{"id":9396,"date":"2017-05-02T22:12:05","date_gmt":"2017-05-02T22:12:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9396"},"modified":"2017-09-23T19:28:59","modified_gmt":"2017-09-23T19:28:59","slug":"simplifying-and-evaluating-expressions-with-integers-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/simplifying-and-evaluating-expressions-with-integers-2\/","title":{"raw":"Simplifying and Evaluating Expressions With Integers","rendered":"Simplifying and Evaluating Expressions With Integers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify integer expressions involving subtraction<\/li>\r\n \t<li>Substitute and simplify integer expressions involving subtraction<\/li>\r\n<\/ul>\r\n<\/div>\r\nNow that you have seen subtraction modeled with color counters,\u00a0we can move on to performing subtraction of integers without the models.\r\n<ul id=\"fs-id2376804\">\r\n \t<li>Subtract [latex]-23 - 7[\/latex]. Think: We start with [latex]23[\/latex] negative counters. We have to subtract [latex]7[\/latex] positives, but there are no positives to take away. So we add [latex]7[\/latex] neutral pairs to get the [latex]7[\/latex] positives. Now we take away the [latex]7[\/latex] positives. So what\u2019s left? We have the original [latex]23[\/latex] negatives plus [latex]7[\/latex] more negatives from the neutral pair. The result is [latex]30[\/latex] negatives. [latex]-23 - 7=-30[\/latex] Notice, that to subtract [latex]\\text{7,}[\/latex] we added [latex]7[\/latex] negatives.<\/li>\r\n \t<li>Subtract [latex]30-\\left(-12\\right)[\/latex]. Think: We start with [latex]30[\/latex] positives. We have to subtract [latex]12[\/latex] negatives, but there are no negatives to take away. So we add [latex]12[\/latex] neutral pairs to the [latex]30[\/latex] positives. Now we take away the [latex]12[\/latex] negatives. What\u2019s left? We have the original [latex]30[\/latex] positives plus [latex]12[\/latex] more positives from the neutral pairs. The result is [latex]42[\/latex] positives. [latex]30-\\left(-12\\right)=42[\/latex] Notice that to subtract [latex]-12[\/latex], we added [latex]12[\/latex].<\/li>\r\n<\/ul>\r\nWhile we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.\r\n\r\nHave you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows:\r\n<div class=\"textbox shaded\">\r\n<h3>Subtraction Property<\/h3>\r\nSubtracting a number is the same as adding it's opposite.\r\n<p style=\"text-align: center;\">[latex]a-b=a+(-b)[\/latex]<\/p>\r\n\r\n<\/div>\r\nLook at these two examples.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220432\/CNX_BMath_Figure_03_03_033_img.png\" alt=\"This figure has two columns. The first column has 6 minus 4. Underneath, there is a row of 6 blue circles, with the first 4 separated from the last 2. The first 4 are circled. Under this row there is 2. The second column has 6 plus negative 4. Underneath there is a row of 6 blue circles with the first 4 separated from the last 2. The first 4 are circled. Under the first four is a row of 4 red circles. Under this there is 2.\" \/>\r\nWe see that [latex]6 - 4[\/latex] gives the same answer as [latex]6+\\left(-4\\right)[\/latex].<\/p>\r\nOf course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract [latex]6 - 4[\/latex] long ago. But knowing that [latex]6 - 4[\/latex] gives the same answer as [latex]6+\\left(-4\\right)[\/latex] helps when we are subtracting negative numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]13 - 8\\text{ and }13+\\left(-8\\right)[\/latex]<\/li>\r\n \t<li>[latex]-17 - 9\\text{ and }-17+\\left(-9\\right)[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1168469839872\" 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><\/td>\r\n<td>[latex]13 - 8[\/latex] and [latex]13+\\left(-8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract to simplify.<\/td>\r\n<td>[latex]13 - 8=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add to simplify.<\/td>\r\n<td>[latex]13+\\left(-8\\right)=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtracting [latex]8[\/latex] from [latex]13[\/latex] is the same as adding [latex]\u22128[\/latex] to [latex]13[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1164272171485\" 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><\/td>\r\n<td>[latex]-17 - 9[\/latex] and [latex]-17+\\left(-9\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract to simplify.<\/td>\r\n<td>[latex]-17 - 9=-26[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add to simplify.<\/td>\r\n<td>[latex]-17+\\left(-9\\right)=-26[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtracting [latex]9[\/latex] from [latex]\u221217[\/latex] is the same as adding [latex]\u22129[\/latex] to [latex]\u221217[\/latex].<\/td>\r\n<td><\/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\n[ohm_question]145193[\/ohm_question]\r\n\r\n<\/div>\r\nNow look what happens when we subtract 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\/24220434\/CNX_BMath_Figure_03_03_024_img.png\" alt=\"This figure has two columns. The first column has 8 minus negative 5. Underneath, there is a row of 13 blue circles. The first 8 are separated from the next 5. Under the last 5 blue circles there is a row of 5 red circles. They are circled. Under this there is 13. The second column has 8 plus 5. Underneath there is a row of 13 blue circles. The first 8 are separated from the last 5. Under this there is 13.