568580863\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8___-|-8|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]8___ - 8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Order.<\/td>\r\n<td>[latex]8&gt;-8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1166567977584\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-9___-|-9|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-9___ - 9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Order.<\/td>\r\n<td>[latex]-9=-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1166569017628\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>4.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-|-7|___ - 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-7___ - 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Order.<\/td>\r\n<td>[latex]-7=-7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144979&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\nIn the video below we show more examples of how to compare expressions that include absolute value and integers.\r\n\r\nhttps:\/\/youtu.be\/TendEcSaM3w\r\n\r\nAbsolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]|9 - 3|[\/latex]<\/li>\r\n \t<li>[latex]4|-2|[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"47658\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"47658\"]\r\n\r\nSolution:\r\nFor each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.\r\n<table id=\"eip-id1168466096398\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|9\u22123|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify inside the absolute value sign.<\/td>\r\n<td>[latex]|6|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the absolute value.<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469668431\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4|\u22122|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the absolute value.<\/td>\r\n<td>[latex]4\u22c52[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom7\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145007&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]|8+7|-|5+6|[\/latex].\r\n[reveal-answer q=\"743144\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"743144\"]\r\n\r\nSolution:\r\nFor each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.\r\n<table id=\"eip-id1168469785725\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|8+7|\u2212|5+6|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify inside each absolute value sign.<\/td>\r\n<td>[latex]|15|\u2212|11|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom8\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145009&amp;theme=oea&amp;iframe_resize_id=mom8\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]24-|19 - 3\\left(6 - 2\\right)|[\/latex].\r\n[reveal-answer q=\"949242\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"949242\"]\r\n\r\nSolution:\r\nWe use the order of operations. Remember to simplify grouping symbols first, so parentheses inside absolute value symbols would be first.\r\n<table id=\"eip-id1168467296922\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]24-|19 - 3\\left(6 - 2\\right)|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify in the parentheses first.<\/td>\r\n<td>[latex]24-|19 - 3\\left(4\\right)|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]3\\left(4\\right)[\/latex] .<\/td>\r\n<td>[latex]24-|19 - 12|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract inside the absolute value sign.<\/td>\r\n<td>[latex]24-|7|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the absolute value.<\/td>\r\n<td>[latex]24 - 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]17[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n&nbsp;\r\n\r\n[ohm_question]145010[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<span style=\"color: #000000;\">Watch the following video to see more examples of how to simplify expressions that contain absolute value.<\/span>\r\n\r\nhttps:\/\/youtu.be\/cd24nT7mAi0\r\n<h2 data-type=\"title\">Simplify Expressions Using the Order of Operations<\/h2>\r\nWe\u2019ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.\r\n\r\nFor example, consider the expression:\r\n<p style=\"text-align: center;\">[latex]4+3\\cdot 7[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccc}\\hfill \\text{Some students say it simplifies to 49.}\\hfill &amp; &amp; &amp; \\hfill \\text{Some students say it simplifies to 25.}\\hfill \\\\ \\begin{array}{ccc}&amp; &amp; \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }4+3\\text{ gives 7.}\\hfill &amp; &amp; \\hfill 7\\cdot 7\\hfill \\\\ \\text{And }7\\cdot 7\\text{ is 49.}\\hfill &amp; &amp; \\hfill 49\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}&amp; &amp; \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }3\\cdot 7\\text{ is 21.}\\hfill &amp; &amp; \\hfill 4+21\\hfill \\\\ \\text{And }21+4\\text{ makes 25.}\\hfill &amp; &amp; \\hfill 25\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/p>\r\nImagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.\r\n<div class=\"textbox shaded\">\r\n<h3>Order of Operations<\/h3>\r\nWhen simplifying mathematical expressions perform the operations in the following order:\r\n1. <strong>P<\/strong>arentheses and other Grouping Symbols\r\n<ul id=\"fs-id1171104029952\" data-bullet-style=\"bullet\">\r\n \t<li>Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.<\/li>\r\n<\/ul>\r\n2. <strong>E<\/strong>xponents\r\n<ul id=\"fs-id1171104407077\" data-bullet-style=\"bullet\">\r\n \t<li>Simplify all expressions with exponents.<\/li>\r\n<\/ul>\r\n3. <strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision\r\n<ul id=\"fs-id1171103140103\" data-bullet-style=\"bullet\">\r\n \t<li>Perform all multiplication and division in order from left to right. These operations have equal priority.<\/li>\r\n<\/ul>\r\n4. <strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction\r\n<ul id=\"fs-id1171104002792\" data-bullet-style=\"bullet\">\r\n \t<li>Perform all addition and subtraction in order from left to right. These operations have equal priority.<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\nStudents often ask, \"How will I remember the order?\" Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. <strong>P<\/strong>lease <strong>E<\/strong>xcuse <strong>M<\/strong>y <strong>D<\/strong>ear <strong>A<\/strong>unt <strong>S<\/strong>ally.\r\n<table id=\"fs-id1786633\" class=\"unnumbered\" summary=\"This table has five rows and two columns. The first row spans both columns and is a header reading \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"2\" data-align=\"center\"><strong>Order of Operations<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\"><strong>P<\/strong>lease<\/td>\r\n<td data-align=\"left\"><strong>P<\/strong>arentheses<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\"><strong>E<\/strong>xcuse<\/td>\r\n<td data-align=\"left\"><strong>E<\/strong>xponents<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\"><strong>M<\/strong>y <strong>D<\/strong>ear<\/td>\r\n<td data-align=\"left\"><strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\"><strong>A<\/strong>unt <strong>S<\/strong>ally<\/td>\r\n<td data-align=\"left\"><strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt\u2019s good that \u2018<strong>M<\/strong>y <strong>D<\/strong>ear\u2019 goes together, as this reminds us that <strong>m<\/strong>ultiplication and <strong>d<\/strong>ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.\r\nSimilarly, \u2018<strong>A<\/strong>unt <strong>S<\/strong>ally\u2019 goes together and so reminds us that <strong>a<\/strong>ddition and <strong>s<\/strong>ubtraction also have equal priority and we do them in order from left to right.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify the expressions:\r\n<ol>\r\n \t<li>[latex]4+3\\cdot 7[\/latex]<\/li>\r\n \t<li>[latex]\\left(4+3\\right)\\cdot 7[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1164750479370\" class=\"unnumbered unstyled\" summary=\"The table shows the expression four plus three times seven. On the next line it states are there any parentheses in the expression? No. The line below that states are there any exponents in the expression? No. The next line states is there any multiplication or division in the expression? Yes. The next line states Multiply first and is followed by the expression of four plus three times seven. The expression is now four plus twenty-one. The last operation is addition. Add four and twenty-one to get twenty-five.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4+3\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply first.<\/td>\r\n<td>[latex]4+\\color{red}{3\\cdot 7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]4+21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]25[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1164754514704\" class=\"unnumbered unstyled\" summary=\"The image shows the expression four plus three in parentheses times seven. Are there any parentheses? Yes, simplify inside the parentheses by adding four and three to get seven. The expression is now seven times seven. Is there any multiplication or division? Yes, multiply seven by seven to get forty-nine.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](4+3)\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>p<\/strong>arentheses? Yes.<\/td>\r\n<td>[latex]\\color{red}{(4+3)}\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify inside the parentheses.<\/td>\r\n<td>[latex](7)7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]49[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144748[\/ohm_question]\r\n\r\n[ohm_question]144751[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]\\text{18}\\div \\text{9}\\cdot \\text{2}[\/latex]<\/li>\r\n \t<li>[latex]\\text{18}\\cdot \\text{9}\\div \\text{2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"604459\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604459\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754213884\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen divided by nine times two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, divide first. Divide eighteen by nine to get two. The expression is now two times two. The last operation is multiplication. Multiply two by two to get four.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]18\\div 9\\cdot 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply and divide from left to right. Divide.<\/td>\r\n<td>[latex]\\color{red}{2}\\cdot 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1164752720001\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen times nine divided by two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, multiply first. Multiply eighteen by nine to get one hundred sixty-two. The expression is now one hundred sixty-two divided by two. The last operation is division. divide one hundred sixty-two by two to get eighty-one.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]18\\cdot 9\\div 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply and divide from left to right.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\color{red}{162}\\div 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144756[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]18\\div 6+4\\left(5 - 2\\right)[\/latex].\r\n[reveal-answer q=\"841846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"841846\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754213917\" class=\"unnumbered unstyled\" summary=\"The image shows the expression eighteen divided by six plus four, in parentheses, five minus two. Are there any parentheses? Yes, perform the subtraction inside the parentheses. Five minus two becomes three inside the parentheses. The expression is now eighteen divided by six plus four, in parentheses, three. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Divide first because multiplication and division are performed left to right. Divide eighteen by six to get three. The expression is now three plus four, in parentheses, three. Now multiply four by the three in parentheses to get twelve. The expression becomes three plus twelve. Add three and twelve to get fifteen.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]18\\div 6+4(5-2)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Parentheses? Yes, subtract first.<\/td>\r\n<td>[latex]18\\div 6+4(\\color{red}{3})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Exponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide first because we multiply and divide left to right.<\/td>\r\n<td>[latex]\\color{red}{3}+4(3)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Any other multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]3+\\color{red}{12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Any other multiplication or division? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Any addition or subtraction? Yes.<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144758[\/ohm_question]\r\n\r\n<\/div>\r\nIn the video below we show another example of how to use the order of operations to simplify a mathematical expression.\r\n\r\nhttps:\/\/youtu.be\/qFUvF5-w9o0\r\n\r\n&nbsp;\r\n\r\nWhen there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\text{Simplify: }5+{2}^{3}+3\\left[6 - 3\\left(4 - 2\\right)\\right][\/latex].