{"id":17462,"date":"2020-03-29T05:56:54","date_gmt":"2020-03-29T05:56:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/?post_type=chapter&#038;p=17462"},"modified":"2020-10-22T09:09:02","modified_gmt":"2020-10-22T09:09:02","slug":"solving-equations-containing-absolute-values","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/solving-equations-containing-absolute-values\/","title":{"raw":"7.1.g - Solving Equations Containing Absolute Values","rendered":"7.1.g &#8211; Solving Equations Containing Absolute Values"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve equations containing absolute values<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"title3\">Solving One-Step Equations Containing Absolute Values with Addition<\/h2>\r\nThe <b>absolute value<\/b> of a number or expression describes its distance from [latex]0[\/latex] on a number line. Since the absolute value expresses only the distance, not the direction of the number on a number line, it is always expressed as a positive number or [latex]0[\/latex].\r\n\r\nFor example, [latex]\u22124[\/latex] and \u00a0[latex]4[\/latex] both have an absolute value of \u00a0[latex]4[\/latex] because they are each [latex]4[\/latex] units from [latex]0[\/latex] on a number line\u2014though they are located in opposite directions from [latex]0[\/latex] on the number line.\r\n\r\nWhen solving absolute value <b>equations<\/b> and <b>inequalities<\/b>, you have to consider both the behavior of absolute value and the properties of equality and inequality.\r\n\r\nBecause both positive and negative values have a positive absolute value, solving absolute value equations means finding the solution for both the positive and the negative values.\r\n\r\nLet\u2019s first look at a very basic example.\r\n<p align=\"center\">[latex] \\displaystyle \\left| x \\right|=5[\/latex]<\/p>\r\nThis equation is read \u201cthe absolute value of \u00a0[latex]x[\/latex] is equal to five.\u201d The solution is the value(s) that are five units away from [latex]0[\/latex] on a number line.\r\n\r\nYou might think of [latex]5[\/latex] right away; that is one solution to the equation. Notice that [latex]\u22125[\/latex] is also a solution because [latex]\u22125[\/latex] is [latex]5[\/latex] units away from \u00a0[latex]0[\/latex] in the opposite direction. So, the solution to this equation [latex] \\displaystyle \\left| x \\right|=5[\/latex] is [latex]x = \u22125[\/latex] or [latex]x = 5[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Solving Equations of the Form [latex]|x|=a[\/latex]<\/h3>\r\nFor any positive number [latex]a[\/latex], the solution of [latex]\\left|x\\right|=a[\/latex]\u00a0is\r\n<p style=\"text-align: center\">[latex]x=a[\/latex]\u00a0 or\u00a0 [latex]x=\u2212a[\/latex]<\/p>\r\n[latex]x[\/latex] can be a single variable or any algebraic expression.\r\n\r\n<\/div>\r\nYou can solve a more complex absolute value problem in a similar fashion.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex] \\displaystyle \\left| x+5\\right|=15[\/latex].\r\n[reveal-answer q=\"624457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624457\"]\r\n\r\nThis equation asks you to find what number plus [latex]5[\/latex] has an absolute value of [latex]15[\/latex]. Since 15 and [latex]\u221215[\/latex] both have an absolute value of [latex]15[\/latex], the absolute value equation is true when the quantity [latex]x + 5[\/latex]\u00a0is [latex]15[\/latex] <i>or<\/i>\u00a0[latex]x + 5[\/latex] is [latex]\u221215[\/latex], since [latex]|15|=15[\/latex] and [latex]|\u221215|=15[\/latex]. So, you need to find out what value for \u00a0[latex]x[\/latex] will make this expression equal to\u00a0[latex]15[\/latex], as well as what value for [latex]x[\/latex] will make the expression equal to [latex]\u221215[\/latex]. Solving the two equations you get\r\n<p align=\"center\">[latex] \\displaystyle \\begin{array}{l}x+5=15\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,{x+5=-15}\\\\\\underline{\\,\\,\\,\\,\\,-5\\,\\,\\,\\,-5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-5\\,\\,\\,\\,\\,\\,\\,\\,\\,-5}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,10\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,\\,=-20\\end{array}[\/latex]<\/p>\r\nYou can check these two solutions in the absolute value equation to see if [latex]x=10[\/latex] and [latex]x = \u221220[\/latex] are correct.\r\n<p align=\"center\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\left| x+5 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| x+5 \\right|=15\\\\\\left| 10+5 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -20+5 \\right|=15\\\\\\,\\,\\,\\,\\,\\,\\,\\left| 15 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -15 \\right|=15\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,15=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15=15\\end{array}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nThe following video provides worked examples of solving linear equations with absolute value terms.