{"id":18673,"date":"2022-04-22T18:00:18","date_gmt":"2022-04-22T18:00:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18673"},"modified":"2022-04-28T23:18:40","modified_gmt":"2022-04-28T23:18:40","slug":"cr-26-absolute-value","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-26-absolute-value\/","title":{"raw":"CR.26: Absolute Value","rendered":"CR.26: Absolute Value"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Solve equations containing absolute values<\/li>\r\n \t<li>Recognize when a linear equation that contains absolute value does not have a solution<\/li>\r\n \t<li>Solve inequalities containing absolute values<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Review of Absolute Value<\/h3>\r\nRemember that absolute value is telling you how far away a number is from 0. Therefore, [latex]|7|=7[\/latex] and [latex]|-7|=7[\/latex] since the distance from -7 to 0 and the distance from 0 to 7 are both 7.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Absolute Values<\/h3>\r\nEvaluate:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\left|-\\frac{3}{14}\\right|[\/latex]<\/li>\r\n \t<li>[latex]|7+10(-2)|[\/latex]<\/li>\r\n<\/ol>\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q67590\">Solution<\/span>\r\n<div id=\"q67590\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\na. [latex]\\left|-\\frac{3}{14}\\right|[\/latex]\r\n\r\nThe absolute value of a real number [latex]a[\/latex], denoted by [latex]|a|[\/latex], is the distance from 0 to [latex]a[\/latex] on the number line. Because absolute value describes a distance, it is never negative.\r\n\r\n[latex]\\left|-\\frac{3}{14}\\right|[\/latex] is read as \"the absolute value of [latex]-\\frac{3}{14}[\/latex]\". The absolute value of [latex]-\\frac{3}{14}[\/latex] is equal to the distance from 0 to [latex]-\\frac{3}{14}[\/latex] on a number line.\r\n\r\nThe distance from 0 to [latex]-\\frac{3}{14}[\/latex] on the number line is [latex]\\frac{3}{14}[\/latex].\r\n\r\nTherefore, [latex]\\left|-\\frac{3}{14}\\right|=\\frac{3}{14}[\/latex].\r\n\r\nb. [latex]|7+10(-2)|[\/latex]\r\n\r\nPerform the operations inside the absolute value. Following the order of operations, first perform the multiplication.\r\n[latex]|7+10(-2)|=|7+(-20)|[\/latex]\r\n\r\nNow perform the addition.\r\n[latex]|7+10(-2)|=|7+(-20)|=|-13|[\/latex]\r\n\r\nFinish by evaluating the absolute value.\r\n[latex]|7+10(-2)|=|7+(-20)|=|-13|=13[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h3>Solving an Absolute Value Equation<\/h3>\r\nNext, we will learn how to slve an <strong>absolute value equation<\/strong>. To solve an equation such as [latex]|2x - 6|=8[\/latex], we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is [latex]8[\/latex] or [latex]-8[\/latex]. This leads to two different equations we can solve independently.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}2x - 6=8\\hfill &amp; \\text{ or }\\hfill &amp; 2x - 6=-8\\hfill \\\\ 2x=14\\hfill &amp; \\hfill &amp; 2x=-2\\hfill \\\\ x=7\\hfill &amp; \\hfill &amp; x=-1\\hfill \\end{array}[\/latex]<\/div>\r\nKnowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Absolute Value Equations<\/h3>\r\nThe absolute value of <em>x <\/em>is written as [latex]|x|[\/latex]. It has the following properties:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{If } x\\ge 0,\\text{ then }|x|=x.\\hfill \\\\ \\text{If }x&lt;0,\\text{ then }|x|=-x.\\hfill \\end{array}[\/latex]<\/div>\r\nFor real numbers [latex]A[\/latex] and [latex]B[\/latex], an equation of the form [latex]|A|=B[\/latex], with [latex]B\\ge 0[\/latex], will have solutions when [latex]A=B[\/latex] or [latex]A=-B[\/latex]. If [latex]B&lt;0[\/latex], the equation [latex]|A|=B[\/latex] has no solution.\r\n\r\nAn <strong>absolute value equation<\/strong> in the form [latex]|ax+b|=c[\/latex] has the following properties:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{If }c&lt;0,|ax+b|=c\\text{ has no solution}.\\hfill \\\\ \\text{If }c=0,|ax+b|=c\\text{ has one solution}.\\hfill \\\\ \\text{If }c&gt;0,|ax+b|=c\\text{ has two solutions}.\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an absolute value equation, solve it.<\/h3>\r\n<ol>\r\n \t<li>Isolate the absolute value expression on one side of the equal sign.<\/li>\r\n \t<li>If [latex]c&gt;0[\/latex], write and solve two equations: [latex]ax+b=c[\/latex] and [latex]ax+b=-c[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the next video, we show examples of solving a simple absolute value equation.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/4g-o_-mAFpc?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving Absolute Value Equations<\/h3>\r\nSolve the following absolute value equations:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]|6x+4|=8[\/latex]<\/li>\r\n \t<li>[latex]|3x+4|=-9[\/latex]<\/li>\r\n \t<li>[latex]|3x - 5|-4=6[\/latex]<\/li>\r\n \t<li>[latex]|11x-9|=|9x+4|[\/latex]<\/li>\r\n \t<li>[latex]|-5x+10|=0[\/latex]<\/li>\r\n<\/ol>\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q67591\">Solution<\/span>\r\n<div id=\"q67591\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\na. [latex]|6x+4|=8[\/latex] Write two equations and solve each:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}6x+4\\hfill&amp;=8\\hfill&amp; 6x+4\\hfill&amp;=-8\\hfill \\\\ 6x\\hfill&amp;=4\\hfill&amp; 6x\\hfill&amp;=-12\\hfill \\\\ x\\hfill&amp;=\\frac{2}{3}\\hfill&amp; x\\hfill&amp;=-2\\hfill \\end{array}[\/latex]<\/p>\r\nThe two solutions are [latex]x=\\frac{2}{3}[\/latex], [latex]x=-2[\/latex].