{"id":9418,"date":"2017-05-02T22:21:40","date_gmt":"2017-05-02T22:21:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9418"},"modified":"2024-04-30T21:22:45","modified_gmt":"2024-04-30T21:22:45","slug":"evaluating-expressions-using-the-distributive-property","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/evaluating-expressions-using-the-distributive-property\/","title":{"raw":"Evaluating Expressions Using the Distributive Property","rendered":"Evaluating Expressions Using the Distributive Property"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate expressions using the distributive property<\/li>\r\n \t<li>Evaluate expressions containing absolute value<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Using the Distributive Property With the Order of Operations<\/h2>\r\nSometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]8 - 2\\left(x+3\\right)[\/latex]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466085523\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 8 minus 2 times the quantity x plus 3 in parentheses. Simplify by distributing the 2 through the parentheses. Notice that the 2 is being subtracted from 8 so the subtraction must also be distributed with the 2. Once the minus 2 is distributed the expression becomes 8 minus 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 8 minus 2 x minus 6. Since 8 and 6 are like terms combine them to get 8 minus 6 is 2. The simplified expression is negative 2 x plus 2.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8--2(x+3)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]8--2\\cdot x--2\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]8--2x--6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]--2x+2[\/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]146971[\/ohm_question]\r\n\r\n[ohm_question]146972[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]4\\left(x - 8\\right)-\\left(x+3\\right)[\/latex]\r\n[reveal-answer q=\"162606\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"162606\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469612939\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 4 times the quantity x minus 8 in parentheses minus the quantity x plus 3 in parentheses. Simplify by performing two distributions. Distribute the 4 through the parentheses to get 4 times x minus 4 times 8. Notice that the quantity x plus 3 in parentheses is being subtracted so write it as a negative 1 to distribute. Distribute negative through the parentheses to get negative 1 times x plus negative 1 times 3. Perform the multiplication and the expression becomes 4 x minus 32 minus x minus 3. Combine the like terms 4 x minus x and negative 32 minus 3. The simplified expression is 3 x minus 35.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4(x--8)--(x+3)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]4x--32--x--3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]3x--35[\/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]146973[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following example, we simplify more expressions that require the distributive property.\r\n\r\nhttps:\/\/youtu.be\/STfLvYhDhwk\r\n<h2>Evaluate Expressions Using the Distributive Property<\/h2>\r\nSome students need to be convinced that the Distributive Property always works.\r\n\r\nIn the examples below, we will practice evaluating some of the expressions from previous examples; in part 1, we will evaluate the form with parentheses, and in part 2 we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWhen [latex]y=10[\/latex] evaluate:\r\n1. [latex]6\\left(5y+1\\right)[\/latex]\r\n2. [latex]6\\cdot 5y+6\\cdot 1[\/latex]\r\n\r\n[reveal-answer q=\"370094\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"370094\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466248176\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 10 substituted into the equation for y to get the expression 6 times the quantity 5 times 10 plus 1 in parentheses. Simplify inside the parentheses. 5 times 10 plus 1 is 51 and the expression becomes 6 times the quantity 51 in parentheses. Multiply to get 306.\">\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]6\\left(5y+1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitue [latex]\\color{red}{10}[\/latex] for y.<\/td>\r\n<td>[latex]6(5\\cdot\\color{red}{10}+1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify in the parentheses.<\/td>\r\n<td>[latex]6\\left(51\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]306[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468230676\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 10 substituted into the equation for y to get the expression the quantity of 6 times 5 times 10 in parentheses plus the quantity 6 times 1 in parentheses. Simplify inside the two sets of parentheses. 6 times 5 times 10 is 300 and 6 times 1 is 6. The expression becomes 300 plus 6. Add to get 306.\">\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]6\\cdot 5y+6\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color {red}{10}[\/latex] for y.<\/td>\r\n<td>[latex]6\\cdot 5\\cdot\\color {red}{10}+6\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]300+6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]306[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice, the answers are the same. When [latex]y=10[\/latex],\r\n<p style=\"text-align: center;\">[latex]6\\left(5y+1\\right)=6\\cdot 5y+6\\cdot 1[\/latex]<\/p>\r\nTry it yourself for a different value of [latex]y[\/latex].