\r\n[reveal-answer q=\"221697\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"221697\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754129434\" class=\"unnumbered unstyled\" summary=\"The image shows the expression five plus two cubed plus three, open bracket, six minus three, open parentheses, four minus two close parentheses, close bracket. Are there any parentheses (or other grouping symbols)? Yes, there are parentheses within brackets. Start with the innermost grouping, the parentheses. Inside the parentheses subtract two from four to get two. The expression becomes five plus two cubed plus three, open bracket, six minus three, in parentheses, two, close bracket. Within the brackets is six minus three, in parentheses, two. Multiply three by two since multiplication comes before subtraction. The expression becomes five plus two cubed plus three, in brackets, six minus six. Finish simplifying inside the brackets by performing the subtraction. Six minus six leaves zero. The expression is now five plus two cubed plus three, in brackets, zero. Are there exponents? Yes, two cubed is eight and the expression becomes five plus eight plus three, in brackets, zero. Is there any multiplication or division? Yes, multiply three by zero to get zero. The expression is now five plus eight plus zero. The remaining operations are both addition. Perform the addition from left to right. five plus eight is thirteen. Thirteen plus zero is thirteen.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]5+{2}^{3}+ 3[6-3(4-2)][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any parentheses (or other grouping symbol)? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Focus on the parentheses that are inside the brackets.<\/td>\r\n<td>[latex]5+{2}^{3}+ 3[6-3\\color{red}{(4-2)}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{3(2)}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Continue inside the brackets and multiply.<\/td>\r\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{6}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Continue inside the brackets and subtract.<\/td>\r\n<td>[latex]5+{2}^{3}+3[\\color{red}{0}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The expression inside the brackets requires no further simplification.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any exponents? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify exponents.<\/td>\r\n<td>[latex]5+\\color{red}{{2}^{3}}+3[0][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]5+8+\\color{red}{3[0]}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any addition or subtraction? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\color{red}{5+8}+0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\color{red}{13+0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144759[\/ohm_question]\r\n\r\n<\/div>\r\nIn the video below we show another example of how to use the order of operations to simplify an expression that contains exponents and grouping symbols.\r\n\r\nhttps:\/\/youtu.be\/8b-rf2AW3Ac\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex].\r\n[reveal-answer q=\"199030\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"199030\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754324162\" class=\"unnumbered unstyled\" summary=\"The image shows the expression two cubed plus three to the fourth divided by three minus five squared. Are there any parentheses? No. Are there any exponents? Yes, several. Simplify each exponent. Two cubed is eight, three to the fourth is eighty-one, and five squared is twenty-five. The expression becomes eight plus eighty-one divided by three minus twenty-five. Is there any multiplication or division? Yes, just division. Divide eighty-one by three to get twenty-seven. The expression is now eight plus twenty-seven minus five. There is both addition and subtraction left. Perform these from left to right. Eight plus twenty-seven is thirty-five. Now the expression is thirty-five minus twenty five which leaves ten.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>If an expression has several exponents, they may be simplified in the same step.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify exponents.<\/td>\r\n<td>[latex]\\color{red}{{2}^{3}}+\\color{red}{{3}^{4}}\\div 3-\\color{red}{{5}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]8+\\color{red}{82\\div 3}-25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\color{red}{8+27}-25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]\\color{red}{35-25}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144762[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\text{Simplify: }7\\left(-2\\right)+4\\left(-7\\right)-6[\/latex].\r\n\r\nSolution:\r\nWe use the order of operations. Multiply first and then add and subtract from left to right.\r\n<table id=\"eip-id1168468659486\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]7\\left(-2\\right)+4\\left(-7\\right)-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply first.<\/td>\r\n<td>[latex]-14+\\left(-28\\right)-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]-42 - 6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]-48[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145332&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n[ohm_question]145332[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see another example of how to use the order of operations to simplify an expression that contains integers.\r\n\r\nhttps:\/\/youtu.be\/RJ7uU9HbdqA\r\n\r\nIn our next example we will simplify expressions with integers that also contain exponents.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1.\r\n[latex]{\\left(-2\\right)}^{4}[\/latex]\r\n2.\r\n[latex]{-2}^{4}[\/latex]\r\n[reveal-answer q=\"699809\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"699809\"]\r\n\r\nSolution:\r\nThe exponent tells how many times to multiply the base.\r\n1. The exponent is [latex]4[\/latex] and the base is [latex]-2[\/latex]. We raise [latex]-2[\/latex] to the fourth power.\r\n<table id=\"eip-id1168468584362\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(-2\\right)}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write in expanded form.<\/td>\r\n<td>[latex]\\left(-2\\right)\\left(-2\\right)\\left(-2\\right)\\left(-2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]4\\left(-2\\right)\\left(-2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-8\\left(-2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]16[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n2. The exponent is [latex]4[\/latex] and the base is [latex]2[\/latex]. We raise [latex]2[\/latex] to the fourth power and then take the opposite.\r\n<table id=\"eip-id1168466711166\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-{2}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write in expanded form.<\/td>\r\n<td>[latex]-\\left(2\\cdot 2\\cdot 2\\cdot 2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-\\left(4\\cdot 2\\cdot 2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-\\left(8\\cdot 2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-16[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you try it.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145337&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n[ohm_question]145337[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following example we provide an explanation for the difference between [latex](-3)^2[\/latex] and [latex]-3^2[\/latex].\r\n\r\nhttps:\/\/youtu.be\/qB5PZzmjenI\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\text{Simplify: }12 - 3\\left(9 - 12\\right)[\/latex].\r\n[reveal-answer q=\"843317\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"843317\"]\r\n\r\nSolution:\r\nAccording to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.\r\n<table id=\"eip-id1168466156737\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]12 - 3\\left(9 - 12\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract the parentheses first.<\/td>\r\n<td>[latex]12 - 3\\left(-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]12-\\left(-9\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]\\text{21}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145340&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n[ohm_question]145340[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]8\\left(-9\\right)\\div {\\left(-2\\right)}^{3}[\/latex].\r\n[reveal-answer q=\"388448\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"388448\"]\r\n\r\nSolution:\r\nWe simplify the exponent first, then multiply and divide.\r\n<table id=\"eip-id1168466276311\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8\\left(-9\\right)\\div {\\left(-2\\right)}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the exponent.<\/td>\r\n<td>[latex]8\\left(-9\\right)\\div \\left(-8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-72\\div \\left(-8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]\\text{9}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145345&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n[ohm_question]145345[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\text{Simplify:}-30\\div 2+\\left(-3\\right)\\left(-7\\right)[\/latex].\r\n[reveal-answer q=\"523980\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"523980\"]\r\n\r\nSolution:\r\nFirst we will multiply and divide from left to right. Then we will add.\r\n<table id=\"eip-id1168469759923\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-30\\div 2+\\left(-3\\right)\\left(-7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]-15+\\left(-3\\right)\\left(-7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-15+21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\text{6}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145348&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n[ohm_question]145348[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to evaluate expressions with integers using the order of operations.\r\n\r\nhttps:\/\/youtu.be\/UlfXxzuJIfs","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Model addition of integers with color counters<\/li>\n<li>Model integer subtraction with color counters<\/li>\n<li>Multiply integers with different signs<\/li>\n<li>Multiply integers with the same sign<\/li>\n<li>Multiply an integer by -1<\/li>\n<li>Divide integers with the same sign<\/li>\n<li>Divide integers with different signs<\/li>\n<li>Divide integers by -1<\/li>\n<li>Use correct notation to indicate absolute value<\/li>\n<li>Simplify expressions that contain\u00a0absolute value<\/li>\n<li>Evaluate expressions that contain absolute value<\/li>\n<li>Use the order of operations to simplify expressions that involve integer multiplication, division, addition, and subtraction<\/li>\n<li>Evaluate integer expressions using the order of operations<\/li>\n<\/ul>\n<\/div>\n<h2>Modeling Addition of Integers<\/h2>\n<p>We will first discuss arithmetic operations with integers.<\/p>\n<p>Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.<\/p>\n<p>The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of [latex]2 - 1[\/latex]. Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.<\/p>\n<p>Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.<\/p>\n<p>We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220151\/CNX_BMath_Figure_03_02_024_img.png\" alt=\"This figure has a blue circle labeled positive and a red circle labeled negative.\" \/><br \/>\nIf we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in the figure below.<\/p>\n<p>A blue counter represents [latex]+1[\/latex]. A red counter represents [latex]-1[\/latex]. Together they add to zero.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220152\/CNX_BMath_Figure_03_02_001.png\" alt=\"This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.\" \/><br \/>\nDoing the Manipulative Mathematics activity &#8220;Addition of signed Numbers&#8221; will help you develop a better understanding of adding integers.<br \/>\nWe will model four addition facts using the numbers [latex]5,-5\\text{ and }3,-3[\/latex].<\/p>\n<p>[latex]5+3 - 5+\\left(-3\\right)-5+35+\\left(-3\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]5+3[\/latex].<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168469436825\" class=\"unnumbered unstyled\" summary=\"This figure has five rows. The first row states interpret the expression and has 5 plus 3 means the sum of 5 and 3. The second row states to model the first number and start with 5 positives. It has 5 blue circles. The third row states model the second number and add 3 positives. It has 5 blue circles followed by 3 blue circles. The third row states count the total number of counters. It is followed by 8 blue circles. The fifth row states the sum of 5 and 3 is 8 and is followed by the expression 5 plus 3 equals 8.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]5+3[\/latex] means the sum of [latex]5[\/latex] and [latex]3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220153\/CNX_BMath_Figure_03_02_025_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Model the second number. Add [latex]3[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220154\/CNX_BMath_Figure_03_02_025_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the total number of counters.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220155\/CNX_BMath_Figure_03_02_025_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>The sum of [latex]5[\/latex] and [latex]3[\/latex] is [latex]8[\/latex].<\/td>\n<td>[latex]5+3=8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression.<\/p>\n<p>[latex]2+4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q510181\">Show Solution<\/span><\/p>\n<div id=\"q510181\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220157\/CNX_BMath_Figure_03_02_005_img.png\" alt=\"This figure has six pink circles in a row, representing positive counters. The first two circles are separated from the following four circles.\" \/><\/p>\n<p>[latex]6[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression.<\/p>\n<p>[latex]2+5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q565122\">Show Solution<\/span><\/p>\n<div id=\"q565122\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220158\/CNX_BMath_Figure_03_02_006_img.png\" alt=\"This figure has seven pink circles in a row, representing positive counters. Two circles are separated from the following 5 circles.