\r\n\r\n&nbsp;\r\n\r\nb. [latex]|3x+4|=-9[\/latex]\r\n\r\nThere is no solution as an absolute value cannot be negative.\r\n\r\n&nbsp;\r\n\r\nc. [latex]|3x - 5|-4=6[\/latex]\r\n\r\nIsolate the absolute value expression and then write two equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}\\hfill &amp; |3x - 5|-4=6\\hfill &amp; \\hfill \\\\ \\hfill &amp; |3x - 5|=10\\hfill &amp; \\hfill \\\\ \\hfill &amp; \\hfill &amp; \\hfill \\\\ 3x - 5=10\\hfill &amp; \\hfill &amp; 3x - 5=-10\\hfill \\\\ 3x=15\\hfill &amp; \\hfill &amp; 3x=-5\\hfill \\\\ x=5\\hfill &amp; \\hfill &amp; x=-\\frac{5}{3}\\hfill \\end{array}[\/latex]<\/div>\r\nThere are two solutions: [latex]x=5[\/latex], [latex]x=-\\frac{5}{3}[\/latex].\r\n\r\n&nbsp;\r\n\r\nd. [latex]|11x-9|=|9x+4|[\/latex]\r\n\r\nThe absolute value on the left is already isolated, so now we can write two equations.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}\\hfill &amp; \\hfill &amp; \\hfill \\\\ 11x-9=9x+4\\hfill &amp; \\hfill &amp; 11x-9=-(9x+4)\\hfill \\\\ 11x-9=9x+4\\hfill &amp; \\hfill &amp; 11x-9=-9x-4\\hfill \\\\2x=13\\hfill &amp; \\hfill &amp; 20x=5\\hfill \\\\ x=\\frac{13}{2}\\hfill &amp; \\hfill &amp; x=\\frac{5}{20}=\\frac{1}{4}\\hfill \\end{array}[\/latex]<\/div>\r\nThere are two solutions: [latex]x=\\frac{13}{2}[\/latex], [latex]x=\\frac{1}{4}[\/latex].\r\n\r\n&nbsp;\r\n\r\ne. [latex]|-5x+10|=0[\/latex]\r\n\r\nThe equation is set equal to zero, so we have to write only one equation.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}-5x+10\\hfill&amp;=0\\hfill \\\\ -5x\\hfill&amp;=-10\\hfill \\\\ x\\hfill&amp;=2\\hfill \\end{array}[\/latex]<\/div>\r\nThere is one solution: [latex]x=2[\/latex].\r\n\r\n<\/div>\r\n<\/div>\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\n<iframe src=\"https:\/\/www.youtube.com\/embed\/-HrOMkIiSfU?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/2bEA7HoDfpk?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the absolute value equation: [latex]|1 - 4x|+8=13[\/latex].\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q567620\">Solution<\/span>\r\n<div id=\"q567620\" class=\"hidden-answer\" style=\"display: none;\">[latex]x=-1[\/latex], [latex]x=\\frac{3}{2}[\/latex]<\/div>\r\n<\/div>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=60839&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Absolute value equations with no solutions<\/h3>\r\nAs we are solving absolute value equations it is important to be aware of special cases. An absolute value is defined as the distance from 0 on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>. [latex]7+\\left|2x-5\\right|=4[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q173733\">Show Solution<\/span>\r\n<div id=\"q173733\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nNotice absolute value is not alone. Subtract [latex]7[\/latex] from each side to isolate the absolute value.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}7+\\left|2x-5\\right|=4\\,\\,\\,\\,\\\\\\underline{\\,-7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,}\\\\\\left|2x-5\\right|=-3\\end{array}[\/latex]<\/p>\r\nResult of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or DNE.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>. [latex]-\\frac{1}{2}\\left|x+3\\right|=6[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q173738\">Show Solution<\/span>\r\n<div id=\"q173738\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nNotice absolute value is not alone, multiply both sides by the reciprocal of [latex]-\\frac{1}{2}[\/latex], which is [latex]-2[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-\\frac{1}{2}\\left|x+3\\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\left(-2\\right)-\\frac{1}{2}\\left|x+3\\right|=\\left(-2\\right)6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left|x+3\\right|=-12\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAgain, we have a result where an absolute value is negative!\r\n\r\nThere is no solution to this equation, or DNE.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn this last video, we show more examples of absolute value equations that have no solutions.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/T-z5cQ58I_g?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h3>Solve inequalities containing absolute values<\/h3>\r\nLet\u2019s apply what you know about solving equations that contain absolute values and what you know about inequalities to solve inequalities that contain absolute values. Let\u2019s start with a simple inequality.\r\n<p style=\"text-align: center;\">[latex]\\left|x\\right|\\leq 4[\/latex]<\/p>\r\nThis inequality is read, \u201cthe absolute value of <i>x <\/i>is less than or equal to 4.\u201d If you are asked to solve for <i>x<\/i>, you want to find out what values of <i>x <\/i>are 4 units or less away from 0 on a number line. You could start by thinking about the number line and what values of <i>x <\/i>would satisfy this equation.\r\n\r\n4 and [latex]\u22124[\/latex] are both four units away from 0, so they are solutions. 3 and [latex]\u22123[\/latex] are also solutions because each of these values is less than 4 units away from 0. So are 1 and [latex]\u22121[\/latex], 0.5 and [latex]\u22120.5[\/latex], and so on\u2014there are an infinite number of values for <i>x<\/i> that will satisfy this inequality.