{"id":9416,"date":"2017-05-02T22:21:27","date_gmt":"2017-05-02T22:21:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9416"},"modified":"2025-10-31T13:36:13","modified_gmt":"2025-10-31T13:36:13","slug":"simplifying-expressions-using-hte-distributive-property","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/simplifying-expressions-using-hte-distributive-property\/","title":{"raw":"Simplifying Expressions With Different Forms of the Distributive Property","rendered":"Simplifying Expressions With Different Forms of the Distributive Property"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Apply the distributive property to simplify an algebraic expression involving whole numbers, integers, fractions and decimals<\/li>\r\n \t<li>Apply the distributive property in different forms<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Simplify Expressions Using the Distributive Property<\/h2>\r\nSuppose three friends are going to the movies. They each need [latex]$9.25[\/latex]; that is, [latex]9[\/latex] dollars and [latex]1[\/latex] quarter. How much money do they need all together? You can think about the dollars separately from the quarters.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222341\/CNX_BMath_Figure_07_03_001_img.png\" alt=\"The image shows the equation 3 times 9 equal to 27. Below the 3 is an image of three people. Below the 9 is an image of 9 one dollar bills. Below the 27 is an image of three groups of 9 one dollar bills for a total of 27 one dollar bills.\" \/>\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222342\/CNX_BMath_Figure_07_03_002_img.png\" alt=\"The image shows the equation 3 times 25 cents equal to 75 cents. Below the 3 is an image of three people. Below the 25 cents is an image of a quarter. Below the 75 cents is an image of three quarters.\" \/>\r\nThey need [latex]3[\/latex] times [latex]$9[\/latex], so [latex]$27[\/latex], and [latex]3[\/latex] times [latex]1[\/latex] quarter, so [latex]75[\/latex] cents. In total, they need [latex]$27.75[\/latex].\r\n\r\nIf you think about doing the math in this way, you are using the Distributive Property.\r\n<div class=\"textbox shaded\">\r\n<h3>Distributive Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are real numbers, then\r\n\r\n[latex]a\\left(b+c\\right)=ab+ac[\/latex]\r\n\r\n<\/div>\r\n<p style=\"text-align: left;\">Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:<\/p>\r\n<p style=\"text-align: center;\">[latex]3(9.25)\\\\3(9\\quad+\\quad0.25)\\\\3(9)\\quad+\\quad3(0.25)\\\\27\\quad+\\quad0.75\\\\27.75[\/latex]<\/p>\r\nIn algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression [latex]3\\left(x+4\\right)[\/latex], the order of operations says to work in the parentheses first. But we cannot add [latex]x[\/latex] and [latex]4[\/latex], since they are not like terms. So we use the Distributive Property, as shown in the next example.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]3\\left(x+4\\right)[\/latex]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468605528\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]3\\left(x+4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]3\\cdot x+3\\cdot 4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]3x+12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nSome students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in the previous example would look like this:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222343\/CNX_BMath_Figure_07_03_003_img.png\" alt=\"The image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4.\" \/>\r\n<p style=\"text-align: center;\">[latex]3\\cdot x+3\\cdot 4[\/latex]<\/p>\r\nNow you try.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146473[\/ohm_question]\r\n\r\n<\/div>\r\nIn our next example, there is a coefficient on the variable y. When you use the distributive property, you multiply the two numbers together, just like simplifying any product. You will also see another example where the expression in parentheses is subtraction, rather than addition. \u00a0You will need to be careful to change the sign of your product.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]6\\left(5y+1\\right)[\/latex]\r\n\r\n[reveal-answer q=\"645849\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"645849\"]\r\n\r\n&nbsp;\r\n\r\nSolution:\r\n<table id=\"eip-id1168466125869\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 5 y plus 1 in parentheses with the number 6 outside the parentheses on the left. Simplify by distributing the 6 through the parentheses to get the expression 6 times 5 y plus 6 times 1. Simplify further by multiplying 6 times 5 y to get 30 y and 6 times 1 to get 6. The expression simplifies to 30 y plus 6.\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222344\/CNX_BMath_Figure_07_03_025_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]6\\cdot 5y+6\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]30y+6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\nSimplify: [latex]2\\left(x - 3\\right)[\/latex]\r\n[reveal-answer q=\"877652\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"877652\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466277015\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression x minus 3 in parentheses with the number 2 outside the parentheses on the left. Simplify by distributing the 2 through the parentheses to get the expression 2 times x minus 2 times 3. Simplify further by multiplying 2 times x to get 2 x and 2 times 3 to get 6. The expression simplifies to 2 x minus 6.\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222347\/CNX_BMath_Figure_07_03_026_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]2\\cdot x-2\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]2x-6[\/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]146474[\/ohm_question]\r\n\r\n[ohm_question]146475[\/ohm_question]\r\n\r\n<\/div>\r\nThe distributive property can be used to simplify expressions that look slightly different from [latex]a\\left(b+c\\right)[\/latex]. Here are two other forms.