[\/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]146974[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWhen [latex]y=3[\/latex], evaluate\r\n1. [latex]-2\\left(4y+1\\right)[\/latex]\r\n2. [latex]-2\\cdot 4y+\\left(-2\\right)\\cdot 1[\/latex]\r\n[reveal-answer q=\"518398\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"518398\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467251696\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 3 substituted into the equation for y to get the expression negative 2 times the quantity 4 times 3 plus 1 in parentheses. Simplify inside the parentheses. 4 times 3 plus 1 is 13 and the expression becomes negative 2 times the quantity 13 in parentheses. Multiply to get negative 26.\">\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]-2\\left(4y+1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{3}[\/latex] for y.<\/td>\r\n<td>[latex]--2(4\\cdot\\color{red}{3}+1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify in the parentheses.<\/td>\r\n<td>[latex]-2\\left(13\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-26[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466177901\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 3 substituted into the equation for y to get the expression negative 2 times the quantity 4 times 3 plus 1 in parentheses. Simplify inside the parentheses. 4 times 3 plus 1 is 13 and the expression becomes negative 2 times the quantity 13 in parentheses. Multiply to get negative 26.\">\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\\cdot 4y+\\left(-2\\right)\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{3}[\/latex] for y.<\/td>\r\n<td>[latex]--2\\cdot 4\\cdot\\color{red}{3}+(--2)\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-24 - 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]-26[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The answers are the same. When [latex]y=3[\/latex],<\/td>\r\n<td>[latex]-2\\left(4y+1\\right)=-8y - 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\n[ohm_question]146975[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWhen [latex]y=35[\/latex] evaluate\r\n1. [latex]-\\left(y+5\\right)[\/latex]\r\n\r\n2. [latex]-y-5[\/latex] to show that [latex]-\\left(y+5\\right)=-y-5[\/latex]\r\n[reveal-answer q=\"169389\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"169389\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469875474\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 35 substituted into the equation for y to get the expression negative quantity 35 plus 5 in parentheses. Simplify inside the parentheses. 35 plus 5 is 40 and the expression becomes negative quantity 40 in parentheses. Multiply to get negative 40.\">\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]-\\left(y+5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{35}[\/latex] for y.<\/td>\r\n<td>[latex]--(\\color{red}{35}+5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add in the parentheses.<\/td>\r\n<td>[latex]-\\left(40\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-40[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466055024\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 35 substituted into the equation for y to get the expression negative 35 minus 5. Subtract to get negative 40.\">\r\n<tbody>\r\n<tr style=\"height: 15.5156px;\">\r\n<td style=\"height: 15.5156px;\">2.<\/td>\r\n<td style=\"height: 15.5156px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]-y - 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Substitute [latex]\\color{red}{35}[\/latex] for y.<\/td>\r\n<td style=\"height: 15px;\">[latex]--\\color{red}{35}--5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Simplify.<\/td>\r\n<td style=\"height: 15px;\">[latex]-40[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">The answers are the same when [latex]y=35[\/latex], demonstrating that<\/td>\r\n<td style=\"height: 15px;\">[latex]-\\left(y+5\\right)=-y-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\n[ohm_question]146986[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video provides another way to show that the distributive property works.\r\n\r\nhttps:\/\/youtu.be\/05jYrJn7W-M\r\n<h2>Absolute Value<\/h2>\r\nAbsolute value expressions are another method of grouping that you may see. Recall that the absolute value of a quantity is always positive or [latex]0[\/latex].\r\n\r\nWhen you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]\\dfrac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}[\/latex]\r\n\r\n[reveal-answer q=\"572632\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"572632\"]This problem has absolute values, decimals, multiplication, subtraction, and addition in it.\r\n\r\nGrouping symbols, including absolute value, are handled first. Simplify the numerator, then the denominator.\r\n\r\nEvaluate [latex]\\left|2\u20136\\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\dfrac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\\\\\\\\\dfrac{3+\\left|-4\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\end{array}[\/latex]<\/p>\r\nTake the absolute value of [latex]\\left|\u22124\\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\dfrac{3+4}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}[\/latex]<\/p>\r\nAdd the numbers in the numerator.