\r\n\r\nThe graph of this inequality will have two closed circles, at 4 and [latex]\u22124[\/latex]. The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality.\r\n<div class=\"wp-nocaption aligncenter\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182900\/image097-1.jpg\" alt=\"Number line. Closed blue circles on negative 4 and 4. Blue line between closed blue circles.\" width=\"575\" height=\"53\" \/><\/div>\r\nThe solution can be written this way:\r\n\r\nInequality: [latex]-4\\leq x\\leq4[\/latex]\r\n\r\nInterval: [latex]\\left[-4,4\\right][\/latex]\r\n\r\nThe situation is a little different when the inequality sign is \u201cgreater than\u201d or \u201cgreater than or equal to.\u201d Consider the simple inequality [latex]\\left|x\\right|&gt;3[\/latex]. Again, you could think of the number line and what values of <i>x<\/i> are greater than 3 units away from zero. This time, 3 and [latex]\u22123[\/latex] are not included in the solution, so there are open circles on both of these values. 2 and [latex]\u22122[\/latex] would not be solutions because they are not more than 3 units away from 0. But 5 and [latex]\u22125[\/latex] would work, and so would all of the values extending to the left of [latex]\u22123[\/latex] and to the right of 3. The graph would look like the one below.\r\n<div class=\"wp-nocaption aligncenter\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182902\/image098-1.jpg\" alt=\"Number line. Open blue circles on negative three and three. Blue arrow through all numbers less than negative 3. Blue arrow through all numbers greater than 3.\" width=\"575\" height=\"53\" \/><\/div>\r\nThe solution to this inequality can be written this way:\r\n\r\nInequality<i>:<\/i> [latex]x&lt;\u22123[\/latex] or [latex]x&gt;3[\/latex].\r\n\r\nInterval: [latex]\\left(-\\infty, -3\\right)\\cup\\left(3,\\infty\\right)[\/latex]\r\n\r\nIn the following video, you will see examples of how to solve and express the solution to absolute value inequalities involving both AND and OR.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/0cXxATY2S-k?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h3>Writing Solutions to Absolute Value Inequalities<\/h3>\r\nFor any positive value of <i>a <\/i>and <em>x,<\/em> a single variable, or any algebraic expression:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Absolute Value Inequality<\/strong><\/td>\r\n<td><strong>Equivalent Inequality<\/strong><\/td>\r\n<td><strong>Interval Notation<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left|{ x }\\right|\\le{ a}[\/latex]<\/td>\r\n<td>[latex]{ -a}\\le{x}\\le{ a}[\/latex]<\/td>\r\n<td>[latex]\\left[-a, a\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left| x \\right|\\lt{a}[\/latex]<\/td>\r\n<td>[latex]{ -a}\\lt{x}\\lt{ a}[\/latex]<\/td>\r\n<td>[latex]\\left(-a, a\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left| x \\right|\\ge{ a}[\/latex]<\/td>\r\n<td>[latex]{x}\\le\\text{\u2212a}[\/latex] or [latex]{x}\\ge{ a}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty,-a\\right]\\cup\\left[a,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left| x \\right|\\gt\\text{a}[\/latex]<\/td>\r\n<td>[latex]\\displaystyle{x}\\lt\\text{\u2212a}[\/latex] or [latex]{x}\\gt{ a}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty,-a\\right)\\cup\\left(a,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet\u2019s look at a few more examples of inequalities containing absolute values.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>. [latex]\\left|x+3\\right|\\gt4[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q867809\">Show Solution<\/span>\r\n<div id=\"q867809\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nSince this is a \u201cgreater than\u201d inequality, the solution can be rewritten according to the \u201cgreater than\u201d rule.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle x+3&lt;-4\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,x+3&gt;4[\/latex]<\/p>\r\nSolve each inequality.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x+3&lt;-4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x+3&gt;4\\\\\\underline{\\,\\,\\,\\,-3\\,\\,\\,\\,\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,-3\\,\\,-3}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,&lt;-7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,\\,\\,&gt;1\\\\\\\\x&lt;-7\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x&gt;1\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nCheck the solutions in the original equation to be sure they work. Check the end point of the first related equation, [latex]\u22127[\/latex] and the end point of the second related equation, 1.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\left| x+3 \\right|&gt;4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| x+3 \\right|&gt;4\\\\\\left| -7+3 \\right|=4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 1+3 \\right|=4\\\\\\,\\,\\,\\,\\,\\,\\,\\left| -4 \\right|=4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 4 \\right|=4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4=4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4=4\\end{array}[\/latex]<\/p>\r\nTry [latex]\u221210[\/latex], a value less than [latex]\u22127[\/latex], and 5, a value greater than 1, to check the inequality.