\" \/><\/p>\n<p>[latex]7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]-5+\\left(-3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q78976\">Show Solution<\/span><\/p>\n<div id=\"q78976\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466019221\" class=\"unnumbered unstyled\" summary=\"This figure has five rows. The first row states to interpret the expression and is followed by negative 5 plus negative 3 means the sum of negative 5 and negative 3. The second row states to model the first number and start with 5 negatives followed by 5 red circles. The third row states model the second number and add 3 negatives followed by a row with 5 red circles and 3 red circles. The fourth row states count the total number of counters followed by 8 red circles. The fifth row states the sum of negative 5 and negative 3 is negative 8, and is followed by the expression negative 5 plus negative 3 equals negative 8.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]-5+\\left(-3\\right)[\/latex] means the sum of [latex]-5[\/latex] and [latex]-3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220159\/CNX_BMath_Figure_03_02_026_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Model the second number. Add [latex]3[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220201\/CNX_BMath_Figure_03_02_026_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the total number of counters.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220202\/CNX_BMath_Figure_03_02_026_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>The sum of [latex]\u22125[\/latex] and [latex]\u22123[\/latex] is [latex]\u22128[\/latex].<\/td>\n<td>[latex]-5+-3=-8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression.<\/p>\n<p>[latex]-2+\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q818374\">Show Solution<\/span><\/p>\n<div id=\"q818374\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220204\/CNX_BMath_Figure_03_02_010_img.png\" alt=\"This figure shows a row of 6 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 4 circles.\" \/><\/p>\n<p>[latex]\u22126[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression.<\/p>\n<p>[latex]-2+\\left(-5\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q875730\">Show Solution<\/span><\/p>\n<div id=\"q875730\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220204\/CNX_BMath_Figure_03_02_011_img.png\" alt=\"This figure shows a row of 7 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 5 circles.\" \/><\/p>\n<p>[latex]\u22127[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The first and second examples\u00a0are very similar. The first example adds [latex]5[\/latex] positives and [latex]3[\/latex] positives\u2014both positives. The second example adds [latex]5[\/latex] negatives and [latex]3[\/latex] negatives\u2014both negatives. In each case, we got a result of [latex]\\text{8-either}8[\/latex] positives or [latex]8[\/latex] negatives. When the signs are the same, the counters are all the same color.<br \/>\nNow let\u2019s see what happens when the signs are different.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]-5+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q229832\">Show Solution<\/span><\/p>\n<div id=\"q229832\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468228028\" class=\"unnumbered unstyled\" summary=\"This figure has six rows. The first row states interpret the expression and is followed by negative 5 plus 3 means the sum of negative 5 and 3. The second row states model the first number and start with 5 negatives. It is followed by 5 red circles. The third row states model the second number and add 3 positives. It is followed by 5 red circles and 3 blue circles. The fourth row states remove any neutral pairs. It is followed by a row of 5 red circles and under that a row of 3 blue circles. The first three red and blue circles are grouped together. The fifth row states count the result and has 2 red circles. The sixth row states the sum of negative 5 and 3 is negative 2. It is followed by the expressions negative 5 plus 3 equals negative 2\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]-5+3[\/latex] means the sum of [latex]-5[\/latex] and [latex]3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220206\/CNX_BMath_Figure_03_02_027_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Model the second number. Add [latex]3[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220207\/CNX_BMath_Figure_03_02_027_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove any neutral pairs.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220209\/CNX_BMath_Figure_03_02_027_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the result.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220210\/CNX_BMath_Figure_03_02_027_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>The sum of [latex]\u22125[\/latex] and [latex]3[\/latex] is [latex]\u22122[\/latex].<\/td>\n<td>[latex]-5+3=-2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Notice that there were more negatives than positives, so the result is negative.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression, and then simplify:<br \/>\n[latex]2+\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q637985\">Show Solution<\/span><\/p>\n<div id=\"q637985\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220213\/CNX_BMath_Figure_03_02_016_img.png\" alt=\"This figure shows two rows of counter circles. The first row has 2 dark pink circles, representing negative counters. The second row has 4 light pink circles, representing positive counters.\" \/><\/p>\n<p>[latex]\u22122[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression, and then simplify:<br \/>\n[latex]2+\\left(-5\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q955006\">Show Solution<\/span><\/p>\n<div id=\"q955006\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220215\/CNX_BMath_Figure_03_02_023_img.png\" alt=\"This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 5 dark pink circles, representing negative counters.\" \/><\/p>\n<p>[latex]\u22123[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]5+\\left(-3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q862636\">Show Solution<\/span><\/p>\n<div id=\"q862636\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468738226\" class=\"unnumbered unstyled\" summary=\"This figure has six rows. The first row states interpret the expression and is followed by 5 plus negative 3 means the sum of 5 and negative 3. The second row states model the first number and start with 5 positives. It is followed by 5 blue circles. The third row states model the second number and add 3 negatives. It is followed by 5 blue circles and 3 red circles. The fourth row states remove any neutral pairs. It is followed by a row of 5 blue circles and under that a row of 3 red circles. The first three red and blue circles are grouped together. The fifth row states count the result and has 2 blue circles. The sixth row states the sum of 5 and negative 3 is 2. It is followed by the expression five plus negative 3 equals 2.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]5+\\left(-3\\right)[\/latex] means the sum of [latex]5[\/latex] and [latex]-3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220216\/CNX_BMath_Figure_03_02_028_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Model the second number. Add [latex]3[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220218\/CNX_BMath_Figure_03_02_028_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove any neutral pairs.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220219\/CNX_BMath_Figure_03_02_028_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the result.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220220\/CNX_BMath_Figure_03_02_028_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>The sum of [latex]5[\/latex] and [latex]\u22123[\/latex] is [latex]2[\/latex].<\/td>\n<td>[latex]5+\\left(-3\\right)=2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression, and then simplify:<\/p>\n<p>[latex]2+\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q512582\">Show Solution<\/span><\/p>\n<div id=\"q512582\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220221\/CNX_BMath_Figure_03_02_022_img.png\" alt=\"This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 4 dark pink circles, representing negative counters.\" \/><\/p>\n<p>[latex]\u22122[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression:<br \/>\n[latex]\\left(-2\\right)+5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q556256\">Show Solution<\/span><\/p>\n<div id=\"q556256\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220222\/CNX_BMath_Figure_03_02_017_img.png\" alt=\"This figure shows two rows of counters. The first row shows 2 dark pink circles, representing negative counters. The second row has 5 light pink circles, representing negative counters. The neutral pairs of one positive and one negative counter are circled leaving three positive counters.\" \/><\/p>\n<p>[latex]3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model each addition.<\/p>\n<ol>\n<li>[latex]4 + 2[\/latex]<\/li>\n<li>[latex]\u22123 + 6[\/latex]<\/li>\n<li>[latex]4 + (\u22125)[\/latex]<\/li>\n<li>[latex]-2 + (\u22123)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q480477\">Show Solution<\/span><\/p>\n<div id=\"q480477\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468294641\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]4[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220223\/CNX_BMath_Figure_03_02_035_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add two positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220224\/CNX_BMath_Figure_03_02_035_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many do you have?<\/td>\n<td>[latex]6[\/latex]. [latex]4+2=6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468679001\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-3+6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]3[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220225\/CNX_BMath_Figure_03_02_036_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]6[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220226\/CNX_BMath_Figure_03_02_036_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove neutral pairs.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220227\/CNX_BMath_Figure_03_02_036_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220228\/CNX_BMath_Figure_03_02_036_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]3[\/latex]. [latex]-3+6=3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467662609\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4+\\left(-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]4[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220229\/CNX_BMath_Figure_03_02_037_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]5[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220230\/CNX_BMath_Figure_03_02_037_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove neutral pairs.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220231\/CNX_BMath_Figure_03_02_037_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220232\/CNX_BMath_Figure_03_02_037_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-1[\/latex]. [latex]4+\\left(-5\\right)=-1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468388113\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-2+\\left(-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]2[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220233\/CNX_BMath_Figure_03_02_038_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]3[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220234\/CNX_BMath_Figure_03_02_038_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many do you have?<\/td>\n<td>[latex]-5[\/latex]. [latex]-2+\\left(-3\\right)=-5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model each addition.<\/p>\n<p>1. [latex]3 + 4[\/latex]<br \/>\n2. [latex]\u22121 + 4[\/latex]<br \/>\n3. [latex]4 + (\u22126)[\/latex]<br \/>\n4. [latex]\u22122 + (\u22122)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q767954\">Show Solution<\/span><\/p>\n<div id=\"q767954\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220235\/CNX_BMath_Figure_03_02_039_img.png\" alt=\".\" \/><\/p>\n<p>2.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220236\/CNX_BMath_Figure_03_02_040_img.png\" alt=\".\" \/><\/p>\n<p>3.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220237\/CNX_BMath_Figure_03_02_041_img.png\" alt=\".\" \/><\/p>\n<p>4.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220238\/CNX_BMath_Figure_03_02_042_img.png\" alt=\".\" \/><\/p>\n<\/div>\n<\/div>\n<p>1. [latex]5 + 1[\/latex]<br \/>\n2. [latex]\u22123 + 7[\/latex]<br \/>\n3. [latex]2 + (\u22128)[\/latex]<br \/>\n4. [latex]\u22123 + (\u22124)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q388859\">Show Solution<\/span><\/p>\n<div id=\"q388859\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220239\/CNX_BMath_Figure_03_02_043_img.png\" alt=\".\" \/><\/p>\n<p>2.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220241\/CNX_BMath_Figure_03_02_044_img.png\" alt=\".\" \/><\/p>\n<p>3.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220242\/CNX_BMath_Figure_03_02_045_img.png\" alt=\".\" \/><\/p>\n<p>4.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220243\/CNX_BMath_Figure_03_02_046_img.png\" alt=\".\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we present more examples of using color counters to model addition of integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Adding Integers with the Same Sign Using Color Counters\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/beI5bQB9a7Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Modeling Subtraction of Integers<\/h2>\n<p>You learn as a child how to subtract numbers through everyday experiences. For example, if you have 10 animal cookies and eat 6 of them, you will have 4 animal cookies left.<\/p>\n<p>Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.<br \/>\nWe will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.<\/p>\n<p>Perhaps when you were younger, you read [latex]5 - 3[\/latex] as <em>five take away three<\/em>. When we use counters, we can think of subtraction the same way.<br \/>\nWe will model four subtraction scenarios\u00a0using the numbers [latex]5[\/latex] and [latex]3[\/latex].