\r\n<div class=\"textbox shaded\">\r\n<h3>different Forms of the Distributive Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are real numbers, then\r\n<p style=\"padding-left: 60px;\">[latex]a\\left(b+c\\right)=ab+ac[\/latex]<\/p>\r\nOther forms\r\n<p style=\"padding-left: 60px;\">[latex]a\\left(b-c\\right)=ab-ac[\/latex]\r\n[latex]\\left(b+c\\right)a=ba+ca[\/latex]<\/p>\r\n\r\n<\/div>\r\nIn the following video we show more examples of using the distributive property.\r\n\r\nhttps:\/\/youtu.be\/Nt8V5cEvAz8\r\n<h2>Using the Distributive Property With Fractions and Decimals<\/h2>\r\nDo you remember how to multiply a fraction by a whole number? We\u2019ll need to do that in the next two examples. The distributive property comes in all shapes and sizes, and can include fractions or decimals as well.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\Large\\frac{3}{4}\\normalsize\\left(n+12\\right)[\/latex]\r\n[reveal-answer q=\"843482\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"843482\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467480771\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression x minus 3 in parentheses with the number 2 outside the parentheses on the left. Simplify by distributing the 2 through the parentheses to get the expression 2 times x minus 2 times 3. Simplify further by multiplying 2 times x to get 2 x and 2 times 3 to get 6. The expression simplifies to 2 x minus 6.\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222349\/CNX_BMath_Figure_07_03_027_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]\\Large\\frac{3}{4}\\normalsize\\cdot n+\\Large\\frac{3}{4}\\normalsize\\cdot 12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{3}{4}\\normalsize n+9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\nSimplify: [latex]8\\Large\\left(\\frac{3}{8}\\normalsize x+\\Large\\frac{1}{4}\\right)[\/latex].\r\n[reveal-answer q=\"689117\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"689117\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468690552\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 3 eights x plus 1 fourth in parentheses with the number 8 outside the parentheses on the left. Simplify by distributing the 8 through the parentheses to get the expression 8 times 3 eights x plus 8 times 1 fourth. Simplify further by multiplying 8 times 3 eights x to get 3 x and 8 times 1 fourth to get 2. The expression simplifies to 3 x plus 2.\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222352\/CNX_BMath_Figure_07_03_028_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]8\\cdot\\Large\\frac{3}{8}\\normalsize x+8\\cdot\\Large\\frac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]3x+2[\/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]146476[\/ohm_question]\r\n\r\n[ohm_question]146479[\/ohm_question]\r\n\r\n<\/div>\r\nUsing the Distributive Property as shown in the next example will be very useful when we solve money applications later.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]100\\left(0.3+0.25q\\right)[\/latex]\r\n[reveal-answer q=\"393263\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"393263\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469752687\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 0.3 plus 0.25 q in parentheses with the number 100 outside the parentheses on the left. Simplify by distributing the 100 through the parentheses to get the expression 100 times 0.3 plus 100 times 0.25 q. Simplify further by multiplying 100 times 0.3 to get 30 and 100 times 0.25 q to get 25 q. The expression simplifies to 30 plus 25 q.\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222355\/CNX_BMath_Figure_07_03_029_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]100(0.3)+100(0.25q)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]30+25q[\/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]146505[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Distributing a Variable<\/h2>\r\nIn the next example we\u2019ll multiply by a variable. We\u2019ll need to do this in a later chapter.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]m\\left(n - 4\\right)[\/latex]\r\n[reveal-answer q=\"288725\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"288725\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468371675\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression n minus 4 in parentheses with the letter m outside the parentheses on the left. Simplify by distributing the m through the parentheses to get the expression m times n minus m times 4. Simplify further by multiplying m times n to get m n and m times 4 to get 4m. The expression simplifies to m n minus 4 m.\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222357\/CNX_BMath_Figure_07_03_030_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]m\\cdot n-m\\cdot 4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]mn-4m[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that we wrote [latex]m\\cdot 4\\text{ as }4m[\/latex]. We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.\r\n\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]146506[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>The Backwards Form of the Distributive Property<\/h2>\r\nThe next example will use the \u2018backwards\u2019 form of the Distributive Property, [latex]\\left(b+c\\right)a=ba+ca[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\left(x+8\\right)p[\/latex]\r\n[reveal-answer q=\"390066\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"390066\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469684869\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression x plus 8 in parentheses with the letter p outside the parentheses on the right Simplify by distributing the p through the parentheses to get the expression p times x plus p times 8. Simplify further by multiplying p times x to get p x and p times 8 to get 8p. The expression simplifies to p x plus 8 p.