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,\\left| x+3 \\right|&gt;4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| x+3 \\right|&gt;4\\\\\\left| -10+3 \\right|&gt;4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 5+3 \\right|&gt;4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -7 \\right|&gt;4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 8 \\right|&gt;4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7&gt;4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8&gt;4\\end{array}[\/latex]<\/p>\r\nBoth solutions check!\r\n<h4>Answer<\/h4>\r\nInequality: [latex] \\displaystyle x&lt;-7\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,x&gt;1[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty, -7\\right)\\cup\\left(1,\\infty\\right)[\/latex]\r\n\r\nGraph:\r\n<div class=\"wp-nocaption wp-image-3966 aligncenter\"><img class=\"wp-image-3966 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182903\/Screen-Shot-2016-05-10-at-5.13.59-PM-300x37.png\" alt=\"x 1\" width=\"551\" height=\"68\" \/><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>y. <\/i>[latex] \\displaystyle \\mathsf{3}\\left| \\mathsf{2}\\mathrm{y}\\mathsf{+6} \\right|-\\mathsf{9&lt;27}[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q632256\">Show Solution<\/span>\r\n<div id=\"q632256\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nBegin to isolate the absolute value by adding 9 to both sides of the inequality.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}3\\left| 2y+6 \\right|-9&lt;27\\\\\\underline{\\,\\,+9\\,\\,\\,+9}\\\\3\\left| 2y+6 \\right|\\,\\,\\,\\,\\,\\,\\,\\,&lt;36\\end{array}[\/latex]<\/p>\r\nDivide both sides by 3 to isolate the absolute value.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underline{3\\left| 2y+6 \\right|}\\,&lt;\\underline{36}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 2y+6 \\right|&lt;12\\end{array}[\/latex]<\/p>\r\nWrite the absolute value inequality using the \u201cless than\u201d rule. Subtract 6 from each part of the inequality.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-12&lt;2y+6&lt;12\\\\\\underline{\\,\\,-6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6\\,\\,\\,-6}\\\\-18\\,&lt;\\,2y\\,\\,\\,\\,\\,\\,\\,\\,\\,&lt;\\,\\,6\\,\\end{array}[\/latex]<\/p>\r\nDivide by 2 to isolate the variable.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underline{-18}&lt;\\underline{2y}&lt;\\underline{\\,6\\,}\\\\2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\-9&lt;\\,\\,y\\,\\,\\,\\,&lt;\\,3\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex] \\displaystyle -9&lt;\\,\\,y\\,\\,&lt;3[\/latex]\r\n\r\nInterval: [latex]\\left(-9,3\\right)[\/latex]\r\n\r\nGraph:\r\n<div class=\"wp-nocaption aligncenter wp-image-3967\"><img class=\"aligncenter wp-image-3967\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182905\/Screen-Shot-2016-05-10-at-5.17.57-PM-300x30.png\" alt=\"Open dot on negative 9 and open dot on 3, with a line through all numbers between 9 and 3.\" width=\"620\" height=\"62\" \/><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the following video, you will see an example of solving multi-step absolute value inequalities involving an OR situation.\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/d-hUviSkmqE?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\nIn the following video you will see an example of solving multi-step absolute value inequalities involving an AND situation.\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/ttUaRf-GzpM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\nIn the last video that follows, you will see an example of solving an absolute value inequality where you need to isolate the absolute value first.\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/5jRUuiMUxWQ?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h3>Identify cases of inequalities containing absolute values that have no solutions<\/h3>\r\nAs with equations, there may be instances in which there is no solution to an inequality.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x. <\/i>[latex]\\left|2x+3\\right|+9\\leq 7[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q931656\">Show Solution<\/span>\r\n<div id=\"q931656\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nIsolate the absolute value by subtracting 9 from both sides of the inequality.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\left| 2x+3 \\right|+9\\,\\le \\,\\,\\,7\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-9\\,\\,\\,\\,\\,-9}\\\\\\,\\,\\,\\,\\,\\,\\,\\left| 2x+3 \\right|\\,\\,\\,\\le -2\\,\\end{array}[\/latex]<\/p>\r\nThe absolute value of a quantity can never be a negative number, so there is no solution to the inequality.\r\n<h4>Answer<\/h4>\r\nNo solution\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Summary<\/h2>\r\nAbsolute inequalities can be solved by rewriting them using compound inequalities. The first step to solving absolute inequalities is to isolate the absolute value. The next step is to decide whether you are working with an OR inequality or an AND inequality. If the inequality is greater than a number, we will use OR. If the inequality is less than a number, we will use AND. Remember that if we end up with an absolute value greater than or less than a negative number, there is no solution.