<\/p>\n<ul>\n<li>[latex]5 - 3[\/latex]<\/li>\n<li>[latex]- 5-\\left(-3\\right)[\/latex]<\/li>\n<li>[latex]-5 - 3[\/latex]<\/li>\n<li>[latex]5-\\left(-3\\right)[\/latex]<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]5 - 3[\/latex].<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168469608926\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus 3 means 5 take away 3. The second row states model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so take away 3 positives. It is followed by 5 blue circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 blue circles that are labeled two positives. The fifth row states the difference of 5 and 3 is 2. It is followed by the expression 5 minus 3 equals 2.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]5 - 3[\/latex] means [latex]5[\/latex] take away [latex]3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220309\/CNX_BMath_Figure_03_03_027_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away the second number. So take away [latex]3[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220311\/CNX_BMath_Figure_03_03_027_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the counters that are left.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220313\/CNX_BMath_Figure_03_03_027_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]5 - 3=2[\/latex] .<\/p>\n<p>The difference between [latex]5[\/latex] and [latex]3[\/latex] is [latex]2[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression:<\/p>\n<p>[latex]6 - 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q49141\">Show Solution<\/span><\/p>\n<div id=\"q49141\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220315\/CNX_BMath_Figure_03_03_003_img.png\" alt=\"This figure shows a row of 6 light pink circles, representing positive counters. The first four are circled.\" \/><\/p>\n<p>[latex]2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression:<\/p>\n<p>[latex]7 - 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q400845\">Show Solution<\/span><\/p>\n<div id=\"q400845\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220318\/CNX_BMath_Figure_03_03_004_img.png\" alt=\"This figure shows a row of 7 light pink circles representing positive counters. The first four counters are circled.\" \/><\/p>\n<p>[latex]3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In teh previous example we subtracted [latex]3[\/latex] positives from positive [latex]5[\/latex]. Now we will subtract\u00a0[latex]3[\/latex] negatives from\u00a0negative [latex]5[\/latex]. Compare the results of this example to the previous example after you read through it.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]-5-\\left(-3\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q511456\">Show Solution<\/span><\/p>\n<div id=\"q511456\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469693755\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so take away 3 negatives. It is followed by 5 red circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 red circles which are labeled two negatives. The fifth row states the difference of negative 5 and negative 3 is negative 2. Next to this is the expression negative 5 minus negative 3 equals negative two.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]-5-\\left(-3\\right)[\/latex] means [latex]-5[\/latex] take away [latex]-3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220319\/CNX_BMath_Figure_03_03_028_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away the second number. So take away [latex]3[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220320\/CNX_BMath_Figure_03_03_028_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the number of counters that are left.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220321\/CNX_BMath_Figure_03_03_028_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5-\\left(-3\\right)=-2[\/latex] .<\/p>\n<p>The difference between [latex]-5[\/latex] and [latex]-3[\/latex] is [latex]-2[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>You can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression:<\/p>\n<p>[latex]-6-\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q720789\">Show Solution<\/span><\/p>\n<div id=\"q720789\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220324\/CNX_BMath_Figure_03_03_008_img.png\" alt=\"This figure shows a row of 6 dark pink circles, representing negative counters. The last four counters are circled.\" \/><\/p>\n<p>[latex]\u22122[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression:<\/p>\n<p>[latex]-7-\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q188555\">Show Solution<\/span><\/p>\n<div id=\"q188555\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220325\/CNX_BMath_Figure_03_03_009_img.png\" alt=\"This figure is a row of 7 dark pink circles representing negative counters. The first four counters are circled.\" \/><\/p>\n<p>[latex]\u22123[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that the previous two examples\u00a0are very much alike.<\/p>\n<ul id=\"fs-id801766\">\n<li>First, we subtracted [latex]3[\/latex] positives from [latex]5[\/latex] positives to get [latex]2[\/latex] positives.<\/li>\n<li>Then we subtracted [latex]3[\/latex] negatives from [latex]5[\/latex] negatives to get [latex]2[\/latex] negatives.<\/li>\n<\/ul>\n<p>Each example used counters of only one color, and the &#8220;take away&#8221; model of subtraction was easy to apply.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220326\/CNX_BMath_Figure_03_03_040_img.png\" alt=\"This figure has a row of 5 blue circles. The first three are circled. Above the row is 5 minus 3 equals 2. Next to this is a row of 5 red circles. The first three are circled. Above the row is negative 5 minus negative 3 equals negative 2.\" \/><br \/>\nNow let\u2019s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]-5 - 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q662198\">Show Solution<\/span><\/p>\n<div id=\"q662198\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468477542\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by negative 5 minus 3 means negative 5 take away 3. The second row states to model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so we need to take away 3 positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The fourth row states there are no positives to take away so add neutral pairs until you have 3 positives. The fifth row states now take away three positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The blue shaded circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 red circles. The seventh row states the difference of negative five and three is negative 8. It is followed by the expression negative 5 minus 3 equals negative 8.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]-5 - 3[\/latex] means [latex]-5[\/latex] take away [latex]3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220328\/CNX_BMath_Figure_03_03_029_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away the second number.<\/p>\n<p>So we need to take away [latex]3[\/latex] positives.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>But there are no positives to take away.<\/p>\n<p>Add neutral pairs until you have [latex]3[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220329\/CNX_BMath_Figure_03_03_029_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Now take away [latex]3[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220330\/CNX_BMath_Figure_03_03_029_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the number of counters that are left.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220332\/CNX_BMath_Figure_03_03_029_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5 - 3=-8[\/latex] .<\/p>\n<p>The difference of [latex]-5[\/latex] and [latex]3[\/latex] is [latex]-8[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression:<\/p>\n<p>[latex]-6 - 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q647548\">Show Solution<\/span><\/p>\n<div id=\"q647548\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220334\/CNX_BMath_Figure_03_03_012_img.png\" alt=\"This figure shows a row of 10 dark pink circles, representing negative counters. The first six counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters. These four positive counters are circled.\" \/><\/p>\n<p>[latex]\u221210[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression:<\/p>\n<p>[latex]-7 - 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q391259\">Show Solution<\/span><\/p>\n<div id=\"q391259\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220336\/CNX_BMath_Figure_03_03_013_img.png\" alt=\"This figure shows a row of 11 dark pink circles, representing negative counters. The first seven counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters. These four positive counters are circled.\" \/><\/p>\n<p>[latex]\u221211[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now we will subtract a negative number from a positive number. Think of this as taking away the negative.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]5-\\left(-3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q943073\">Show Solution<\/span><\/p>\n<div id=\"q943073\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467234371\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states to model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so we need to take away 3 negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The fourth row states there are no negatives to take away so add neutral pairs until you have 3 negatives. The fifth row states now take away three negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The red circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 blue circles. The seventh row states the difference of 5 and negative 3 is eight, beside it the expression 5 minus negative 3 equals eight.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]5-\\left(-3\\right)[\/latex] means [latex]5[\/latex] take away [latex]-3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220337\/CNX_BMath_Figure_03_03_030_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away the second number, so take away [latex]3[\/latex] negatives.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>But there are no negatives to take away.<\/p>\n<p>Add neutral pairs until you have [latex]3[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220338\/CNX_BMath_Figure_03_03_030_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away [latex]3[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220339\/CNX_BMath_Figure_03_03_030_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the number of counters that are left.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220341\/CNX_BMath_Figure_03_03_030_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The difference of [latex]5[\/latex] and [latex]-3[\/latex] is [latex]8[\/latex] .<\/p>\n<p>[latex]5-\\left(-3\\right)=8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression:<\/p>\n<p>[latex]6-\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q459252\">Show Solution<\/span><\/p>\n<div id=\"q459252\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220342\/CNX_BMath_Figure_03_03_016_img.png\" alt=\"This figure shows a row of 10 light pink circles, representing positive counters. The first six counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.\" \/><\/p>\n<p>[latex]10[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression:<br \/>\n[latex]7-\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q632721\">Show Solution<\/span><\/p>\n<div id=\"q632721\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220343\/CNX_BMath_Figure_03_03_017_img.png\" alt=\"This figure shows a row of 11 light pink circles, representing positive counters. The first seven counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.\" \/><\/p>\n<p>[latex]11[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Now we will do an example that summarizes the situations above, with different numbers. Recall the different scenarios:<\/p>\n<ul>\n<li>subtracting a positive number from a positive number<\/li>\n<li>subtracting a positive number from a negative number<\/li>\n<li>subtracting a negative number from a positive number<\/li>\n<li>subtracting a negative number from a negative number<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model each subtraction.<\/p>\n<p>[latex]8 \u2212 2[\/latex]<br \/>\n[latex]\u22125 \u2212 4[\/latex]<br \/>\n[latex]6 \u2212 (\u22126)[\/latex]<br \/>\n[latex]\u22128 \u2212 (\u22123)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q214895\">Show Solution<\/span><\/p>\n<div id=\"q214895\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168468308805\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]8 - 2[\/latex]<\/p>\n<p>This means [latex]8[\/latex] take away [latex]2[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]8[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220345\/CNX_BMath_Figure_03_03_041_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]2[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220346\/CNX_BMath_Figure_03_03_041_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]8 - 2=6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466328467\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5 - 4[\/latex]<\/p>\n<p>This means [latex]-5[\/latex] take away [latex]4[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]5[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220347\/CNX_BMath_Figure_03_03_042_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>You need to take away [latex]4[\/latex] positives.<\/p>\n<p>Add [latex]4[\/latex] neutral pairs to get [latex]4[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220349\/CNX_BMath_Figure_03_03_042_img-02.png\" alt=\".\" \/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220350\/CNX_BMath_Figure_03_03_042_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]4[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220352\/CNX_BMath_Figure_03_03_042_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220354\/CNX_BMath_Figure_03_03_042_img-05.