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222400\/CNX_BMath_Figure_07_03_031_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222400\/CNX_BMath_Figure_07_03_031_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146507[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Distributing a Negative Term<\/h2>\r\nWhen you distribute a negative number, you need to be extra careful to get the signs correct.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]-2\\left(4y+1\\right)[\/latex]\r\n[reveal-answer q=\"31628\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"31628\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468516237\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 4 y plus 1 in parentheses with the number negative 2 outside the parentheses on the left. Simplify by distributing the negative 2 through the parentheses to get the expression negative 2 times 4 y plus negative 2 times 1. Simplify further by multiplying negative 2 times 4 y to get negative 8 y and negative 2 times 1 to get negative 2. The expression becomes negative 8 y plus negative 2. Simplify further by writing plus a negative as subtraction. The expression simplifies to negative 8 y minus 2.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222401\/CNX_BMath_Figure_07_03_032_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]-2\\cdot 4y+(-2)\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-8y-2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\nSimplify: [latex]-11\\left(4 - 3a\\right)[\/latex]\r\n[reveal-answer q=\"149539\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149539\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466284406\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 4 minus 3 a in parentheses with the number negative 11 outside the parentheses on the left. Simplify by distributing the negative 11 through the parentheses to get the expression negative 11 times 4 minus negative 11 times 3 a. Simplify further by multiplying negative 11 times 4 to get negative 44 and negative 11 times 3 a to get negative 33 a. The expression becomes negative 44 plus negative 33 a. Simplify further by writing plus a negative as subtraction. The expression simplifies to negative 44 minus 33 a.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-11(4-3a)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]-11\\cdot 4-(-11)\\cdot 3a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-44+(-33a)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-44+33a[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nYou could also write the result as [latex]33a - 44[\/latex]. Do you know why?\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146512[\/ohm_question]\r\n\r\n[ohm_question]146511[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, [latex]-a=-1\\cdot a[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]-\\left(y+5\\right)[\/latex]\r\n[reveal-answer q=\"833119\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"833119\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466218813\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression y plus 5 in parentheses with a negative sign outside the parentheses on the left. The negative sign outside the parentheses means the opposite and the opposite can be obtained by multiplying by negative 1. Rewrite the expression as y plus 5 in parentheses with a negative 1 outside the parentheses on the left. Simplify by distributing the negative 1 through the parentheses to get the expression negative 1 times y plus negative 1 times 5. Simplify further by multiplying negative 1 times y to get negative y and negative 1 times 5 to get negative 5. The expression becomes negative y plus negative 5. Simplify further by writing plus a negative as subtraction. The expression simplifies to negative y minus 5.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-(y+5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiplying by \u22121 results in the opposite.<\/td>\r\n<td>[latex]-1(y+5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]-1\\cdot y+(-1)\\cdot 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-y+(-5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-y-5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146513[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Apply the distributive property to simplify an algebraic expression involving whole numbers, integers, fractions and decimals<\/li>\n<li>Apply the distributive property in different forms<\/li>\n<\/ul>\n<\/div>\n<h2>Simplify Expressions Using the Distributive Property<\/h2>\n<p>Suppose three friends are going to the movies. They each need [latex]$9.25[\/latex]; that is, [latex]9[\/latex] dollars and [latex]1[\/latex] quarter. How much money do they need all together? You can think about the dollars separately from the quarters.<\/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\/24222341\/CNX_BMath_Figure_07_03_001_img.png\" alt=\"The image shows the equation 3 times 9 equal to 27. Below the 3 is an image of three people. Below the 9 is an image of 9 one dollar bills. Below the 27 is an image of three groups of 9 one dollar bills for a total of 27 one dollar bills.\" \/><\/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\/24222342\/CNX_BMath_Figure_07_03_002_img.png\" alt=\"The image shows the equation 3 times 25 cents equal to 75 cents. Below the 3 is an image of three people. Below the 25 cents is an image of a quarter. Below the 75 cents is an image of three quarters.