\" width=\"789\" height=\"137\" \/>\r\nWe see that [latex]8-\\left(-5\\right)[\/latex] gives the same result as [latex]8+5[\/latex]. Subtracting a negative number is like adding a positive. In the next example, we will see more examples of this concept.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]9-\\left(-15\\right)\\text{ and }9+15[\/latex]<\/li>\r\n \t<li>[latex]-7-\\left(-4\\right)\\text{ and }-7+4[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"815868\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"815868\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468712402\" 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><\/td>\r\n<td>[latex]9-\\left(-15\\right)[\/latex] and [latex]9+15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract to simplify.<\/td>\r\n<td>[latex]9-\\left(-15\\right)=-24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add to simplify.<\/td>\r\n<td>[latex]9+15=24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtracting [latex]\u221215[\/latex] from [latex]9[\/latex] is the same as adding [latex]15[\/latex] to [latex]9[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1164270694692\" 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><\/td>\r\n<td>[latex]-7-\\left(-4\\right)[\/latex] and [latex]-7+4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract to simplify.<\/td>\r\n<td>[latex]-7-\\left(-4\\right)=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add to simplify.<\/td>\r\n<td>[latex]-7+4=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtracting [latex]\u22124[\/latex] from [latex]\u22127[\/latex] is the same as adding [latex]4[\/latex] to [latex]\u22127[\/latex].<\/td>\r\n<td><\/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\n[ohm_question]145195[\/ohm_question]\r\n\r\n<\/div>\r\nThe table below summarizes the four different scenarios we encountered in the previous examples, and how you would use counters to simplify.\r\n<table id=\"eip-703\" style=\"width: 85%;\" summary=\".\"><caption>Subtraction of Integers<\/caption>\r\n<tbody>\r\n<tr>\r\n<td>[latex]5 - 3[\/latex]<\/td>\r\n<td>[latex]-5-\\left(-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex] positives<\/td>\r\n<td>[latex]2[\/latex] negatives<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">When there would be enough counters of the color to take away, subtract.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-5 - 3[\/latex]<\/td>\r\n<td>[latex]5-\\left(-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-8[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex] negatives, want to subtract [latex]3[\/latex] positives<\/td>\r\n<td>[latex]5[\/latex] positives, want to subtract [latex]3[\/latex] negatives<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>need neutral pairs<\/td>\r\n<td>need neutral pairs<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">When there would not be enough of the counters to take away, add neutral pairs.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn our next example we show how to subtract a negative with two digit numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]-74-\\left(-58\\right)[\/latex].\r\n[reveal-answer q=\"397191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"397191\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468287610\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>We are taking [latex]58[\/latex] negatives away from [latex]74[\/latex] negatives.<\/td>\r\n<td>[latex]-74-\\left(-58\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/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 can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145197[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show another example of subtracting two digit integers.\r\n\r\nhttps:\/\/youtu.be\/IfiN-mJZu2E\r\n\r\nNow let's increase the complexity of the examples a little bit. We will use the order of operations to simplify terms in parentheses before we subtract from left to right.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]7-\\left(-4 - 3\\right)-9[\/latex]\r\n[reveal-answer q=\"769817\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"769817\"]\r\n\r\nSolution:\r\nWe use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right.\r\n<table id=\"eip-id1168469609684\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Simplify inside the parentheses first.<\/td>\r\n<td>[latex]7-\\left(-4 - 3\\right)-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract from left to right.<\/td>\r\n<td>[latex]7-\\left(-7\\right)-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]14--9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]5[\/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[ohm_question]145199[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see more examples of simplifying integer expressions that involve subtraction.\r\n\r\nhttps:\/\/youtu.be\/mDkSpz0BPPc\r\n\r\nNow we will add another operation to an expression. Because multiplication and division come before addition and subtraction, we will multiply, then subtract.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]3\\cdot 7 - 4\\cdot 7 - 5\\cdot 8[\/latex].