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5 - 4=-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469609690\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]6-\\left(-6\\right)[\/latex]<\/p>\n<p>This means [latex]6[\/latex] take away [latex]-6[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]6[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220355\/CNX_BMath_Figure_03_03_043_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]6[\/latex] neutrals to get [latex]6[\/latex] negatives to take away.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220357\/CNX_BMath_Figure_03_03_043_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove [latex]6[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220359\/CNX_BMath_Figure_03_03_043_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220401\/CNX_BMath_Figure_03_03_043_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]6-\\left(-6\\right)=12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468534936\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-8-\\left(-3\\right)[\/latex]<\/p>\n<p>This means [latex]-8[\/latex] take away [latex]-3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]8[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220403\/CNX_BMath_Figure_03_03_044_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]3[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220404\/CNX_BMath_Figure_03_03_044_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220405\/CNX_BMath_Figure_03_03_044_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-8-\\left(-3\\right)=-5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model each subtraction.<\/p>\n<p>1. [latex]7 - (-8)[\/latex]<br \/>\n2. [latex]-7 - (-2)[\/latex]<br \/>\n3. [latex]4 - 1[\/latex]<br \/>\n4. [latex]-6 - 8[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q324876\">Show Solution<\/span><\/p>\n<div id=\"q324876\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220406\/CNX_BMath_Figure_03_03_045_img.png\" alt=\".\" \/><\/p>\n<p>2.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220408\/CNX_BMath_Figure_03_03_046_img.png\" alt=\".\" \/><\/p>\n<p>3.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220409\/CNX_BMath_Figure_03_03_047_img.png\" alt=\".\" \/><\/p>\n<p>4.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220410\/CNX_BMath_Figure_03_03_048_img.png\" alt=\".\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model each subtraction.<\/p>\n<p>1. [latex]4 - (-6)[\/latex]<br \/>\n2. [latex]-8 - (-1)[\/latex]<br \/>\n3. [latex]7 - 3[\/latex]<br \/>\n4. [latex]-4 - 2[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q677150\">Show Solution<\/span><\/p>\n<div id=\"q677150\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220411\/CNX_BMath_Figure_03_03_049_img.png\" alt=\".\" \/><\/p>\n<p>2.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220412\/CNX_BMath_Figure_03_03_050_img.png\" alt=\".\" \/><\/p>\n<p>3.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220413\/CNX_BMath_Figure_03_03_051_img.png\" alt=\".\" \/><\/p>\n<p>4.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220414\/CNX_BMath_Figure_03_03_052_img.png\" alt=\".\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Each of the examples so far have been carefully constructed so that the sign of the answer matched the sign of the first number in the expression. \u00a0For example, in\u00a0\u00a0[latex]\u22125 \u2212 4[\/latex], the result is [latex]-9[\/latex], which is the same sign as [latex]-5[\/latex]. Now we will see subtraction where the sign of the result is different from the starting number.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model each subtraction expression:<\/p>\n<ol id=\"eip-id1168468264062\" class=\"circled\">\n<li>[latex]2 - 8[\/latex]<\/li>\n<li>[latex]-3-\\left(-8\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q485889\">Show Solution<\/span><\/p>\n<div id=\"q485889\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466285116\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has six rows. The first row states we start with 2 positives. It is followed by 2 blue circles. The second row states we need to take away 8 positives but we have only 2. The third row states add neutral pairs until there are 8 positives to take away. It is followed by a row of 8 blue circles. The first two are separated from the last 6. Under the last six there is a row of 6 red circles. The fourth row states then take away the 8 positives. It is followed by a row of 8 blue circles. They are circled. The fifth row states find the number of counters that are left. There are six negatives. It is followed by 6 red circles labeled six negatives. The sixth row has 2 minus 8 equals negative 6.\">\n<tbody>\n<tr>\n<td>1.<\/p>\n<p>We start with [latex]2[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220415\/CNX_BMath_Figure_03_03_031_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>We need to take away [latex]8[\/latex] positives, but we have only [latex]2[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Add neutral pairs until there are [latex]8[\/latex] positives to take away.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220416\/CNX_BMath_Figure_03_03_031_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away [latex]8[\/latex]\u00a0positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220418\/CNX_BMath_Figure_03_03_031_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the number of counters that are left.<\/p>\n<p>There are [latex]6[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220419\/CNX_BMath_Figure_03_03_031_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]2 - 8=-6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468388588\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has six rows. The first row states we start with 3 negatives. It is followed by 3 red circles. The second row states we need to take away 8 negatives but we have only 3. The third row states add neutral pairs until there are 8 negatives to take away. It is followed by a row of 8 red circles. The first three are separated from the last 5. Under the last 5 there is a row of 5 blue circles. The fourth row states then take away the 8 negatives. It is followed by a row of 8 red circles. They are circled. The fifth row states find the number of counters that are left. There are five positives. It is followed by 5 blue circles. The sixth row has negative 3 minus negative 8 equals 5.\">\n<tbody>\n<tr>\n<td>2.<\/p>\n<p>We start with [latex]3[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220422\/CNX_BMath_Figure_03_03_032_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>We need to take away [latex]8[\/latex] negatives, but we have only [latex]3[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Add neutral pairs until there are [latex]8[\/latex] negatives to take away.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220423\/CNX_BMath_Figure_03_03_032_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away the [latex]8[\/latex] negatives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220425\/CNX_BMath_Figure_03_03_032_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the number of counters that are left.<\/p>\n<p>There are [latex]5[\/latex] positives.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220426\/CNX_BMath_Figure_03_03_032_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-3-\\left(-8\\right)=5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model each subtraction expression.<\/p>\n<ol id=\"eip-id1168466205125\" class=\"circled\">\n<li>[latex]7 - 9[\/latex]<\/li>\n<li>[latex]-5-(-9)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q678756\">Show Solution<\/span><\/p>\n<div id=\"q678756\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220427\/CNX_BMath_Figure_03_03_020_img.png\" alt=\"This figure shows a row of 9 light pink circles, representing positive counters. The first seven are separated from the last two. The entire row is circled. Below the last two light pink circles is a row of two dark pink circles, representing negative counters.\" \/><\/p>\n<p>[latex]\u22122[\/latex]<\/p>\n<p>2.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220428\/CNX_BMath_Figure_03_03_021_img.png\" alt=\"This figure shows a row of 9 dark pink circles, representing negative counters. The first five are separated from the last four. The entire row is circled. Below the last four dark pink circles is a row of four light pink circles, representing positive counters.\" \/><\/p>\n<p>[latex]4[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model each subtraction expression.<\/p>\n<ol id=\"eip-id1168468470083\" class=\"circled\">\n<li>[latex]4 - 7[\/latex]<\/li>\n<li>[latex]-7-\\left(-10\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q164098\">Show Solution<\/span><\/p>\n<div id=\"q164098\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220429\/CNX_BMath_Figure_03_03_022_img.png\" alt=\"This figure shows a row of seven light pink circles, representing positive counters. The first four are separated from the last three. The entire row is circled. Below the last three light pink circles is a row of three dark pink circles, representing negative counters.\" \/><\/p>\n<p>[latex]\u22123[\/latex]<\/p>\n<p>2.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220430\/CNX_BMath_Figure_03_03_023_img.png\" alt=\"This figure shows a row of 10 dark pink circles, representing negative counters. The first seven are separated from the last three. The entire row is circled. Below the last three dark pink circles is a row of three light pink circles, representing positive counters.\" \/><\/p>\n<p>[latex]3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When you subtract two integers, there are two possibilities, either the result will have a different sign from the starting number, or it will have the same sign.<\/p>\n<p>Watch the video below to see more examples of modeling integer subtraction with color counters.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Subtracting Integers with Color Counters (No Extra Zeros Needed)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ZjiL7GfxgiI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Modeling Multiplication of Integers<\/h2>\n<p>Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.<br \/>\nWe remember that [latex]a\\cdot b[\/latex] means add [latex]a,b[\/latex] times. Here, we are using the model shown in the graphic below\u00a0just to help us discover the pattern.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220519\/CNX_BMath_Figure_03_04_001.png\" alt=\"This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.\" \/><br \/>\nNow consider what it means to multiply [latex]5[\/latex] by [latex]-3[\/latex]. It means subtract [latex]5,3[\/latex] times. Looking at subtraction as <em>taking away<\/em>, it means to take away [latex]5,3[\/latex] times. But there is nothing to take away, so we start by adding neutral pairs as shown in the graphic below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220522\/CNX_BMath_Figure_03_04_004-1.png\" alt=\"This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives. Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.\" \/><br \/>\nIn both cases, we started with [latex]\\mathbf{\\text{15}}[\/latex] neutral pairs. In the case on the left, we took away [latex]\\mathbf{\\text{5}},\\mathbf{\\text{3}}[\/latex] times and the result was [latex]-\\mathbf{\\text{15}}[\/latex]. To multiply [latex]\\left(-5\\right)\\left(-3\\right)[\/latex], we took away [latex]-\\mathbf{\\text{5}},\\mathbf{\\text{3}}[\/latex] times and the result was [latex]\\mathbf{\\text{15}}[\/latex]. So we found that<\/p>\n<p>[latex]\\begin{array}{ccc}5\\cdot 3=15\\hfill & & -5\\left(3\\right)=-15\\hfill \\\\ 5\\left(-3\\right)=-15\\hfill & & \\left(-5\\right)\\left(-3\\right)=15\\hfill \\end{array}[\/latex]<\/p>\n<p>Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplication of Signed Numbers<\/h3>\n<p>The sign of the product of two numbers depends on their signs.<\/p>\n<table id=\"fs-id1385641\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"This is a table with two columns. The first column states same signs. The second column states product. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under product, it states negative.\">\n<thead>\n<tr valign=\"top\">\n<th>Same signs<\/th>\n<th>Product<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u2022Two positives<\/p>\n<p>\u2022Two negatives<\/td>\n<td>Positive<\/p>\n<p>Positive<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-id2295230\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"This is a table with two columns. The first column states same signs. The second column states product. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under product, it states negative.\">\n<thead>\n<tr valign=\"top\">\n<th>Different signs<\/th>\n<th>Product<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u2022Positive \u2022 negative<\/p>\n<p>\u2022Negative \u2022 positive<\/td>\n<td>Negative<\/p>\n<p>Negative<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply each of the following:<\/p>\n<ol>\n<li>[latex]-9\\cdot 3[\/latex]<\/li>\n<li>[latex]-2\\left(-5\\right)[\/latex]<\/li>\n<li>[latex]4\\left(-8\\right)[\/latex]<\/li>\n<li>[latex]7\\cdot 6[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1168469461529\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-9\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply, noting that the signs are different and so the product is negative.<\/td>\n<td>[latex]-27[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467249422\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-2\\left(-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply, noting that the signs are the same and so the product is positive.