\r\n<h2><\/h2>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Solve equations containing absolute values<\/li>\n<li>Recognize when a linear equation that contains absolute value does not have a solution<\/li>\n<li>Solve inequalities containing absolute values<\/li>\n<\/ul>\n<\/div>\n<h3>Review of Absolute Value<\/h3>\n<p>Remember that absolute value is telling you how far away a number is from 0. Therefore, [latex]|7|=7[\/latex] and [latex]|-7|=7[\/latex] since the distance from -7 to 0 and the distance from 0 to 7 are both 7.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Absolute Values<\/h3>\n<p>Evaluate:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\left|-\\frac{3}{14}\\right|[\/latex]<\/li>\n<li>[latex]|7+10(-2)|[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q67590\">Solution<\/span><\/p>\n<div id=\"q67590\" class=\"hidden-answer\" style=\"display: none;\">\n<p>a. [latex]\\left|-\\frac{3}{14}\\right|[\/latex]<\/p>\n<p>The absolute value of a real number [latex]a[\/latex], denoted by [latex]|a|[\/latex], is the distance from 0 to [latex]a[\/latex] on the number line. Because absolute value describes a distance, it is never negative.<\/p>\n<p>[latex]\\left|-\\frac{3}{14}\\right|[\/latex] is read as &#8220;the absolute value of [latex]-\\frac{3}{14}[\/latex]&#8220;. The absolute value of [latex]-\\frac{3}{14}[\/latex] is equal to the distance from 0 to [latex]-\\frac{3}{14}[\/latex] on a number line.<\/p>\n<p>The distance from 0 to [latex]-\\frac{3}{14}[\/latex] on the number line is [latex]\\frac{3}{14}[\/latex].<\/p>\n<p>Therefore, [latex]\\left|-\\frac{3}{14}\\right|=\\frac{3}{14}[\/latex].<\/p>\n<p>b. [latex]|7+10(-2)|[\/latex]<\/p>\n<p>Perform the operations inside the absolute value. Following the order of operations, first perform the multiplication.<br \/>\n[latex]|7+10(-2)|=|7+(-20)|[\/latex]<\/p>\n<p>Now perform the addition.<br \/>\n[latex]|7+10(-2)|=|7+(-20)|=|-13|[\/latex]<\/p>\n<p>Finish by evaluating the absolute value.<br \/>\n[latex]|7+10(-2)|=|7+(-20)|=|-13|=13[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Solving an Absolute Value Equation<\/h3>\n<p>Next, we will learn how to slve an <strong>absolute value equation<\/strong>. To solve an equation such as [latex]|2x - 6|=8[\/latex], we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is [latex]8[\/latex] or [latex]-8[\/latex]. This leads to two different equations we can solve independently.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}2x - 6=8\\hfill & \\text{ or }\\hfill & 2x - 6=-8\\hfill \\\\ 2x=14\\hfill & \\hfill & 2x=-2\\hfill \\\\ x=7\\hfill & \\hfill & x=-1\\hfill \\end{array}[\/latex]<\/div>\n<p>Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Absolute Value Equations<\/h3>\n<p>The absolute value of <em>x <\/em>is written as [latex]|x|[\/latex]. It has the following properties:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{If } x\\ge 0,\\text{ then }|x|=x.\\hfill \\\\ \\text{If }x<0,\\text{ then }|x|=-x.\\hfill \\end{array}[\/latex]<\/div>\n<p>For real numbers [latex]A[\/latex] and [latex]B[\/latex], an equation of the form [latex]|A|=B[\/latex], with [latex]B\\ge 0[\/latex], will have solutions when [latex]A=B[\/latex] or [latex]A=-B[\/latex]. If [latex]B<0[\/latex], the equation [latex]|A|=B[\/latex] has no solution.\n\nAn <strong>absolute value equation<\/strong> in the form [latex]|ax+b|=c[\/latex] has the following properties:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{If }c<0,|ax+b|=c\\text{ has no solution}.\\hfill \\\\ \\text{If }c=0,|ax+b|=c\\text{ has one solution}.\\hfill \\\\ \\text{If }c>0,|ax+b|=c\\text{ has two solutions}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an absolute value equation, solve it.<\/h3>\n<ol>\n<li>Isolate the absolute value expression on one side of the equal sign.<\/li>\n<li>If [latex]c>0[\/latex], write and solve two equations: [latex]ax+b=c[\/latex] and [latex]ax+b=-c[\/latex].<\/li>\n<\/ol>\n<\/div>\n<p>In the next video, we show examples of solving a simple absolute value equation.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/4g-o_-mAFpc?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Absolute Value Equations<\/h3>\n<p>Solve the following absolute value equations:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]|6x+4|=8[\/latex]<\/li>\n<li>[latex]|3x+4|=-9[\/latex]<\/li>\n<li>[latex]|3x - 5|-4=6[\/latex]<\/li>\n<li>[latex]|11x-9|=|9x+4|[\/latex]<\/li>\n<li>[latex]|-5x+10|=0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q67591\">Solution<\/span><\/p>\n<div id=\"q67591\" class=\"hidden-answer\" style=\"display: none;\">\n<p>a. [latex]|6x+4|=8[\/latex] Write two equations and solve each:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}6x+4\\hfill&=8\\hfill& 6x+4\\hfill&=-8\\hfill \\\\ 6x\\hfill&=4\\hfill& 6x\\hfill&=-12\\hfill \\\\ x\\hfill&=\\frac{2}{3}\\hfill& x\\hfill&=-2\\hfill \\end{array}[\/latex]<\/p>\n<p>The two solutions are [latex]x=\\frac{2}{3}[\/latex], [latex]x=-2[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>b. [latex]|3x+4|=-9[\/latex]<\/p>\n<p>There is no solution as an absolute value cannot be negative.