\r\nhttps:\/\/youtu.be\/U-7fF-W8_xE\r\n<h2 id=\"title3\">Solving One-Step Equations Containing Absolute Values With Multiplication<\/h2>\r\nIn the last section, we saw examples of solving equations with absolute values where the only operation was addition or subtraction. Now we will see how to solve equations with absolute value that include multiplication.\r\n\r\nRemember that absolute value refers to the distance from zero. You can use the same technique of first isolating the absolute value, then setting up and solving two equations to solve an absolute value equation involving multiplication.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex] \\displaystyle \\left| 2x\\right|=6[\/latex].\r\n[reveal-answer q=\"624455\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624455\"]\r\n\r\nThis equation asks you to find what number times [latex]2[\/latex] has an absolute value of [latex]6[\/latex].\r\n\r\nSince [latex]6[\/latex] and [latex]\u22126[\/latex] both have an absolute value of [latex]6[\/latex], the absolute value equation is true when the quantity [latex]2x[\/latex]\u00a0is [latex]6[\/latex] <i>or<\/i>\u00a0[latex]2x[\/latex] is [latex]\u22126[\/latex], since [latex]|6|=6[\/latex] and [latex]|\u22126|=6[\/latex].\r\n\r\nSo, you need to find out what value for [latex]x[\/latex] will make this expression equal to [latex]6[\/latex], as well as what value for [latex]x[\/latex] will make the expression equal to [latex]\u22126[\/latex].\r\n\r\nSolving the two equations you get\r\n<p align=\"center\">[latex]2x=6\\text { or } 2x=-6[\/latex]<\/p>\r\n<p align=\"center\">[latex]\\frac{2x}{2}=\\frac{6}{2}\\text { or } \\frac{2x}{2}=\\frac{-6}{2}[\/latex]<\/p>\r\n<p align=\"center\">[latex]x=3\\text { or } x=-3[\/latex]<\/p>\r\nYou can check these two solutions in the absolute value equation to see if [latex]x=3[\/latex] and [latex]x=\u22123[\/latex] are correct.\r\n<p align=\"center\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\left|3\\cdot2 \\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -3\\cdot2 \\right|=6\\\\\\left| 6 \\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -6 \\right|=6\\\\\\end{array}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve [latex] \\displaystyle\\frac{1}{3}\\left|k\\right|=12[\/latex].\r\n[reveal-answer q=\"604455\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604455\"]\r\n\r\nNotice how this example is different from the last; [latex] \\displaystyle\\frac{1}{3}[\/latex] is outside the absolute value grouping symbols. This means we need to isolate the absolute value first, then apply the definition of absolute value.\r\n\r\nFirst, isolate the absolute value term by multiplying by the inverse of [latex] \\displaystyle\\frac{1}{3}[\/latex]:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\frac{1}{3}\\left|k\\right|=12\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\left(3\\right)\\frac{1}{3}\\left|k\\right|=\\left(3\\right)12\\\\\\left|k\\right|=36\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Apply the definition of absolute value:<\/p>\r\n<p style=\"text-align: center\">[latex] \\displaystyle{k }=36\\text { or } {k }=-36[\/latex]<\/p>\r\nYou can check these two solutions in the absolute value equation to see if [latex]k=36[\/latex] and [latex]k=\u221236[\/latex] are correct.\r\n<p align=\"center\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\frac{1}{3}\\left|36 \\right|=12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{1}{3}\\left|-36 \\right|=12\\\\\\left| 12 \\right|=12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -12 \\right|=12\\\\\\end{array}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nIn the following video, you will see two examples of how to solve an absolute value equation, one with integers and one with fractions.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=CTLnJ955xzc&amp;feature=youtu.be\r\n<h2>Solving Multi-Step Equations With Absolute Value<\/h2>\r\nWe can apply the same techniques we used for solving a one-step equation which contains absolute value to an equation that will take more than one step to solve. \u00a0Let's start with an example where the first step is to write two equations: one equal to positive \u00a0[latex]26[\/latex] and one equal to negative [latex]26[\/latex].\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]p[\/latex].\u00a0[latex]\\left|2p\u20134\\right|=26[\/latex]\r\n\r\n[reveal-answer q=\"371950\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"371950\"]\r\n\r\nWrite the two equations that will give an absolute value of [latex]26[\/latex].\r\n<p style=\"text-align: center\">[latex] \\displaystyle 2p-4=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,2p-4=\\,-26[\/latex]<\/p>\r\nSolve each equation for [latex]p[\/latex] by isolating the variable<i>.