<\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466068278\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4\\left(-8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply, noting that the signs are different and so the product is negative.<\/td>\n<td>[latex]-32[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466077286\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]7\\cdot 6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The signs are the same, so the product is positive.<\/td>\n<td>[latex]42[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145306&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145306\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145306&theme=oea&iframe_resize_id=ohm145306&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video for more examples of how to multiply integers with different signs, and the same sign.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Multiplying Integers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QY-Za42DItw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>When we multiply a number by [latex]1[\/latex], the result is the same number. What happens when we multiply a number by [latex]-1?[\/latex] Let\u2019s multiply a positive number and then a negative number by [latex]-1[\/latex] to see what we get.<\/p>\n<p>[latex]\\begin{array}{ccc}\\hfill -1\\cdot 4\\hfill & & \\hfill -1\\left(-3\\right)\\hfill \\\\ \\hfill -4\\hfill & & \\hfill 3\\hfill \\\\ \\hfill -4\\text{ is the opposite of }\\mathbf{\\text{4}}\\hfill & & \\hfill \\mathbf{\\text{3}}\\text{ is the opposite of }-3\\hfill \\end{array}[\/latex]<br \/>\nEach time we multiply a number by [latex]-1[\/latex], we get its opposite.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplication by [latex]-1[\/latex]<\/h3>\n<p>Multiplying a number by [latex]-1[\/latex] gives its opposite.<\/p>\n<p>[latex]-1a=-a[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply each of the following:<\/p>\n<ol>\n<li>[latex]-1\\cdot 7[\/latex]<\/li>\n<li>[latex]-1\\left(-11\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q459967\">Show Solution<\/span><\/p>\n<div id=\"q459967\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468284281\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>The signs are different, so the product will be negative.<\/td>\n<td>[latex]-1\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Notice that [latex]\u22127[\/latex] is the opposite of [latex]7[\/latex].<\/td>\n<td>[latex]-7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469855058\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>The signs are the same, so the product will be positive.<\/td>\n<td>[latex]-1\\left(-11\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Notice that [latex]11[\/latex] is the opposite of [latex]\u221211[\/latex].<\/td>\n<td>[latex]11[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145319&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145319\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145319&theme=oea&iframe_resize_id=ohm145319&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Division with Integers<\/h2>\n<p>Division is the inverse operation of multiplication. So, [latex]15\\div 3=5[\/latex] because [latex]5\\cdot 3=15[\/latex] In words, this expression says that [latex]\\mathbf{\\text{15}}[\/latex] can be divided into [latex]\\mathbf{\\text{3}}[\/latex] groups of [latex]\\mathbf{\\text{5}}[\/latex] each because adding five three times gives [latex]\\mathbf{\\text{15}}[\/latex]. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.<\/p>\n<p>[latex]\\begin{array}{ccccc}5\\cdot 3=15\\text{ so }15\\div 3=5\\hfill & & & & -5\\left(3\\right)=-15\\text{ so }-15\\div 3=-5\\hfill \\\\ \\left(-5\\right)\\left(-3\\right)=15\\text{ so }15\\div \\left(-3\\right)=-5\\hfill & & & & 5\\left(-3\\right)=-15\\text{ so }-15\\div -3=5\\hfill \\end{array}[\/latex]<br \/>\nDivision of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division of Signed Numbers<\/h3>\n<p>The sign of the quotient of two numbers depends on their signs.<\/p>\n<table id=\"fs-id2593939\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"This is a table with two columns. The first column states same signs. The second column states quotient. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under quotient, it states negative.\">\n<thead>\n<tr valign=\"top\">\n<th>Same signs<\/th>\n<th>Quotient<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u2022Two positives<\/p>\n<p>\u2022Two negatives<\/td>\n<td>Positive<\/p>\n<p>Positive<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-id2379645\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"This is a table with two columns. The first column states same signs. The second column states quotient. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under quotient, it states negative.\">\n<thead>\n<tr valign=\"top\">\n<th>Different signs<\/th>\n<th>Quotient<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u2022Positive &amp; negative<\/p>\n<p>\u2022Negative &amp; positive<\/td>\n<td>Negative<\/p>\n<p>Negative<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Remember, you can always check the answer to a division problem by multiplying.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide each of the following:<\/p>\n<ol>\n<li>[latex]-27\\div 3[\/latex]<\/li>\n<li>[latex]-100\\div \\left(-4\\right)[\/latex]<\/li>\n<\/ol>\n<p>Solution<\/p>\n<table id=\"eip-id1168467157728\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr style=\"height: 15.7812px;\">\n<td style=\"height: 15.7812px;\">1.<\/td>\n<td style=\"height: 15.7812px;\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]-27\\div 3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Divide, noting that the signs are different and so the quotient is negative.<\/td>\n<td style=\"height: 15px;\">[latex]-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467157714\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-100\\div \\left(-4\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide, noting that the signs are the same and so the quotient is positive.<\/td>\n<td>[latex]25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145323&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145323\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145323&theme=oea&iframe_resize_id=ohm145323&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Just as we saw with multiplication, when we divide a number by [latex]1[\/latex], the result is the same number. What happens when we divide a number by [latex]-1?[\/latex] Let\u2019s divide a positive number and then a negative number by [latex]-1[\/latex] to see what we get.<\/p>\n<p>[latex]\\begin{array}{cccc}8\\div \\left(-1\\right)\\hfill & & & -9\\div \\left(-1\\right)\\hfill \\\\ -8\\hfill & & & 9\\hfill \\\\ \\hfill \\text{-8 is the opposite of 8}\\hfill & & & \\hfill \\text{9 is the opposite of -9}\\hfill \\end{array}[\/latex]<br \/>\nWhen we divide a number by [latex]-1[\/latex] we get its opposite.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division by [latex]-1[\/latex]<\/h3>\n<p>Dividing a number by [latex]-1[\/latex] gives its opposite.<\/p>\n<p>[latex]a\\div \\left(-1\\right)=-a[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide each of the following:<\/p>\n<ol>\n<li>[latex]16\\div \\left(-1\\right)[\/latex]<\/li>\n<li>[latex]-20\\div \\left(-1\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q343427\">Show Solution<\/span><\/p>\n<div id=\"q343427\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466004748\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]16\\div \\left(-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The dividend, [latex]16[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\n<td>[latex]-16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Notice that the signs were different, so the result was negative.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469488934\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-20\\div \\left(-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The dividend, [latex]\u201320[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\n<td>[latex]20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that the signs were the same, so the quotient was positive.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145326\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145326&theme=oea&iframe_resize_id=ohm145326&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video for more examples of how to divide integers with the same and different signs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex:  Dividing Integers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/z5ZFiyLi5Y0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Absolute Value<\/h2>\n<p>We saw that numbers such as [latex]5[\/latex] and [latex]-5[\/latex] are opposites because they are the same distance from [latex]0[\/latex] on the number line. They are both five units from [latex]0[\/latex]. The distance between [latex]0[\/latex] and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative.<br \/>\nThe symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[\/latex] is written as [latex]|5|[\/latex], and the absolute value of [latex]-5[\/latex] is written as [latex]|-5|[\/latex] as shown below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220135\/CNX_BMath_Figure_03_01_019.png\" alt=\"This figure is a number line. The points negative 5 and 5 are labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units. Also above the number line the distance from 0 to 5 is labeled as 5 units.\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Absolute Value<\/h3>\n<p>The absolute value of a number is its distance from [latex]0[\/latex] on the number line.<br \/>\nThe absolute value of a number [latex]n[\/latex] is written as [latex]|n|[\/latex].<\/p>\n<p>[latex]|n|\\ge 0\\text{ for all numbers}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>[latex]|3|[\/latex]<br \/>\n[latex]|-44|[\/latex]<br \/>\n[latex]|0|[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1165118025864\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]|3|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex] is [latex]3[\/latex] units from zero.<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1171842979184\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]|-44|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u221244[\/latex] is [latex]44[\/latex] units from zero.<\/td>\n<td>[latex]44[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1171842583307\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]|0|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex] is already at zero.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144930&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144931&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144932&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144933&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<p>In the video below we show more example of how to find the absolute value of an integer.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex 1: Determine the Absolute Value of an Integer\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/I8bTqGmkqGI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Evaluate:<\/p>\n<ol>\n<li>[latex]|x|\\text{ when }x=-35[\/latex]<\/li>\n<li>[latex]|\\mathit{\\text{-y}}|\\text{ when }y=-20[\/latex]<\/li>\n<li>[latex]-|u|\\text{ when }u=12[\/latex]<\/li>\n<li>[latex]-|p|\\text{ when }p=-14[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q727778\">Show Solution<\/span><\/p>\n<div id=\"q727778\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466007339\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has three rows. The first row has the absolute value of x. The second row states to substitute negative 35 for x and has the absolute value symbol with negative 35. The third row states to take the absolute value and has 35.\">\n<tbody>\n<tr>\n<td>1. To find [latex]|x|[\/latex] when [latex]x=-35:[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]|x|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{-35}[\/latex] for x.<\/td>\n<td>[latex]\\mid\\color{red}{-35}\\mid[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the absolute value.<\/td>\n<td>[latex]35[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469585161\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has four rows. The first row has the absolute value of y. The second row states to substitute negative 20 for y and has inside of the absolute value symbol a negative sign with negative 20 inside of parentheses. The third row states to simplify and has 20 inside of the absolute value symbol. The fourth row states to take the absolute value and has 20.\">\n<tbody>\n<tr>\n<td>2. To find [latex]|-y|[\/latex] when [latex]y=-20:[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]|-y|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{-20}[\/latex] for y.<\/td>\n<td>[latex]\\mid-(\\color{red}{-20})\\mid[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]|20|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the absolute value.<\/td>\n<td>[latex]20[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466911241\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has three rows. The first row has a negative sign in front of the absolute value of u. The second row states to substitute 12 for u and has a negative sign in front of the absolute value symbol with 12 on the inside. The third row states to take the absolute value and has negative 12.\">\n<tbody>\n<tr>\n<td>3. To find [latex]-|u|[\/latex] when [latex]u=12:[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-|u|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{12}[\/latex] for u.<\/td>\n<td>[latex]-\\mid\\color{red}{12}\\mid[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the absolute value.