\" \/><br \/>\nThey need [latex]3[\/latex] times [latex]$9[\/latex], so [latex]$27[\/latex], and [latex]3[\/latex] times [latex]1[\/latex] quarter, so [latex]75[\/latex] cents. In total, they need [latex]$27.75[\/latex].<\/p>\n<p>If you think about doing the math in this way, you are using the Distributive Property.<\/p>\n<div class=\"textbox shaded\">\n<h3>Distributive Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are real numbers, then<\/p>\n<p>[latex]a\\left(b+c\\right)=ab+ac[\/latex]<\/p>\n<\/div>\n<p style=\"text-align: left;\">Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:<\/p>\n<p style=\"text-align: center;\">[latex]3(9.25)\\\\3(9\\quad+\\quad0.25)\\\\3(9)\\quad+\\quad3(0.25)\\\\27\\quad+\\quad0.75\\\\27.75[\/latex]<\/p>\n<p>In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression [latex]3\\left(x+4\\right)[\/latex], the order of operations says to work in the parentheses first. But we cannot add [latex]x[\/latex] and [latex]4[\/latex], since they are not like terms. So we use the Distributive Property, as shown in the next example.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]3\\left(x+4\\right)[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168468605528\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]3\\left(x+4\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]3\\cdot x+3\\cdot 4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]3x+12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in the previous example would look like this:<\/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\/24222343\/CNX_BMath_Figure_07_03_003_img.png\" alt=\"The image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4.\" \/><\/p>\n<p style=\"text-align: center;\">[latex]3\\cdot x+3\\cdot 4[\/latex]<\/p>\n<p>Now you try.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146473\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146473&theme=oea&iframe_resize_id=ohm146473&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In our next example, there is a coefficient on the variable y. When you use the distributive property, you multiply the two numbers together, just like simplifying any product. You will also see another example where the expression in parentheses is subtraction, rather than addition. \u00a0You will need to be careful to change the sign of your product.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]6\\left(5y+1\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q645849\">Show Solution<\/span><\/p>\n<div id=\"q645849\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168466125869\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 5 y plus 1 in parentheses with the number 6 outside the parentheses on the left. Simplify by distributing the 6 through the parentheses to get the expression 6 times 5 y plus 6 times 1. Simplify further by multiplying 6 times 5 y to get 30 y and 6 times 1 to get 6. The expression simplifies to 30 y plus 6.\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222344\/CNX_BMath_Figure_07_03_025_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]6\\cdot 5y+6\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]30y+6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>Simplify: [latex]2\\left(x - 3\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q877652\">Show Solution<\/span><\/p>\n<div id=\"q877652\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466277015\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression x minus 3 in parentheses with the number 2 outside the parentheses on the left. Simplify by distributing the 2 through the parentheses to get the expression 2 times x minus 2 times 3. Simplify further by multiplying 2 times x to get 2 x and 2 times 3 to get 6. The expression simplifies to 2 x minus 6.\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222347\/CNX_BMath_Figure_07_03_026_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]2\\cdot x-2\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]2x-6[\/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=\"ohm146474\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146474&theme=oea&iframe_resize_id=ohm146474&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146475\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146475&theme=oea&iframe_resize_id=ohm146475&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The distributive property can be used to simplify expressions that look slightly different from [latex]a\\left(b+c\\right)[\/latex]. Here are two other forms.<\/p>\n<div class=\"textbox shaded\">\n<h3>different Forms of the Distributive Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are real numbers, then<\/p>\n<p style=\"padding-left: 60px;\">[latex]a\\left(b+c\\right)=ab+ac[\/latex]<\/p>\n<p>Other forms<\/p>\n<p style=\"padding-left: 60px;\">[latex]a\\left(b-c\\right)=ab-ac[\/latex]<br \/>\n[latex]\\left(b+c\\right)a=ba+ca[\/latex]<\/p>\n<\/div>\n<p>In the following video we show more examples of using the distributive property.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  The Distributive Property\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Nt8V5cEvAz8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using the Distributive Property With Fractions and Decimals<\/h2>\n<p>Do you remember how to multiply a fraction by a whole number? We\u2019ll need to do that in the next two examples. The distributive property comes in all shapes and sizes, and can include fractions or decimals as well.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\Large\\frac{3}{4}\\normalsize\\left(n+12\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q843482\">Show Solution<\/span><\/p>\n<div id=\"q843482\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467480771\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression x minus 3 in parentheses with the number 2 outside the parentheses on the left. Simplify by distributing the 2 through the parentheses to get the expression 2 times x minus 2 times 3. Simplify further by multiplying 2 times x to get 2 x and 2 times 3 to get 6. The expression simplifies to 2 x minus 6.