\r\n[reveal-answer q=\"805962\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"805962\"]\r\n\r\nSolution:\r\nWe use the order of operations to simplify this expression. First we multiply, and then subtract from left to right.\r\n<table id=\"eip-id1168468652720\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Multiply first.<\/td>\r\n<td>[latex]3\\cdot 7 - 4\\cdot 7 - 5\\cdot 8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract from left to right.<\/td>\r\n<td>[latex]21--28--40[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]--7--40[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]--47[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you try.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145200[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see another example of simplifying an integer expression involving multiplication and subtraction.\r\n\r\nhttps:\/\/youtu.be\/42Su4r5UmoE\r\n<h3>Evaluate Variable Expressions with Integers<\/h3>\r\nNow we\u2019ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEvaluate [latex]x - 4\\text{ when}[\/latex]\r\n<ol>\r\n \t<li>[latex]x=3[\/latex]<\/li>\r\n \t<li>[latex]x=-6[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"955990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"955990\"]\r\n\r\nSolution:\r\n1. To evaluate [latex]x - 4[\/latex] when [latex]x=3[\/latex] , substitute [latex]3[\/latex] for [latex]x[\/latex] in the expression.\r\n<table id=\"eip-id1168465998847\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"This figure has 3 rows. The first row has x minus 4. The second row states substitute 3 for x and has 3 minus 4. The third row states subtract and has negative 1.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x--4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{Substitute }\\color{red}{3}\\text{ for }x[\/latex]<\/td>\r\n<td>[latex]\\color{red}{3}--4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]--1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n2. To evaluate [latex]x - 4[\/latex] when [latex]x=-6[\/latex], substitute [latex]-6[\/latex] for [latex]x[\/latex] in the expression.\r\n<table id=\"eip-id1168466833806\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"This figure has 3 rows. The first row has x minus 4. The second row states substitute negative 6 for x and has negative 6 minus 4. The third row states subtract and has negative 10.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x--4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{Substitute }\\color{red}{--6}\\text{ for }x[\/latex]<\/td>\r\n<td>[latex]\\color{red}{--6}--4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/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\nNow you try.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145203[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next example, we will subtract a positive and a negative.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEvaluate [latex]20-z\\text{ when}[\/latex]\r\n<ol>\r\n \t<li>[latex]z=12[\/latex]<\/li>\r\n \t<li>[latex]z=-12[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"34289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"34289\"]\r\n\r\nSolution:\r\n1. To evaluate [latex]20-z\\text{ when }z=12[\/latex], substitute [latex]12[\/latex] for [latex]z[\/latex] in the expression.\r\n<table id=\"eip-id1168469489544\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"This figure has 3 rows. The first row has 20 minus z. The second row states substitute 12 for z and has 20 minus 12. The third row states subtract and has 8.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]20--z[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{Substitute }\\color{red}{12}\\text{ for }z[\/latex]<\/td>\r\n<td>[latex]20--\\color{red}{12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n2. To evaluate [latex]20-z\\text{ when }z=-12,\\text{ substitute }-12\\text{ for }z\\text{ in the expression.}[\/latex]\r\n<table id=\"eip-id1168469804288\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"This figure has 3 rows. The first row has 20 minus z. The second row states substitute negative 12 for z and has 20 minus negative 12. The third row states subtract and has 32.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]20--z[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{Substitute }\\color{red}{--12}\\text{ for }z[\/latex]<\/td>\r\n<td>[latex]20--(\\color{red}{--12})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]32[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you try.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145205[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify integer expressions involving subtraction<\/li>\n<li>Substitute and simplify integer expressions involving subtraction<\/li>\n<\/ul>\n<\/div>\n<p>Now that you have seen subtraction modeled with color counters,\u00a0we can move on to performing subtraction of integers without the models.<\/p>\n<ul id=\"fs-id2376804\">\n<li>Subtract [latex]-23 - 7[\/latex]. Think: We start with [latex]23[\/latex] negative counters. We have to subtract [latex]7[\/latex] positives, but there are no positives to take away. So we add [latex]7[\/latex] neutral pairs to get the [latex]7[\/latex] positives. Now we take away the [latex]7[\/latex] positives. So what\u2019s left? We have the original [latex]23[\/latex] negatives plus [latex]7[\/latex] more negatives from the neutral pair. The result is [latex]30[\/latex] negatives. [latex]-23 - 7=-30[\/latex] Notice, that to subtract [latex]\\text{7,}[\/latex] we added [latex]7[\/latex] negatives.