<\/i>\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}2p-4=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2p-4=\\,-26\\\\\\underline{\\,\\,\\,\\,\\,\\,+4\\,\\,\\,\\,+4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,+4\\,\\,\\,\\,\\,\\,\\,+4}\\\\\\underline{2p}\\,\\,\\,\\,\\,\\,=\\underline{30}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{2p}\\,\\,\\,\\,\\,=\\,\\underline{-22}\\\\2\\,\\,\\,\\,\\,\\,\\,=\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\,\\,\\,2\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,p=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,p=\\,-11\\end{array}[\/latex]<\/p>\r\nCheck the solutions in the original equation.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,\\left| 2p-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 2p-4 \\right|=26\\\\\\left| 2(15)-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\left| 2(-11)-4 \\right|=26\\\\\\,\\,\\,\\,\\,\\left| 30-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -22-4 \\right|=26\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 26 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -26 \\right|=26\\end{array}[\/latex]<\/p>\r\nBoth solutions check!\r\n<h4>Answer<\/h4>\r\n[latex]p=15[\/latex] or [latex]p=-11[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]1293[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video, we show more examples of solving a simple absolute value equation.\r\n\r\nhttps:\/\/youtu.be\/4g-o_-mAFpc\r\n\r\nNow let's look at an example where you need to do an algebraic step or two before you can write your two equations. The goal here is to get the absolute value on one side of the equation by itself. Then we can proceed as we did in the previous example.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]w[\/latex]. [latex]3\\left|4w\u20131\\right|\u20135=10[\/latex]\r\n\r\n[reveal-answer q=\"303228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"303228\"]\r\n\r\nIsolate the term with the absolute value by adding \u00a0[latex]5[\/latex] to both sides.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3\\left|4w-1\\right|-5=10\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+5\\,\\,\\,+5}\\\\ 3\\left|4w-1\\right|=15\\end{array}[\/latex]<\/p>\r\nDivide both sides by [latex]3[\/latex].\u00a0Now the absolute value is isolated.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r} \\underline{3\\left|4w-1\\right|}=\\underline{15}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\,\\\\\\left|4w-1\\right|=\\,\\,5\\end{array}[\/latex]<\/p>\r\nWrite the two equations that will give an absolute value of \u00a0[latex]5[\/latex] and solve them.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}4w-1=5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4w-1=-5\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,+1\\,\\,+1}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,+1\\,\\,\\,\\,\\,+1}\\\\\\,\\,\\,\\,\\,\\underline{4w}=\\underline{6}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{4w}\\,\\,\\,\\,\\,\\,\\,=\\underline{-4}\\\\4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,w=\\frac{3}{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,w=-1\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,w=\\frac{3}{2}\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nCheck the solutions in the original equation.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\\\\\\\3\\left| 4\\left( \\frac{3}{2} \\right)-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\\\\\\\\\,\\,\\,\\,\\,\\,3\\left| \\frac{12}{2}-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 4(-1)-1\\, \\right|-5=10\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 6-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| -4-1\\, \\right|-5=10\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left(5\\right)-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| -5 \\right|-5=10\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15-5=10\\\\10=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,10=10\\end{array}[\/latex]<\/p>\r\nBoth solutions check\r\n<h4>Answer<\/h4>\r\n[latex]w=-1\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,w=\\frac{3}{2}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations.\r\n\r\nhttps:\/\/youtu.be\/-HrOMkIiSfU\r\n\r\nhttps:\/\/youtu.be\/2bEA7HoDfpk\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]38177[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve equations containing absolute values<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"title3\">Solving One-Step Equations Containing Absolute Values with Addition<\/h2>\n<p>The <b>absolute value<\/b> of a number or expression describes its distance from [latex]0[\/latex] on a number line. Since the absolute value expresses only the distance, not the direction of the number on a number line, it is always expressed as a positive number or [latex]0[\/latex].<\/p>\n<p>For example, [latex]\u22124[\/latex] and \u00a0[latex]4[\/latex] both have an absolute value of \u00a0[latex]4[\/latex] because they are each [latex]4[\/latex] units from [latex]0[\/latex] on a number line\u2014though they are located in opposite directions from [latex]0[\/latex] on the number line.