<\/p>\n<p>&nbsp;<\/p>\n<p>c. [latex]|3x - 5|-4=6[\/latex]<\/p>\n<p>Isolate the absolute value expression and then write two equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}\\hfill & |3x - 5|-4=6\\hfill & \\hfill \\\\ \\hfill & |3x - 5|=10\\hfill & \\hfill \\\\ \\hfill & \\hfill & \\hfill \\\\ 3x - 5=10\\hfill & \\hfill & 3x - 5=-10\\hfill \\\\ 3x=15\\hfill & \\hfill & 3x=-5\\hfill \\\\ x=5\\hfill & \\hfill & x=-\\frac{5}{3}\\hfill \\end{array}[\/latex]<\/div>\n<p>There are two solutions: [latex]x=5[\/latex], [latex]x=-\\frac{5}{3}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>d. [latex]|11x-9|=|9x+4|[\/latex]<\/p>\n<p>The absolute value on the left is already isolated, so now we can write two equations.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}\\hfill & \\hfill & \\hfill \\\\ 11x-9=9x+4\\hfill & \\hfill & 11x-9=-(9x+4)\\hfill \\\\ 11x-9=9x+4\\hfill & \\hfill & 11x-9=-9x-4\\hfill \\\\2x=13\\hfill & \\hfill & 20x=5\\hfill \\\\ x=\\frac{13}{2}\\hfill & \\hfill & x=\\frac{5}{20}=\\frac{1}{4}\\hfill \\end{array}[\/latex]<\/div>\n<p>There are two solutions: [latex]x=\\frac{13}{2}[\/latex], [latex]x=\\frac{1}{4}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>e. [latex]|-5x+10|=0[\/latex]<\/p>\n<p>The equation is set equal to zero, so we have to write only one equation.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}-5x+10\\hfill&=0\\hfill \\\\ -5x\\hfill&=-10\\hfill \\\\ x\\hfill&=2\\hfill \\end{array}[\/latex]<\/div>\n<p>There is one solution: [latex]x=2[\/latex].<\/p>\n<\/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\" src=\"https:\/\/www.youtube.com\/embed\/-HrOMkIiSfU?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/2bEA7HoDfpk?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the absolute value equation: [latex]|1 - 4x|+8=13[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q567620\">Solution<\/span><\/p>\n<div id=\"q567620\" class=\"hidden-answer\" style=\"display: none;\">[latex]x=-1[\/latex], [latex]x=\\frac{3}{2}[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=60839&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3>Absolute value equations with no solutions<\/h3>\n<p>As we are solving absolute value equations it is important to be aware of special cases. An absolute value is defined as the distance from 0 on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>. [latex]7+\\left|2x-5\\right|=4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q173733\">Show Solution<\/span><\/p>\n<div id=\"q173733\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Notice absolute value is not alone. Subtract [latex]7[\/latex] from each side to isolate the absolute value.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}7+\\left|2x-5\\right|=4\\,\\,\\,\\,\\\\\\underline{\\,-7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,}\\\\\\left|2x-5\\right|=-3\\end{array}[\/latex]<\/p>\n<p>Result of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or DNE.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>. [latex]-\\frac{1}{2}\\left|x+3\\right|=6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q173738\">Show Solution<\/span><\/p>\n<div id=\"q173738\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Notice absolute value is not alone, multiply both sides by the reciprocal of [latex]-\\frac{1}{2}[\/latex], which is [latex]-2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-\\frac{1}{2}\\left|x+3\\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\left(-2\\right)-\\frac{1}{2}\\left|x+3\\right|=\\left(-2\\right)6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left|x+3\\right|=-12\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Again, we have a result where an absolute value is negative!<\/p>\n<p>There is no solution to this equation, or DNE.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In this last video, we show more examples of absolute value equations that have no solutions.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/T-z5cQ58I_g?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Solve inequalities containing absolute values<\/h3>\n<p>Let\u2019s apply what you know about solving equations that contain absolute values and what you know about inequalities to solve inequalities that contain absolute values. Let\u2019s start with a simple inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\left|x\\right|\\leq 4[\/latex]<\/p>\n<p>This inequality is read, \u201cthe absolute value of <i>x <\/i>is less than or equal to 4.\u201d If you are asked to solve for <i>x<\/i>, you want to find out what values of <i>x <\/i>are 4 units or less away from 0 on a number line. You could start by thinking about the number line and what values of <i>x <\/i>would satisfy this equation.<\/p>\n<p>4 and [latex]\u22124[\/latex] are both four units away from 0, so they are solutions. 3 and [latex]\u22123[\/latex] are also solutions because each of these values is less than 4 units away from 0. So are 1 and [latex]\u22121[\/latex], 0.5 and [latex]\u22120.5[\/latex], and so on\u2014there are an infinite number of values for <i>x<\/i> that will satisfy this inequality.<\/p>\n<p>The graph of this inequality will have two closed circles, at 4 and [latex]\u22124[\/latex]. The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality.<\/p>\n<div class=\"wp-nocaption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182900\/image097-1.jpg\" alt=\"Number line. Closed blue circles on negative 4 and 4. Blue line between closed blue circles.\" width=\"575\" height=\"53\" \/><\/div>\n<p>The solution can be written this way:<\/p>\n<p>Inequality: [latex]-4\\leq x\\leq4[\/latex]<\/p>\n<p>Interval: [latex]\\left[-4,4\\right][\/latex]<\/p>\n<p>The situation is a little different when the inequality sign is \u201cgreater than\u201d or \u201cgreater than or equal to.