<\/li>\n<li>Subtract [latex]30-\\left(-12\\right)[\/latex]. Think: We start with [latex]30[\/latex] positives. We have to subtract [latex]12[\/latex] negatives, but there are no negatives to take away. So we add [latex]12[\/latex] neutral pairs to the [latex]30[\/latex] positives. Now we take away the [latex]12[\/latex] negatives. What\u2019s left? We have the original [latex]30[\/latex] positives plus [latex]12[\/latex] more positives from the neutral pairs. The result is [latex]42[\/latex] positives. [latex]30-\\left(-12\\right)=42[\/latex] Notice that to subtract [latex]-12[\/latex], we added [latex]12[\/latex].<\/li>\n<\/ul>\n<p>While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.<\/p>\n<p>Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows:<\/p>\n<div class=\"textbox shaded\">\n<h3>Subtraction Property<\/h3>\n<p>Subtracting a number is the same as adding it&#8217;s opposite.<\/p>\n<p style=\"text-align: center;\">[latex]a-b=a+(-b)[\/latex]<\/p>\n<\/div>\n<p>Look at these two examples.<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220432\/CNX_BMath_Figure_03_03_033_img.png\" alt=\"This figure has two columns. The first column has 6 minus 4. Underneath, there is a row of 6 blue circles, with the first 4 separated from the last 2. The first 4 are circled. Under this row there is 2. The second column has 6 plus negative 4. Underneath there is a row of 6 blue circles with the first 4 separated from the last 2. The first 4 are circled. Under the first four is a row of 4 red circles. Under this there is 2.\" \/><br \/>\nWe see that [latex]6 - 4[\/latex] gives the same answer as [latex]6+\\left(-4\\right)[\/latex].<\/p>\n<p>Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract [latex]6 - 4[\/latex] long ago. But knowing that [latex]6 - 4[\/latex] gives the same answer as [latex]6+\\left(-4\\right)[\/latex] helps when we are subtracting negative numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]13 - 8\\text{ and }13+\\left(-8\\right)[\/latex]<\/li>\n<li>[latex]-17 - 9\\text{ and }-17+\\left(-9\\right)[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1168469839872\" class=\"unnumbered 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]13 - 8[\/latex] and [latex]13+\\left(-8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract to simplify.<\/td>\n<td>[latex]13 - 8=5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add to simplify.<\/td>\n<td>[latex]13+\\left(-8\\right)=5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtracting [latex]8[\/latex] from [latex]13[\/latex] is the same as adding [latex]\u22128[\/latex] to [latex]13[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1164272171485\" class=\"unnumbered 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]-17 - 9[\/latex] and [latex]-17+\\left(-9\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract to simplify.<\/td>\n<td>[latex]-17 - 9=-26[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add to simplify.<\/td>\n<td>[latex]-17+\\left(-9\\right)=-26[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtracting [latex]9[\/latex] from [latex]\u221217[\/latex] is the same as adding [latex]\u22129[\/latex] to [latex]\u221217[\/latex].<\/td>\n<td><\/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><iframe loading=\"lazy\" id=\"ohm145193\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145193&theme=oea&iframe_resize_id=ohm145193&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Now look what happens when we subtract a negative.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220434\/CNX_BMath_Figure_03_03_024_img.png\" alt=\"This figure has two columns. The first column has 8 minus negative 5. Underneath, there is a row of 13 blue circles. The first 8 are separated from the next 5. Under the last 5 blue circles there is a row of 5 red circles. They are circled. Under this there is 13. The second column has 8 plus 5. Underneath there is a row of 13 blue circles. The first 8 are separated from the last 5. Under this there is 13.\" width=\"789\" height=\"137\" \/><br \/>\nWe see that [latex]8-\\left(-5\\right)[\/latex] gives the same result as [latex]8+5[\/latex]. Subtracting a negative number is like adding a positive. In the next example, we will see more examples of this concept.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]9-\\left(-15\\right)\\text{ and }9+15[\/latex]<\/li>\n<li>[latex]-7-\\left(-4\\right)\\text{ and }-7+4[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q815868\">Show Solution<\/span><\/p>\n<div id=\"q815868\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468712402\" class=\"unnumbered 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-\\left(-15\\right)[\/latex] and [latex]9+15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract to simplify.<\/td>\n<td>[latex]9-\\left(-15\\right)=-24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add to simplify.<\/td>\n<td>[latex]9+15=24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtracting [latex]\u221215[\/latex] from [latex]9[\/latex] is the same as adding [latex]15[\/latex] to [latex]9[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1164270694692\" class=\"unnumbered 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]-7-\\left(-4\\right)[\/latex] and [latex]-7+4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract to simplify.