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\dfrac{3+4}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\\\\\\\\\dfrac{7}{2\\left| 3\\cdot 1.5 \\right|-(-3)}\\end{array}[\/latex]<\/p>\r\nNow that the numerator is simplified, turn to the denominator.\r\n\r\nEvaluate the absolute value expression first. [latex]3 \\cdot 1.5 = 4.5[\/latex], giving\r\n<p style=\"text-align: center;\">\u00a0[latex]\\dfrac{7}{2\\left|{ 4.5}\\right|-(-3)}[\/latex]<\/p>\r\nTake the absolute value of [latex]\\left|4.5\\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\dfrac{7}{2(4.5)-\\left(-3\\right)}[\/latex]<\/p>\r\nThe expression \u201c[latex]2\\left|4.5\\right|[\/latex]\u201d reads \u201c[latex]2[\/latex] times the absolute value of [latex]4.5[\/latex].\u201d Multiply [latex]2[\/latex] times [latex]4.5[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\dfrac{7}{9-\\left(-3\\right)}[\/latex]<\/p>\r\nSubtract. [latex]9-(-3)=9+3=12[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\dfrac{7}{9-\\left(-3\\right)}\\\\\\\\\\dfrac{7}{12}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\dfrac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-3\\left(-3\\right)}=\\dfrac{7}{12}[\/latex]\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]109962[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video uses the order of operations to simplify an expression in fraction form that contains absolute value terms. Note how the absolute values are treated like parentheses and brackets when using the order of operations.\r\n\r\nhttps:\/\/youtu.be\/6wmCQprxlnU\r\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate expressions using the distributive property<\/li>\n<li>Evaluate expressions containing absolute value<\/li>\n<\/ul>\n<\/div>\n<h2>Using the Distributive Property With the Order of Operations<\/h2>\n<p>Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]8 - 2\\left(x+3\\right)[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168466085523\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 8 minus 2 times the quantity x plus 3 in parentheses. Simplify by distributing the 2 through the parentheses. Notice that the 2 is being subtracted from 8 so the subtraction must also be distributed with the 2. Once the minus 2 is distributed the expression becomes 8 minus 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 8 minus 2 x minus 6. Since 8 and 6 are like terms combine them to get 8 minus 6 is 2. The simplified expression is negative 2 x plus 2.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]8--2(x+3)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]8--2\\cdot x--2\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]8--2x--6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]--2x+2[\/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=\"ohm146971\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146971&theme=oea&iframe_resize_id=ohm146971&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146972\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146972&theme=oea&iframe_resize_id=ohm146972&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]4\\left(x - 8\\right)-\\left(x+3\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q162606\">Show Solution<\/span><\/p>\n<div id=\"q162606\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469612939\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 4 times the quantity x minus 8 in parentheses minus the quantity x plus 3 in parentheses. Simplify by performing two distributions. Distribute the 4 through the parentheses to get 4 times x minus 4 times 8. Notice that the quantity x plus 3 in parentheses is being subtracted so write it as a negative 1 to distribute. Distribute negative through the parentheses to get negative 1 times x plus negative 1 times 3. Perform the multiplication and the expression becomes 4 x minus 32 minus x minus 3. Combine the like terms 4 x minus x and negative 32 minus 3. The simplified expression is 3 x minus 35.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]4(x--8)--(x+3)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]4x--32--x--3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]3x--35[\/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=\"ohm146973\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146973&theme=oea&iframe_resize_id=ohm146973&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following example, we simplify more expressions that require the distributive property.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 3:  Combining Like Terms Requiring Distribution\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/STfLvYhDhwk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Evaluate Expressions Using the Distributive Property<\/h2>\n<p>Some students need to be convinced that the Distributive Property always works.