\u201d Consider the simple inequality [latex]\\left|x\\right|>3[\/latex]. Again, you could think of the number line and what values of <i>x<\/i> are greater than 3 units away from zero. This time, 3 and [latex]\u22123[\/latex] are not included in the solution, so there are open circles on both of these values. 2 and [latex]\u22122[\/latex] would not be solutions because they are not more than 3 units away from 0. But 5 and [latex]\u22125[\/latex] would work, and so would all of the values extending to the left of [latex]\u22123[\/latex] and to the right of 3. The graph would look like the one below.<\/p>\n<div class=\"wp-nocaption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182902\/image098-1.jpg\" alt=\"Number line. Open blue circles on negative three and three. Blue arrow through all numbers less than negative 3. Blue arrow through all numbers greater than 3.\" width=\"575\" height=\"53\" \/><\/div>\n<p>The solution to this inequality can be written this way:<\/p>\n<p>Inequality<i>:<\/i> [latex]x<\u22123[\/latex] or [latex]x>3[\/latex].<\/p>\n<p>Interval: [latex]\\left(-\\infty, -3\\right)\\cup\\left(3,\\infty\\right)[\/latex]<\/p>\n<p>In the following video, you will see examples of how to solve and express the solution to absolute value inequalities involving both AND and OR.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/0cXxATY2S-k?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Writing Solutions to Absolute Value Inequalities<\/h3>\n<p>For any positive value of <i>a <\/i>and <em>x,<\/em> a single variable, or any algebraic expression:<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Absolute Value Inequality<\/strong><\/td>\n<td><strong>Equivalent Inequality<\/strong><\/td>\n<td><strong>Interval Notation<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left|{ x }\\right|\\le{ a}[\/latex]<\/td>\n<td>[latex]{ -a}\\le{x}\\le{ a}[\/latex]<\/td>\n<td>[latex]\\left[-a, a\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left| x \\right|\\lt{a}[\/latex]<\/td>\n<td>[latex]{ -a}\\lt{x}\\lt{ a}[\/latex]<\/td>\n<td>[latex]\\left(-a, a\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left| x \\right|\\ge{ a}[\/latex]<\/td>\n<td>[latex]{x}\\le\\text{\u2212a}[\/latex] or [latex]{x}\\ge{ a}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty,-a\\right]\\cup\\left[a,\\infty\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left| x \\right|\\gt\\text{a}[\/latex]<\/td>\n<td>[latex]\\displaystyle{x}\\lt\\text{\u2212a}[\/latex] or [latex]{x}\\gt{ a}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty,-a\\right)\\cup\\left(a,\\infty\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let\u2019s look at a few more examples of inequalities containing absolute values.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>. [latex]\\left|x+3\\right|\\gt4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q867809\">Show Solution<\/span><\/p>\n<div id=\"q867809\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Since this is a \u201cgreater than\u201d inequality, the solution can be rewritten according to the \u201cgreater than\u201d rule.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle x+3<-4\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,x+3>4[\/latex]<\/p>\n<p>Solve each inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x+3<-4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x+3>4\\\\\\underline{\\,\\,\\,\\,-3\\,\\,\\,\\,\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,-3\\,\\,-3}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,<-7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,\\,\\,>1\\\\\\\\x<-7\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x>1\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Check the solutions in the original equation to be sure they work. Check the end point of the first related equation, [latex]\u22127[\/latex] and the end point of the second related equation, 1.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,\\left| x+3 \\right|>4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| x+3 \\right|>4\\\\\\left| -7+3 \\right|=4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 1+3 \\right|=4\\\\\\,\\,\\,\\,\\,\\,\\,\\left| -4 \\right|=4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 4 \\right|=4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4=4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4=4\\end{array}[\/latex]<\/p>\n<p>Try [latex]\u221210[\/latex], a value less than [latex]\u22127[\/latex], and 5, a value greater than 1, to check the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,\\left| x+3 \\right|>4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| x+3 \\right|>4\\\\\\left| -10+3 \\right|>4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 5+3 \\right|>4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -7 \\right|>4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 8 \\right|>4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7>4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8>4\\end{array}[\/latex]<\/p>\n<p>Both solutions check!