<\/td>\n<td>[latex]-7-\\left(-4\\right)=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add to simplify.<\/td>\n<td>[latex]-7+4=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtracting [latex]\u22124[\/latex] from [latex]\u22127[\/latex] is the same as adding [latex]4[\/latex] to [latex]\u22127[\/latex].<\/td>\n<td><\/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><iframe loading=\"lazy\" id=\"ohm145195\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145195&theme=oea&iframe_resize_id=ohm145195&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The table below summarizes the four different scenarios we encountered in the previous examples, and how you would use counters to simplify.<\/p>\n<table id=\"eip-703\" style=\"width: 85%;\" summary=\".\">\n<caption>Subtraction of Integers<\/caption>\n<tbody>\n<tr>\n<td>[latex]5 - 3[\/latex]<\/td>\n<td>[latex]-5-\\left(-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex] positives<\/td>\n<td>[latex]2[\/latex] negatives<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">When there would be enough counters of the color to take away, subtract.<\/td>\n<\/tr>\n<tr>\n<td>[latex]-5 - 3[\/latex]<\/td>\n<td>[latex]5-\\left(-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-8[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex] negatives, want to subtract [latex]3[\/latex] positives<\/td>\n<td>[latex]5[\/latex] positives, want to subtract [latex]3[\/latex] negatives<\/td>\n<\/tr>\n<tr>\n<td>need neutral pairs<\/td>\n<td>need neutral pairs<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">When there would not be enough of the counters to take away, add neutral pairs.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In our next example we show how to subtract a negative with two digit numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]-74-\\left(-58\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q397191\">Show Solution<\/span><\/p>\n<div id=\"q397191\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468287610\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>We are taking [latex]58[\/latex] negatives away from [latex]74[\/latex] negatives.<\/td>\n<td>[latex]-74-\\left(-58\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]-16[\/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><iframe loading=\"lazy\" id=\"ohm145197\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145197&theme=oea&iframe_resize_id=ohm145197&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of subtracting two digit integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Subtract Two Digit Integers (Pos-Neg) Formal Rules and Number Line (Pos Sum)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/IfiN-mJZu2E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Now let&#8217;s increase the complexity of the examples a little bit. We will use the order of operations to simplify terms in parentheses before we subtract from left to right.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]7-\\left(-4 - 3\\right)-9[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q769817\">Show Solution<\/span><\/p>\n<div id=\"q769817\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right.<\/p>\n<table id=\"eip-id1168469609684\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>Simplify inside the parentheses first.<\/td>\n<td>[latex]7-\\left(-4 - 3\\right)-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract from left to right.<\/td>\n<td>[latex]7-\\left(-7\\right)-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]14--9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]5[\/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=\"ohm145199\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145199&theme=oea&iframe_resize_id=ohm145199&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see more examples of simplifying integer expressions that involve subtraction.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1:  Evaluate Expressions Involving Integer Subtraction\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/mDkSpz0BPPc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Now we will add another operation to an expression. Because multiplication and division come before addition and subtraction, we will multiply, then subtract.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]3\\cdot 7 - 4\\cdot 7 - 5\\cdot 8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q805962\">Show Solution<\/span><\/p>\n<div id=\"q805962\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe use the order of operations to simplify this expression. First we multiply, and then subtract from left to right.<\/p>\n<table id=\"eip-id1168468652720\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>Multiply first.<\/td>\n<td>[latex]3\\cdot 7 - 4\\cdot 7 - 5\\cdot 8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract from left to right.<\/td>\n<td>[latex]21--28--40[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]--7--40[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]--47[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you try.