\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222349\/CNX_BMath_Figure_07_03_027_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]\\Large\\frac{3}{4}\\normalsize\\cdot n+\\Large\\frac{3}{4}\\normalsize\\cdot 12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{3}{4}\\normalsize n+9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>Simplify: [latex]8\\Large\\left(\\frac{3}{8}\\normalsize x+\\Large\\frac{1}{4}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q689117\">Show Solution<\/span><\/p>\n<div id=\"q689117\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468690552\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 3 eights x plus 1 fourth in parentheses with the number 8 outside the parentheses on the left. Simplify by distributing the 8 through the parentheses to get the expression 8 times 3 eights x plus 8 times 1 fourth. Simplify further by multiplying 8 times 3 eights x to get 3 x and 8 times 1 fourth to get 2. The expression simplifies to 3 x plus 2.\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222352\/CNX_BMath_Figure_07_03_028_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]8\\cdot\\Large\\frac{3}{8}\\normalsize x+8\\cdot\\Large\\frac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]3x+2[\/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=\"ohm146476\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146476&theme=oea&iframe_resize_id=ohm146476&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146479\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146479&theme=oea&iframe_resize_id=ohm146479&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]100\\left(0.3+0.25q\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q393263\">Show Solution<\/span><\/p>\n<div id=\"q393263\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469752687\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 0.3 plus 0.25 q in parentheses with the number 100 outside the parentheses on the left. Simplify by distributing the 100 through the parentheses to get the expression 100 times 0.3 plus 100 times 0.25 q. Simplify further by multiplying 100 times 0.3 to get 30 and 100 times 0.25 q to get 25 q. The expression simplifies to 30 plus 25 q.\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222355\/CNX_BMath_Figure_07_03_029_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]100(0.3)+100(0.25q)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]30+25q[\/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=\"ohm146505\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146505&theme=oea&iframe_resize_id=ohm146505&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Distributing a Variable<\/h2>\n<p>In the next example we\u2019ll multiply by a variable. We\u2019ll need to do this in a later chapter.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]m\\left(n - 4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q288725\">Show Solution<\/span><\/p>\n<div id=\"q288725\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468371675\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression n minus 4 in parentheses with the letter m outside the parentheses on the left. Simplify by distributing the m through the parentheses to get the expression m times n minus m times 4. Simplify further by multiplying m times n to get m n and m times 4 to get 4m. The expression simplifies to m n minus 4 m.\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222357\/CNX_BMath_Figure_07_03_030_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]m\\cdot n-m\\cdot 4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]mn-4m[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that we wrote [latex]m\\cdot 4\\text{ as }4m[\/latex]. We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.<\/p>\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=\"ohm146506\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146506&theme=oea&iframe_resize_id=ohm146506&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>The Backwards Form of the Distributive Property<\/h2>\n<p>The next example will use the \u2018backwards\u2019 form of the Distributive Property, [latex]\\left(b+c\\right)a=ba+ca[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\left(x+8\\right)p[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q390066\">Show Solution<\/span><\/p>\n<div id=\"q390066\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469684869\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression x plus 8 in parentheses with the letter p outside the parentheses on the right Simplify by distributing the p through the parentheses to get the expression p times x plus p times 8. Simplify further by multiplying p times x to get p x and p times 8 to get 8p. The expression simplifies to p x plus 8 p.\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222400\/CNX_BMath_Figure_07_03_031_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222400\/CNX_BMath_Figure_07_03_031_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146507\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146507&theme=oea&iframe_resize_id=ohm146507&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Distributing a Negative Term<\/h2>\n<p>When you distribute a negative number, you need to be extra careful to get the signs correct.