<\/p>\n<p>When solving absolute value <b>equations<\/b> and <b>inequalities<\/b>, you have to consider both the behavior of absolute value and the properties of equality and inequality.<\/p>\n<p>Because both positive and negative values have a positive absolute value, solving absolute value equations means finding the solution for both the positive and the negative values.<\/p>\n<p>Let\u2019s first look at a very basic example.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\left| x \\right|=5[\/latex]<\/p>\n<p>This equation is read \u201cthe absolute value of \u00a0[latex]x[\/latex] is equal to five.\u201d The solution is the value(s) that are five units away from [latex]0[\/latex] on a number line.<\/p>\n<p>You might think of [latex]5[\/latex] right away; that is one solution to the equation. Notice that [latex]\u22125[\/latex] is also a solution because [latex]\u22125[\/latex] is [latex]5[\/latex] units away from \u00a0[latex]0[\/latex] in the opposite direction. So, the solution to this equation [latex]\\displaystyle \\left| x \\right|=5[\/latex] is [latex]x = \u22125[\/latex] or [latex]x = 5[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Solving Equations of the Form [latex]|x|=a[\/latex]<\/h3>\n<p>For any positive number [latex]a[\/latex], the solution of [latex]\\left|x\\right|=a[\/latex]\u00a0is<\/p>\n<p style=\"text-align: center\">[latex]x=a[\/latex]\u00a0 or\u00a0 [latex]x=\u2212a[\/latex]<\/p>\n<p>[latex]x[\/latex] can be a single variable or any algebraic expression.<\/p>\n<\/div>\n<p>You can solve a more complex absolute value problem in a similar fashion.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]\\displaystyle \\left| x+5\\right|=15[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q624457\">Show Solution<\/span><\/p>\n<div id=\"q624457\" class=\"hidden-answer\" style=\"display: none\">\n<p>This equation asks you to find what number plus [latex]5[\/latex] has an absolute value of [latex]15[\/latex]. Since 15 and [latex]\u221215[\/latex] both have an absolute value of [latex]15[\/latex], the absolute value equation is true when the quantity [latex]x + 5[\/latex]\u00a0is [latex]15[\/latex] <i>or<\/i>\u00a0[latex]x + 5[\/latex] is [latex]\u221215[\/latex], since [latex]|15|=15[\/latex] and [latex]|\u221215|=15[\/latex]. So, you need to find out what value for \u00a0[latex]x[\/latex] will make this expression equal to\u00a0[latex]15[\/latex], as well as what value for [latex]x[\/latex] will make the expression equal to [latex]\u221215[\/latex]. Solving the two equations you get<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}x+5=15\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,{x+5=-15}\\\\\\underline{\\,\\,\\,\\,\\,-5\\,\\,\\,\\,-5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-5\\,\\,\\,\\,\\,\\,\\,\\,\\,-5}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,10\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,\\,=-20\\end{array}[\/latex]<\/p>\n<p>You can check these two solutions in the absolute value equation to see if [latex]x=10[\/latex] and [latex]x = \u221220[\/latex] are correct.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\left| x+5 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| x+5 \\right|=15\\\\\\left| 10+5 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -20+5 \\right|=15\\\\\\,\\,\\,\\,\\,\\,\\,\\left| 15 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -15 \\right|=15\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,15=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15=15\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>The following video provides worked examples of solving linear equations with absolute value terms.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Solving Absolute Value Equations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/U-7fF-W8_xE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title3\">Solving One-Step Equations Containing Absolute Values With Multiplication<\/h2>\n<p>In the last section, we saw examples of solving equations with absolute values where the only operation was addition or subtraction. Now we will see how to solve equations with absolute value that include multiplication.<\/p>\n<p>Remember that absolute value refers to the distance from zero. You can use the same technique of first isolating the absolute value, then setting up and solving two equations to solve an absolute value equation involving multiplication.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]\\displaystyle \\left| 2x\\right|=6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q624455\">Show Solution<\/span><\/p>\n<div id=\"q624455\" class=\"hidden-answer\" style=\"display: none\">\n<p>This equation asks you to find what number times [latex]2[\/latex] has an absolute value of [latex]6[\/latex].