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145200\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145200&theme=oea&iframe_resize_id=ohm145200&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see another example of simplifying an integer expression involving multiplication and subtraction.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Evaluate an Expression Involving Integer Operations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/42Su4r5UmoE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Evaluate Variable Expressions with Integers<\/h3>\n<p>Now we\u2019ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Evaluate [latex]x - 4\\text{ when}[\/latex]<\/p>\n<ol>\n<li>[latex]x=3[\/latex]<\/li>\n<li>[latex]x=-6[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q955990\">Show Solution<\/span><\/p>\n<div id=\"q955990\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\n1. To evaluate [latex]x - 4[\/latex] when [latex]x=3[\/latex] , substitute [latex]3[\/latex] for [latex]x[\/latex] in the expression.<\/p>\n<table id=\"eip-id1168465998847\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"This figure has 3 rows. The first row has x minus 4. The second row states substitute 3 for x and has 3 minus 4. The third row states subtract and has negative 1.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x--4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{Substitute }\\color{red}{3}\\text{ for }x[\/latex]<\/td>\n<td>[latex]\\color{red}{3}--4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]--1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>2. To evaluate [latex]x - 4[\/latex] when [latex]x=-6[\/latex], substitute [latex]-6[\/latex] for [latex]x[\/latex] in the expression.<\/p>\n<table id=\"eip-id1168466833806\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"This figure has 3 rows. The first row has x minus 4. The second row states substitute negative 6 for x and has negative 6 minus 4. The third row states subtract and has negative 10.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x--4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{Substitute }\\color{red}{--6}\\text{ for }x[\/latex]<\/td>\n<td>[latex]\\color{red}{--6}--4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]--10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you try.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145203\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145203&theme=oea&iframe_resize_id=ohm145203&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next example, we will subtract a positive and a negative.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Evaluate [latex]20-z\\text{ when}[\/latex]<\/p>\n<ol>\n<li>[latex]z=12[\/latex]<\/li>\n<li>[latex]z=-12[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q34289\">Show Solution<\/span><\/p>\n<div id=\"q34289\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\n1. To evaluate [latex]20-z\\text{ when }z=12[\/latex], substitute [latex]12[\/latex] for [latex]z[\/latex] in the expression.<\/p>\n<table id=\"eip-id1168469489544\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"This figure has 3 rows. The first row has 20 minus z. The second row states substitute 12 for z and has 20 minus 12. The third row states subtract and has 8.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]20--z[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{Substitute }\\color{red}{12}\\text{ for }z[\/latex]<\/td>\n<td>[latex]20--\\color{red}{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>2. To evaluate [latex]20-z\\text{ when }z=-12,\\text{ substitute }-12\\text{ for }z\\text{ in the expression.}[\/latex]<\/p>\n<table id=\"eip-id1168469804288\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"This figure has 3 rows. The first row has 20 minus z. The second row states substitute negative 12 for z and has 20 minus negative 12. The third row states subtract and has 32.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]20--z[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{Substitute }\\color{red}{--12}\\text{ for }z[\/latex]<\/td>\n<td>[latex]20--(\\color{red}{--12})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]32[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you try.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145205\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145205&theme=oea&iframe_resize_id=ohm145205&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-9396\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Subtract Two Digit Integers (Pos-Neg) Formal Rules and Number Line (Pos Sum). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/IfiN-mJZu2E\">https:\/\/youtu.be\/IfiN-mJZu2E<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 1: Evaluate Expressions Involving Integer Subtraction. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/mDkSpz0BPPc\">https:\/\/youtu.be\/mDkSpz0BPPc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Evaluate an Expression Involving Integer Operations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/42Su4r5UmoE\">https:\/\/youtu.be\/42Su4r5UmoE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID: 145193, 145195, 145197, 145199, 145200, 145203, 145205. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at 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