<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]\\displaystyle x<-7\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,x>1[\/latex]<\/p>\n<p>Interval: [latex]\\left(-\\infty, -7\\right)\\cup\\left(1,\\infty\\right)[\/latex]<\/p>\n<p>Graph:<\/p>\n<div class=\"wp-nocaption wp-image-3966 aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3966 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182903\/Screen-Shot-2016-05-10-at-5.13.59-PM-300x37.png\" alt=\"x 1\" width=\"551\" height=\"68\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>y. <\/i>[latex]\\displaystyle \\mathsf{3}\\left| \\mathsf{2}\\mathrm{y}\\mathsf{+6} \\right|-\\mathsf{9<27}[\/latex]\n\n\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q632256\">Show Solution<\/span><\/p>\n<div id=\"q632256\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Begin to isolate the absolute value by adding 9 to both sides of the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}3\\left| 2y+6 \\right|-9<27\\\\\\underline{\\,\\,+9\\,\\,\\,+9}\\\\3\\left| 2y+6 \\right|\\,\\,\\,\\,\\,\\,\\,\\,<36\\end{array}[\/latex]<\/p>\n<p>Divide both sides by 3 to isolate the absolute value.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underline{3\\left| 2y+6 \\right|}\\,<\\underline{36}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 2y+6 \\right|<12\\end{array}[\/latex]<\/p>\n<p>Write the absolute value inequality using the \u201cless than\u201d rule. Subtract 6 from each part of the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-12<2y+6<12\\\\\\underline{\\,\\,-6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6\\,\\,\\,-6}\\\\-18\\,<\\,2y\\,\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,6\\,\\end{array}[\/latex]<\/p>\n<p>Divide by 2 to isolate the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underline{-18}<\\underline{2y}<\\underline{\\,6\\,}\\\\2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\-9<\\,\\,y\\,\\,\\,\\,<\\,3\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]\\displaystyle -9<\\,\\,y\\,\\,<3[\/latex]\n\nInterval: [latex]\\left(-9,3\\right)[\/latex]\n\nGraph:\n\n\n<div class=\"wp-nocaption aligncenter wp-image-3967\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3967\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182905\/Screen-Shot-2016-05-10-at-5.17.57-PM-300x30.png\" alt=\"Open dot on negative 9 and open dot on 3, with a line through all numbers between 9 and 3.\" width=\"620\" height=\"62\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you will see an example of solving multi-step absolute value inequalities involving an OR situation.<br \/>\n<iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/d-hUviSkmqE?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\nIn the following video you will see an example of solving multi-step absolute value inequalities involving an AND situation.<br \/>\n<iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/ttUaRf-GzpM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\nIn the last video that follows, you will see an example of solving an absolute value inequality where you need to isolate the absolute value first.<br \/>\n<iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/5jRUuiMUxWQ?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Identify cases of inequalities containing absolute values that have no solutions<\/h3>\n<p>As with equations, there may be instances in which there is no solution to an inequality.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x. <\/i>[latex]\\left|2x+3\\right|+9\\leq 7[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q931656\">Show Solution<\/span><\/p>\n<div id=\"q931656\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Isolate the absolute value by subtracting 9 from both sides of the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\left| 2x+3 \\right|+9\\,\\le \\,\\,\\,7\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-9\\,\\,\\,\\,\\,-9}\\\\\\,\\,\\,\\,\\,\\,\\,\\left| 2x+3 \\right|\\,\\,\\,\\le -2\\,\\end{array}[\/latex]<\/p>\n<p>The absolute value of a quantity can never be a negative number, so there is no solution to the inequality.<\/p>\n<h4>Answer<\/h4>\n<p>No solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<h2>Summary<\/h2>\n<p>Absolute inequalities can be solved by rewriting them using compound inequalities. The first step to solving absolute inequalities is to isolate the absolute value. The next step is to decide whether you are working with an OR inequality or an AND inequality. If the inequality is greater than a number, we will use OR. If the inequality is less than a number, we will use AND. Remember that if we end up with an absolute value greater than or less than a negative number, there is no solution.<\/p>\n<h2><\/h2>\n","protected":false},"author":264444,"menu_order":26,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18673","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18673","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":20,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18673\/revisions"}],"predecessor-version":[{"id":18721,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18673\/revisions\/18721"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18673\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18673"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18673"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18673"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18673"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}