<\/td>\n<td>[latex]-12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468381460\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has three rows. The first row has a negative sign in front of the absolute value of p. The second row states to substitute negative 14 for p and has a negative sign in front of the absolute value of negative 14. The third row states to take the absolute value and has negative 14.\">\n<tbody>\n<tr>\n<td>4. To find [latex]-|p|[\/latex] when [latex]p=-14:[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-|p|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{-14}[\/latex] for p.<\/td>\n<td>[latex]-\\mid\\color{red}{-14}\\mid[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the absolute value.<\/td>\n<td>[latex]-14[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Notice that the result is negative only when there is a negative sign outside the absolute value symbol.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144934&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Fill in [latex]\\text{<},\\text{>},\\text{or}=[\/latex] for each of the following:<\/p>\n<ol>\n<li>[latex]|-5|___-|-5|[\/latex]<\/li>\n<li>[latex]8___-|-8|[\/latex]<\/li>\n<li>[latex]-9___-|-9|[\/latex]<\/li>\n<li>[latex]-|-7|___ - 7[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q622738\">Show Solution<\/span><\/p>\n<div id=\"q622738\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nTo compare two expressions, simplify each one first. Then compare.<\/p>\n<table id=\"eip-id1166567974475\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]|-5|___-|-5|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]5___ - 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Order.<\/td>\n<td>[latex]5>-5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1166568580863\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]8___-|-8|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]8___ - 8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Order.<\/td>\n<td>[latex]8>-8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1166567977584\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-9___-|-9|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-9___ - 9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Order.<\/td>\n<td>[latex]-9=-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1166569017628\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-|-7|___ - 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-7___ - 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Order.<\/td>\n<td>[latex]-7=-7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144979&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<p>In the video below we show more examples of how to compare expressions that include absolute value and integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex:  Comparing Absolute Value of Integers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/TendEcSaM3w?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Absolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]|9 - 3|[\/latex]<\/li>\n<li>[latex]4|-2|[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q47658\">Show Solution<\/span><\/p>\n<div id=\"q47658\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nFor each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.<\/p>\n<table id=\"eip-id1168466096398\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]|9\u22123|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify inside the absolute value sign.<\/td>\n<td>[latex]|6|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the absolute value.<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469668431\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4|\u22122|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the absolute value.<\/td>\n<td>[latex]4\u22c52[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom7\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145007&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]|8+7|-|5+6|[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q743144\">Show Solution<\/span><\/p>\n<div id=\"q743144\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nFor each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.<\/p>\n<table id=\"eip-id1168469785725\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]|8+7|\u2212|5+6|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify inside each absolute value sign.<\/td>\n<td>[latex]|15|\u2212|11|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom8\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145009&amp;theme=oea&amp;iframe_resize_id=mom8\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]24-|19 - 3\\left(6 - 2\\right)|[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q949242\">Show Solution<\/span><\/p>\n<div id=\"q949242\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe use the order of operations. Remember to simplify grouping symbols first, so parentheses inside absolute value symbols would be first.<\/p>\n<table id=\"eip-id1168467296922\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]24-|19 - 3\\left(6 - 2\\right)|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify in the parentheses first.<\/td>\n<td>[latex]24-|19 - 3\\left(4\\right)|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]3\\left(4\\right)[\/latex] .<\/td>\n<td>[latex]24-|19 - 12|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract inside the absolute value sign.<\/td>\n<td>[latex]24-|7|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the absolute value.<\/td>\n<td>[latex]24 - 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]17[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145010\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145010&theme=oea&iframe_resize_id=ohm145010&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p><span style=\"color: #000000;\">Watch the following video to see more examples of how to simplify expressions that contain absolute value.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Simplify Absolute Value Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/cd24nT7mAi0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 data-type=\"title\">Simplify Expressions Using the Order of Operations<\/h2>\n<p>We\u2019ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.<\/p>\n<p>For example, consider the expression:<\/p>\n<p style=\"text-align: center;\">[latex]4+3\\cdot 7[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccc}\\hfill \\text{Some students say it simplifies to 49.}\\hfill & & & \\hfill \\text{Some students say it simplifies to 25.}\\hfill \\\\ \\begin{array}{ccc}& & \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }4+3\\text{ gives 7.}\\hfill & & \\hfill 7\\cdot 7\\hfill \\\\ \\text{And }7\\cdot 7\\text{ is 49.}\\hfill & & \\hfill 49\\hfill \\end{array}& & & \\begin{array}{ccc}& & \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }3\\cdot 7\\text{ is 21.}\\hfill & & \\hfill 4+21\\hfill \\\\ \\text{And }21+4\\text{ makes 25.}\\hfill & & \\hfill 25\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/p>\n<p>Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.<\/p>\n<div class=\"textbox shaded\">\n<h3>Order of Operations<\/h3>\n<p>When simplifying mathematical expressions perform the operations in the following order:<br \/>\n1. <strong>P<\/strong>arentheses and other Grouping Symbols<\/p>\n<ul id=\"fs-id1171104029952\" data-bullet-style=\"bullet\">\n<li>Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.<\/li>\n<\/ul>\n<p>2. <strong>E<\/strong>xponents<\/p>\n<ul id=\"fs-id1171104407077\" data-bullet-style=\"bullet\">\n<li>Simplify all expressions with exponents.<\/li>\n<\/ul>\n<p>3. <strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision<\/p>\n<ul id=\"fs-id1171103140103\" data-bullet-style=\"bullet\">\n<li>Perform all multiplication and division in order from left to right. These operations have equal priority.<\/li>\n<\/ul>\n<p>4. <strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction<\/p>\n<ul id=\"fs-id1171104002792\" data-bullet-style=\"bullet\">\n<li>Perform all addition and subtraction in order from left to right. These operations have equal priority.<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Students often ask, &#8220;How will I remember the order?&#8221; Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. <strong>P<\/strong>lease <strong>E<\/strong>xcuse <strong>M<\/strong>y <strong>D<\/strong>ear <strong>A<\/strong>unt <strong>S<\/strong>ally.<\/p>\n<table id=\"fs-id1786633\" class=\"unnumbered\" summary=\"This table has five rows and two columns. The first row spans both columns and is a header reading\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"2\" data-align=\"center\"><strong>Order of Operations<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\"><strong>P<\/strong>lease<\/td>\n<td data-align=\"left\"><strong>P<\/strong>arentheses<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\"><strong>E<\/strong>xcuse<\/td>\n<td data-align=\"left\"><strong>E<\/strong>xponents<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\"><strong>M<\/strong>y <strong>D<\/strong>ear<\/td>\n<td data-align=\"left\"><strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\"><strong>A<\/strong>unt <strong>S<\/strong>ally<\/td>\n<td data-align=\"left\"><strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It\u2019s good that \u2018<strong>M<\/strong>y <strong>D<\/strong>ear\u2019 goes together, as this reminds us that <strong>m<\/strong>ultiplication and <strong>d<\/strong>ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.<br \/>\nSimilarly, \u2018<strong>A<\/strong>unt <strong>S<\/strong>ally\u2019 goes together and so reminds us that <strong>a<\/strong>ddition and <strong>s<\/strong>ubtraction also have equal priority and we do them in order from left to right.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify the expressions:<\/p>\n<ol>\n<li>[latex]4+3\\cdot 7[\/latex]<\/li>\n<li>[latex]\\left(4+3\\right)\\cdot 7[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1164750479370\" class=\"unnumbered unstyled\" summary=\"The table shows the expression four plus three times seven. On the next line it states are there any parentheses in the expression? No. The line below that states are there any exponents in the expression? No. The next line states is there any multiplication or division in the expression? Yes. The next line states Multiply first and is followed by the expression of four plus three times seven. The expression is now four plus twenty-one. The last operation is addition. Add four and twenty-one to get twenty-five.\" data-label=\"\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4+3\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply first.<\/td>\n<td>[latex]4+\\color{red}{3\\cdot 7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]4+21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1164754514704\" class=\"unnumbered unstyled\" summary=\"The image shows the expression four plus three in parentheses times seven. Are there any parentheses? Yes, simplify inside the parentheses by adding four and three to get seven. The expression is now seven times seven. Is there any multiplication or division? Yes, multiply seven by seven to get forty-nine.\" data-label=\"\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex](4+3)\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>p<\/strong>arentheses? Yes.<\/td>\n<td>[latex]\\color{red}{(4+3)}\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify inside the parentheses.<\/td>\n<td>[latex](7)7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]49[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144748\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144748&theme=oea&iframe_resize_id=ohm144748&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm144751\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144751&theme=oea&iframe_resize_id=ohm144751&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]\\text{18}\\div \\text{9}\\cdot \\text{2}[\/latex]<\/li>\n<li>[latex]\\text{18}\\cdot \\text{9}\\div \\text{2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q604459\">Show Solution<\/span><\/p>\n<div id=\"q604459\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754213884\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen divided by nine times two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, divide first. Divide eighteen by nine to get two. The expression is now two times two. The last operation is multiplication. Multiply two by two to get four.\" data-label=\"\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]18\\div 9\\cdot 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply and divide from left to right. Divide.<\/td>\n<td>[latex]\\color{red}{2}\\cdot 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1164752720001\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen times nine divided by two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, multiply first. Multiply eighteen by nine to get one hundred sixty-two. The expression is now one hundred sixty-two divided by two. The last operation is division. divide one hundred sixty-two by two to get eighty-one.\" data-label=\"\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]18\\cdot 9\\div 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply and divide from left to right.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\color{red}{162}\\div 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144756\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144756&theme=oea&iframe_resize_id=ohm144756&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]18\\div 6+4\\left(5 - 2\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q841846\">Show Solution<\/span><\/p>\n<div id=\"q841846\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754213917\" class=\"unnumbered unstyled\" summary=\"The image shows the expression eighteen divided by six plus four, in parentheses, five minus two. Are there any parentheses? Yes, perform the subtraction inside the parentheses. Five minus two becomes three inside the parentheses. The expression is now eighteen divided by six plus four, in parentheses, three. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Divide first because multiplication and division are performed left to right. Divide eighteen by six to get three. The expression is now three plus four, in parentheses, three. Now multiply four by the three in parentheses to get twelve. The expression becomes three plus twelve. Add three and twelve to get fifteen.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]18\\div 6+4(5-2)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Parentheses? Yes, subtract first.<\/td>\n<td>[latex]18\\div 6+4(\\color{red}{3})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Exponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Divide first because we multiply and divide left to right.<\/td>\n<td>[latex]\\color{red}{3}+4(3)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Any other multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]3+\\color{red}{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Any other multiplication or division? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Any addition or subtraction? Yes.<\/td>\n<td>[latex]15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144758\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144758&theme=oea&iframe_resize_id=ohm144758&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video below we show another example of how to use the order of operations to simplify a mathematical expression.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" title=\"Ex: Evaluate an Expression Using the Order of Operations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/qFUvF5-w9o0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>[latex]\\text{Simplify: }5+{2}^{3}+3\\left[6 - 3\\left(4 - 2\\right)\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q221697\">Show Solution<\/span><\/p>\n<div id=\"q221697\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754129434\" class=\"unnumbered unstyled\" summary=\"The image shows the expression five plus two cubed plus three, open bracket, six minus three, open parentheses, four minus two close parentheses, close bracket. Are there any parentheses (or other grouping symbols)? Yes, there are parentheses within brackets. Start with the innermost grouping, the parentheses. Inside the parentheses subtract two from four to get two. The expression becomes five plus two cubed plus three, open bracket, six minus three, in parentheses, two, close bracket. Within the brackets is six minus three, in parentheses, two. Multiply three by two since multiplication comes before subtraction. The expression becomes five plus two cubed plus three, in brackets, six minus six. Finish simplifying inside the brackets by performing the subtraction. Six minus six leaves zero. The expression is now five plus two cubed plus three, in brackets, zero. Are there exponents? Yes, two cubed is eight and the expression becomes five plus eight plus three, in brackets, zero. Is there any multiplication or division? Yes, multiply three by zero to get zero. The expression is now five plus eight plus zero. The remaining operations are both addition. Perform the addition from left to right. five plus eight is thirteen. Thirteen plus zero is thirteen.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]5+{2}^{3}+ 3[6-3(4-2)][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any parentheses (or other grouping symbol)? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Focus on the parentheses that are inside the brackets.<\/td>\n<td>[latex]5+{2}^{3}+ 3[6-3\\color{red}{(4-2)}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{3(2)}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Continue inside the brackets and multiply.<\/td>\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{6}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Continue inside the brackets and subtract.<\/td>\n<td>[latex]5+{2}^{3}+3[\\color{red}{0}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The expression inside the brackets requires no further simplification.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any exponents? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify exponents.<\/td>\n<td>[latex]5+\\color{red}{{2}^{3}}+3[0][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Is there any multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]5+8+\\color{red}{3[0]}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Is there any addition or subtraction? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\color{red}{5+8}+0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\color{red}{13+0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]13[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144759\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144759&theme=oea&iframe_resize_id=ohm144759&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video below we show another example of how to use the order of operations to simplify an expression that contains exponents and grouping symbols.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-9\" title=\"Example 3:  Evaluate An Expression Using The Order of Operation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8b-rf2AW3Ac?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q199030\">Show Solution<\/span><\/p>\n<div id=\"q199030\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754324162\" class=\"unnumbered unstyled\" summary=\"The image shows the expression two cubed plus three to the fourth divided by three minus five squared. Are there any parentheses? No. Are there any exponents? Yes, several. Simplify each exponent. Two cubed is eight, three to the fourth is eighty-one, and five squared is twenty-five. The expression becomes eight plus eighty-one divided by three minus twenty-five. Is there any multiplication or division? Yes, just division. Divide eighty-one by three to get twenty-seven. The expression is now eight plus twenty-seven minus five. There is both addition and subtraction left. Perform these from left to right. Eight plus twenty-seven is thirty-five. Now the expression is thirty-five minus twenty five which leaves ten.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>If an expression has several exponents, they may be simplified in the same step.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify exponents.<\/td>\n<td>[latex]\\color{red}{{2}^{3}}+\\color{red}{{3}^{4}}\\div 3-\\color{red}{{5}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]8+\\color{red}{82\\div 3}-25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\color{red}{8+27}-25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]\\color{red}{35-25}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144762\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144762&theme=oea&iframe_resize_id=ohm144762&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>[latex]\\text{Simplify: }7\\left(-2\\right)+4\\left(-7\\right)-6[\/latex].<\/p>\n<p>Solution:<br \/>\nWe use the order of operations. Multiply first and then add and subtract from left to right.<\/p>\n<table id=\"eip-id1168468659486\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]7\\left(-2\\right)+4\\left(-7\\right)-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply first.<\/td>\n<td>[latex]-14+\\left(-28\\right)-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]-42 - 6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]-48[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145332&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145332\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145332&theme=oea&iframe_resize_id=ohm145332&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see another example of how to use the order of operations to simplify an expression that contains integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-10\" title=\"Ex:  Simplify an Expression With Integers Using the Order of Operations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/RJ7uU9HbdqA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In our next example we will simplify expressions with integers that also contain exponents.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1.<br \/>\n[latex]{\\left(-2\\right)}^{4}[\/latex]<br \/>\n2.<br \/>\n[latex]{-2}^{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q699809\">Show Solution<\/span><\/p>\n<div id=\"q699809\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nThe exponent tells how many times to multiply the base.<br \/>\n1. The exponent is [latex]4[\/latex] and the base is [latex]-2[\/latex]. We raise [latex]-2[\/latex] to the fourth power.<\/p>\n<table id=\"eip-id1168468584362\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(-2\\right)}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write in expanded form.<\/td>\n<td>[latex]\\left(-2\\right)\\left(-2\\right)\\left(-2\\right)\\left(-2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]4\\left(-2\\right)\\left(-2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-8\\left(-2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]16[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>2. The exponent is [latex]4[\/latex] and the base is [latex]2[\/latex]. We raise [latex]2[\/latex] to the fourth power and then take the opposite.<\/p>\n<table id=\"eip-id1168466711166\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-{2}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write in expanded form.<\/td>\n<td>[latex]-\\left(2\\cdot 2\\cdot 2\\cdot 2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-\\left(4\\cdot 2\\cdot 2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-\\left(8\\cdot 2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-16[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you try it.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145337&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145337\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145337&theme=oea&iframe_resize_id=ohm145337&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following example we provide an explanation for the difference between [latex](-3)^2[\/latex] and [latex]-3^2[\/latex].<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-11\" title=\"Ex: Evaluating Negative Numbers Raised to Powers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/qB5PZzmjenI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>[latex]\\text{Simplify: }12 - 3\\left(9 - 12\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q843317\">Show Solution<\/span><\/p>\n<div id=\"q843317\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nAccording to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.<\/p>\n<table id=\"eip-id1168466156737\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]12 - 3\\left(9 - 12\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract the parentheses first.<\/td>\n<td>[latex]12 - 3\\left(-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]12-\\left(-9\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]\\text{21}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145340&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145340\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145340&theme=oea&iframe_resize_id=ohm145340&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]8\\left(-9\\right)\\div {\\left(-2\\right)}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q388448\">Show Solution<\/span><\/p>\n<div id=\"q388448\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe simplify the exponent first, then multiply and divide.<\/p>\n<table id=\"eip-id1168466276311\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]8\\left(-9\\right)\\div {\\left(-2\\right)}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the exponent.<\/td>\n<td>[latex]8\\left(-9\\right)\\div \\left(-8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-72\\div \\left(-8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]\\text{9}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145345&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145345\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145345&theme=oea&iframe_resize_id=ohm145345&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>[latex]\\text{Simplify:}-30\\div 2+\\left(-3\\right)\\left(-7\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q523980\">Show Solution<\/span><\/p>\n<div id=\"q523980\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nFirst we will multiply and divide from left to right. Then we will add.<\/p>\n<table id=\"eip-id1168469759923\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-30\\div 2+\\left(-3\\right)\\left(-7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]-15+\\left(-3\\right)\\left(-7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-15+21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\text{6}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=145348&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145348\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145348&theme=oea&iframe_resize_id=ohm145348&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to evaluate expressions with integers using the order of operations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-12\" title=\"Ex 1:  Order of Operations with Integers\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/UlfXxzuJIfs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n","protected":false},"author":264444,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18145","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18145","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":30,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18145\/revisions"}],"predecessor-version":[{"id":18725,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18145\/revisions\/18725"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18145\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18145"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18145"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18145"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}