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]-2\\left(4y+1\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q31628\">Show Solution<\/span><\/p>\n<div id=\"q31628\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468516237\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 4 y plus 1 in parentheses with the number negative 2 outside the parentheses on the left. Simplify by distributing the negative 2 through the parentheses to get the expression negative 2 times 4 y plus negative 2 times 1. Simplify further by multiplying negative 2 times 4 y to get negative 8 y and negative 2 times 1 to get negative 2. The expression becomes negative 8 y plus negative 2. Simplify further by writing plus a negative as subtraction. The expression simplifies to negative 8 y minus 2.\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222401\/CNX_BMath_Figure_07_03_032_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]-2\\cdot 4y+(-2)\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-8y-2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>Simplify: [latex]-11\\left(4 - 3a\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149539\">Show Solution<\/span><\/p>\n<div id=\"q149539\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466284406\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 4 minus 3 a in parentheses with the number negative 11 outside the parentheses on the left. Simplify by distributing the negative 11 through the parentheses to get the expression negative 11 times 4 minus negative 11 times 3 a. Simplify further by multiplying negative 11 times 4 to get negative 44 and negative 11 times 3 a to get negative 33 a. The expression becomes negative 44 plus negative 33 a. Simplify further by writing plus a negative as subtraction. The expression simplifies to negative 44 minus 33 a.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-11(4-3a)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]-11\\cdot 4-(-11)\\cdot 3a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-44+(-33a)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-44+33a[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>You could also write the result as [latex]33a - 44[\/latex]. Do you know why?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146512\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146512&theme=oea&iframe_resize_id=ohm146512&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146511\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146511&theme=oea&iframe_resize_id=ohm146511&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, [latex]-a=-1\\cdot a[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]-\\left(y+5\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q833119\">Show Solution<\/span><\/p>\n<div id=\"q833119\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466218813\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression y plus 5 in parentheses with a negative sign outside the parentheses on the left. The negative sign outside the parentheses means the opposite and the opposite can be obtained by multiplying by negative 1. Rewrite the expression as y plus 5 in parentheses with a negative 1 outside the parentheses on the left. Simplify by distributing the negative 1 through the parentheses to get the expression negative 1 times y plus negative 1 times 5. Simplify further by multiplying negative 1 times y to get negative y and negative 1 times 5 to get negative 5. The expression becomes negative y plus negative 5. Simplify further by writing plus a negative as subtraction. The expression simplifies to negative y minus 5.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-(y+5)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiplying by \u22121 results in the opposite.<\/td>\n<td>[latex]-1(y+5)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]-1\\cdot y+(-1)\\cdot 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-y+(-5)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-y-5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146513\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146513&theme=oea&iframe_resize_id=ohm146513&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-9416\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 146513, 146511, 146510, 146509, 146506, 146505. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: The Distributive Property. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Nt8V5cEvAz8\">https:\/\/youtu.be\/Nt8V5cEvAz8<\/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 3: Combining Like Terms Requiring Distribution. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/STfLvYhDhwk\">https:\/\/youtu.be\/STfLvYhDhwk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/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 http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Ex 1: The Distributive Property\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Nt8V5cEvAz8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 3: Combining Like Terms Requiring Distribution\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/STfLvYhDhwk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 146513, 146511, 146510, 146509, 146506, 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Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"703ea099-1059-4267-ab80-70b9312b76a5","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-9416","chapter","type-chapter","status-publish","hentry"],"part":7349,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9416","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":39,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9416\/revisions"}],"predecessor-version":[{"id":16160,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9416\/revisions\/16160"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/parts\/7349"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9416\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/media?parent=9416"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=9416"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/contributor?post=9416"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/license?post=9416"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}