<\/p>\n<p>In the examples below, we will practice evaluating some of the expressions from previous examples; in part 1, we will evaluate the form with parentheses, and in part 2 we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>When [latex]y=10[\/latex] evaluate:<br \/>\n1. [latex]6\\left(5y+1\\right)[\/latex]<br \/>\n2. [latex]6\\cdot 5y+6\\cdot 1[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q370094\">Show Solution<\/span><\/p>\n<div id=\"q370094\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466248176\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 10 substituted into the equation for y to get the expression 6 times the quantity 5 times 10 plus 1 in parentheses. Simplify inside the parentheses. 5 times 10 plus 1 is 51 and the expression becomes 6 times the quantity 51 in parentheses. Multiply to get 306.\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]6\\left(5y+1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitue [latex]\\color{red}{10}[\/latex] for y.<\/td>\n<td>[latex]6(5\\cdot\\color{red}{10}+1)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify in the parentheses.<\/td>\n<td>[latex]6\\left(51\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]306[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468230676\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 10 substituted into the equation for y to get the expression the quantity of 6 times 5 times 10 in parentheses plus the quantity 6 times 1 in parentheses. Simplify inside the two sets of parentheses. 6 times 5 times 10 is 300 and 6 times 1 is 6. The expression becomes 300 plus 6. Add to get 306.\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]6\\cdot 5y+6\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color {red}{10}[\/latex] for y.<\/td>\n<td>[latex]6\\cdot 5\\cdot\\color {red}{10}+6\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]300+6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]306[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice, the answers are the same. When [latex]y=10[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]6\\left(5y+1\\right)=6\\cdot 5y+6\\cdot 1[\/latex]<\/p>\n<p>Try it yourself for a different value of [latex]y[\/latex].<\/p><\/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=\"ohm146974\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146974&theme=oea&iframe_resize_id=ohm146974&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>When [latex]y=3[\/latex], evaluate<br \/>\n1. [latex]-2\\left(4y+1\\right)[\/latex]<br \/>\n2. [latex]-2\\cdot 4y+\\left(-2\\right)\\cdot 1[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q518398\">Show Solution<\/span><\/p>\n<div id=\"q518398\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467251696\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 3 substituted into the equation for y to get the expression negative 2 times the quantity 4 times 3 plus 1 in parentheses. Simplify inside the parentheses. 4 times 3 plus 1 is 13 and the expression becomes negative 2 times the quantity 13 in parentheses. Multiply to get negative 26.\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-2\\left(4y+1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{3}[\/latex] for y.<\/td>\n<td>[latex]--2(4\\cdot\\color{red}{3}+1)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify in the parentheses.<\/td>\n<td>[latex]-2\\left(13\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-26[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466177901\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 3 substituted into the equation for y to get the expression negative 2 times the quantity 4 times 3 plus 1 in parentheses. Simplify inside the parentheses. 4 times 3 plus 1 is 13 and the expression becomes negative 2 times the quantity 13 in parentheses. Multiply to get negative 26.\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-2\\cdot 4y+\\left(-2\\right)\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{3}[\/latex] for y.<\/td>\n<td>[latex]--2\\cdot 4\\cdot\\color{red}{3}+(--2)\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-24 - 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]-26[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The answers are the same. When [latex]y=3[\/latex],<\/td>\n<td>[latex]-2\\left(4y+1\\right)=-8y - 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><iframe loading=\"lazy\" id=\"ohm146975\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146975&theme=oea&iframe_resize_id=ohm146975&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>When [latex]y=35[\/latex] evaluate<br \/>\n1. [latex]-\\left(y+5\\right)[\/latex]<\/p>\n<p>2. [latex]-y-5[\/latex] to show that [latex]-\\left(y+5\\right)=-y-5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q169389\">Show Solution<\/span><\/p>\n<div id=\"q169389\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469875474\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 35 substituted into the equation for y to get the expression negative quantity 35 plus 5 in parentheses. Simplify inside the parentheses. 35 plus 5 is 40 and the expression becomes negative quantity 40 in parentheses. Multiply to get negative 40.\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-\\left(y+5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{35}[\/latex] for y.<\/td>\n<td>[latex]--(\\color{red}{35}+5)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add in the parentheses.