<\/p>\n<p>Since [latex]6[\/latex] and [latex]\u22126[\/latex] both have an absolute value of [latex]6[\/latex], the absolute value equation is true when the quantity [latex]2x[\/latex]\u00a0is [latex]6[\/latex] <i>or<\/i>\u00a0[latex]2x[\/latex] is [latex]\u22126[\/latex], since [latex]|6|=6[\/latex] and [latex]|\u22126|=6[\/latex].<\/p>\n<p>So, you need to find out what value for [latex]x[\/latex] will make this expression equal to [latex]6[\/latex], as well as what value for [latex]x[\/latex] will make the expression equal to [latex]\u22126[\/latex].<\/p>\n<p>Solving the two equations you get<\/p>\n<p style=\"text-align: center;\">[latex]2x=6\\text { or } 2x=-6[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2x}{2}=\\frac{6}{2}\\text { or } \\frac{2x}{2}=\\frac{-6}{2}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=3\\text { or } x=-3[\/latex]<\/p>\n<p>You can check these two solutions in the absolute value equation to see if [latex]x=3[\/latex] and [latex]x=\u22123[\/latex] are correct.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\left|3\\cdot2 \\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -3\\cdot2 \\right|=6\\\\\\left| 6 \\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -6 \\right|=6\\\\\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve [latex]\\displaystyle\\frac{1}{3}\\left|k\\right|=12[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q604455\">Show Solution<\/span><\/p>\n<div id=\"q604455\" class=\"hidden-answer\" style=\"display: none\">\n<p>Notice how this example is different from the last; [latex]\\displaystyle\\frac{1}{3}[\/latex] is outside the absolute value grouping symbols. This means we need to isolate the absolute value first, then apply the definition of absolute value.<\/p>\n<p>First, isolate the absolute value term by multiplying by the inverse of [latex]\\displaystyle\\frac{1}{3}[\/latex]:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\frac{1}{3}\\left|k\\right|=12\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\left(3\\right)\\frac{1}{3}\\left|k\\right|=\\left(3\\right)12\\\\\\left|k\\right|=36\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">Apply the definition of absolute value:<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle{k }=36\\text { or } {k }=-36[\/latex]<\/p>\n<p>You can check these two solutions in the absolute value equation to see if [latex]k=36[\/latex] and [latex]k=\u221236[\/latex] are correct.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\frac{1}{3}\\left|36 \\right|=12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{1}{3}\\left|-36 \\right|=12\\\\\\left| 12 \\right|=12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -12 \\right|=12\\\\\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you will see two examples of how to solve an absolute value equation, one with integers and one with fractions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Solving Absolute Value Equation Using Multiplication and Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/CTLnJ955xzc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Solving Multi-Step Equations With Absolute Value<\/h2>\n<p>We can apply the same techniques we used for solving a one-step equation which contains absolute value to an equation that will take more than one step to solve. \u00a0Let&#8217;s start with an example where the first step is to write two equations: one equal to positive \u00a0[latex]26[\/latex] and one equal to negative [latex]26[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]p[\/latex].\u00a0[latex]\\left|2p\u20134\\right|=26[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q371950\">Show Solution<\/span><\/p>\n<div id=\"q371950\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write the two equations that will give an absolute value of [latex]26[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle 2p-4=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,2p-4=\\,-26[\/latex]<\/p>\n<p>Solve each equation for [latex]p[\/latex] by isolating the variable<i>.<\/i><\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}2p-4=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2p-4=\\,-26\\\\\\underline{\\,\\,\\,\\,\\,\\,+4\\,\\,\\,\\,+4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,+4\\,\\,\\,\\,\\,\\,\\,+4}\\\\\\underline{2p}\\,\\,\\,\\,\\,\\,=\\underline{30}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{2p}\\,\\,\\,\\,\\,=\\,\\underline{-22}\\\\2\\,\\,\\,\\,\\,\\,\\,=\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\,\\,\\,2\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,p=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,p=\\,-11\\end{array}[\/latex]<\/p>\n<p>Check the solutions in the original equation.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,\\left| 2p-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 2p-4 \\right|=26\\\\\\left| 2(15)-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\left| 2(-11)-4 \\right|=26\\\\\\,\\,\\,\\,\\,\\left| 30-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -22-4 \\right|=26\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 26 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -26 \\right|=26\\end{array}[\/latex]<\/p>\n<p>Both solutions check!