<\/td>\n<td>[latex]-\\left(40\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-40[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466055024\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows 35 substituted into the equation for y to get the expression negative 35 minus 5. Subtract to get negative 40.\">\n<tbody>\n<tr style=\"height: 15.5156px;\">\n<td style=\"height: 15.5156px;\">2.<\/td>\n<td style=\"height: 15.5156px;\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]-y - 5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Substitute [latex]\\color{red}{35}[\/latex] for y.<\/td>\n<td style=\"height: 15px;\">[latex]--\\color{red}{35}--5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Simplify.<\/td>\n<td style=\"height: 15px;\">[latex]-40[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">The answers are the same when [latex]y=35[\/latex], demonstrating that<\/td>\n<td style=\"height: 15px;\">[latex]-\\left(y+5\\right)=-y-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><iframe loading=\"lazy\" id=\"ohm146986\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146986&theme=oea&iframe_resize_id=ohm146986&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video provides another way to show that the distributive property works.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Evaluate Numerical Expressions Using Order of Operations and Distribution\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/05jYrJn7W-M?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Absolute Value<\/h2>\n<p>Absolute value expressions are another method of grouping that you may see. Recall that the absolute value of a quantity is always positive or [latex]0[\/latex].<\/p>\n<p>When you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\dfrac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q572632\">Show Solution<\/span><\/p>\n<div id=\"q572632\" class=\"hidden-answer\" style=\"display: none\">This problem has absolute values, decimals, multiplication, subtraction, and addition in it.<\/p>\n<p>Grouping symbols, including absolute value, are handled first. Simplify the numerator, then the denominator.<\/p>\n<p>Evaluate [latex]\\left|2\u20136\\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\dfrac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\\\\\\\\\dfrac{3+\\left|-4\\right|}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\end{array}[\/latex]<\/p>\n<p>Take the absolute value of [latex]\\left|\u22124\\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{3+4}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}[\/latex]<\/p>\n<p>Add the numbers in the numerator.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\dfrac{3+4}{2\\left|3\\cdot1.5\\right|-\\left(-3\\right)}\\\\\\\\\\dfrac{7}{2\\left| 3\\cdot 1.5 \\right|-(-3)}\\end{array}[\/latex]<\/p>\n<p>Now that the numerator is simplified, turn to the denominator.<\/p>\n<p>Evaluate the absolute value expression first. [latex]3 \\cdot 1.5 = 4.5[\/latex], giving<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]\\dfrac{7}{2\\left|{ 4.5}\\right|-(-3)}[\/latex]<\/p>\n<p>Take the absolute value of [latex]\\left|4.5\\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{7}{2(4.5)-\\left(-3\\right)}[\/latex]<\/p>\n<p>The expression \u201c[latex]2\\left|4.5\\right|[\/latex]\u201d reads \u201c[latex]2[\/latex] times the absolute value of [latex]4.5[\/latex].\u201d Multiply [latex]2[\/latex] times [latex]4.5[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{7}{9-\\left(-3\\right)}[\/latex]<\/p>\n<p>Subtract. [latex]9-(-3)=9+3=12[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\dfrac{7}{9-\\left(-3\\right)}\\\\\\\\\\dfrac{7}{12}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\dfrac{3+\\left|2-6\\right|}{2\\left|3\\cdot1.5\\right|-3\\left(-3\\right)}=\\dfrac{7}{12}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm109962\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109962&theme=oea&iframe_resize_id=ohm109962&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video uses the order of operations to simplify an expression in fraction form that contains absolute value terms. Note how the absolute values are treated like parentheses and brackets when using the order of operations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Simplify an Expression in Fraction Form with Absolute Values\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/6wmCQprxlnU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-9418\">\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 146986, 146975, 146974, 146973, 146972, 146971. <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>Evaluate Numerical Expressions Using Order of Operations and Distribution. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/05jYrJn7W-M\">https:\/\/youtu.be\/05jYrJn7W-M<\/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":15,"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\":\"Evaluate Numerical Expressions Using Order of Operations and Distribution\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/05jYrJn7W-M\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 146986, 146975, 146974, 146973, 146972, 146971\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"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\":\"\"}]","CANDELA_OUTCOMES_GUID":"05bd420059694f99bb988ef77e0ab132, 976aa95e8ae74fd58387a4db47bd1959, 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