<\/p>\n<h4>Answer<\/h4>\n<p>[latex]p=15[\/latex] or [latex]p=-11[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1293\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1293&theme=oea&iframe_resize_id=ohm1293&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video, we show more examples of solving a simple absolute value equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 2:  Solving Absolute Value Equations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/4g-o_-mAFpc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Now let&#8217;s look at an example where you need to do an algebraic step or two before you can write your two equations. The goal here is to get the absolute value on one side of the equation by itself. Then we can proceed as we did in the previous example.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]w[\/latex]. [latex]3\\left|4w\u20131\\right|\u20135=10[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q303228\">Show Solution<\/span><\/p>\n<div id=\"q303228\" class=\"hidden-answer\" style=\"display: none\">\n<p>Isolate the term with the absolute value by adding \u00a0[latex]5[\/latex] to both sides.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3\\left|4w-1\\right|-5=10\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+5\\,\\,\\,+5}\\\\ 3\\left|4w-1\\right|=15\\end{array}[\/latex]<\/p>\n<p>Divide both sides by [latex]3[\/latex].\u00a0Now the absolute value is isolated.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r} \\underline{3\\left|4w-1\\right|}=\\underline{15}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\,\\\\\\left|4w-1\\right|=\\,\\,5\\end{array}[\/latex]<\/p>\n<p>Write the two equations that will give an absolute value of \u00a0[latex]5[\/latex] and solve them.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}4w-1=5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4w-1=-5\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,+1\\,\\,+1}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,+1\\,\\,\\,\\,\\,+1}\\\\\\,\\,\\,\\,\\,\\underline{4w}=\\underline{6}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{4w}\\,\\,\\,\\,\\,\\,\\,=\\underline{-4}\\\\4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,w=\\frac{3}{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,w=-1\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,w=\\frac{3}{2}\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Check the solutions in the original equation.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\\\\\\\3\\left| 4\\left( \\frac{3}{2} \\right)-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\\\\\\\\\,\\,\\,\\,\\,\\,3\\left| \\frac{12}{2}-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 4(-1)-1\\, \\right|-5=10\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 6-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| -4-1\\, \\right|-5=10\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left(5\\right)-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| -5 \\right|-5=10\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15-5=10\\\\10=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,10=10\\end{array}[\/latex]<\/p>\n<p>Both solutions check<\/p>\n<h4>Answer<\/h4>\n<p>[latex]w=-1\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,w=\\frac{3}{2}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 4:  Solving Absolute Value Equations (Requires Isolating Abs. Value)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/-HrOMkIiSfU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex 5:  Solving Absolute Value Equations (Requires Isolating Abs. Value)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/2bEA7HoDfpk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm38177\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=38177&theme=oea&iframe_resize_id=ohm38177&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n","protected":false},"author":253111,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"0bdb5d89bfce4d0ebafbee429e80baf4, 3a773a653d9e4e52bcb175288d0201fb","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17462","chapter","type-chapter","status-publish","hentry"],"part":7476,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17462","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/users\/253111"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17462\/revisions"}],"predecessor-version":[{"id":20302,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17462\/revisions\/20302"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/7476"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17462\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/media?parent=17462"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=17462"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=17462"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/license?post=17462"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}