{"id":1714,"date":"2021-08-27T19:08:28","date_gmt":"2021-08-27T19:08:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1714"},"modified":"2022-02-01T20:03:19","modified_gmt":"2022-02-01T20:03:19","slug":"absolute-value-equations-and-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/absolute-value-equations-and-inequalities\/","title":{"raw":"Absolute Value Equations and Inequalities","rendered":"Absolute Value Equations and Inequalities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Evaluate expressions that contain absolute value<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Solve absolute value equations<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Solve absolute value inequalities<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe saw that numbers such as [latex]5[\/latex] and [latex]-5[\/latex] are opposites because they are the same distance from [latex]0[\/latex] on the number line. They are both five units from [latex]0[\/latex]. The distance between [latex]0[\/latex] and any number on the number line is called the <strong>absolute value<\/strong> of that number.\r\n\r\nBecause distance is never negative, the absolute value of any number is never negative.\r\n\r\nThe symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[\/latex] is written as [latex]|5|[\/latex], and the absolute value of [latex]-5[\/latex] is written as [latex]|-5|[\/latex] as shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220135\/CNX_BMath_Figure_03_01_019.png\" alt=\"This figure is a number line. The points negative 5 and 5 are labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units. Also above the number line the distance from 0 to 5 is labeled as 5 units.\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Absolute Value<\/h3>\r\nThe absolute value of a number is its distance from [latex]0[\/latex] on the number line.\r\nThe absolute value of a number [latex]n[\/latex] is written as [latex]|n|[\/latex].\r\n\r\n[latex]|n|\\ge 0[\/latex] for all numbers\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3><\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>\u00a0[latex]|3|[\/latex]<\/li>\r\n \t<li>\u00a0[latex]|-44|[\/latex]<\/li>\r\n \t<li>\u00a0[latex]|0|[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"558762\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"558762\"]\r\n<table id=\"eip-id1165118025864\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|3|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex] is [latex]3[\/latex] units from zero.<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1171842979184\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|-44|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u221244[\/latex] is [latex]44[\/latex] units from zero.<\/td>\r\n<td>[latex]44[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1171842583307\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]|0|[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex] is already at zero.<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video below we show another example of how to find the absolute value of an integer.\r\n\r\n<iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7115017&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=I8bTqGmkqGI&amp;video_target=tpm-plugin-24gvgwzn-I8bTqGmkqGI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<h2>Solve Equations Containing Absolute Value<\/h2>\r\nWhen solving absolute value equations, we are asking about when the distance of an expression from [latex]0[\/latex] is a particular value. 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.\r\n\r\nConsider the equation [latex]|x|=5[\/latex]. This equation asks you to find all numbers whose distance is [latex]5[\/latex] units from [latex]0[\/latex]. Its solutions are [latex]5[\/latex] and [latex]-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 [latex]15[\/latex] 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 [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\n<p align=\"center\">The following video provides worked examples of solving linear equations with absolute value terms.<\/p>\r\nhttps:\/\/youtu.be\/U-7fF-W8_xE\r\n\r\nNow we will see how to solve equations with absolute value that include 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<p align=\"center\">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>\r\nhttps:\/\/youtu.be\/CTLnJ955xzc\r\n\r\nSometimes an absolute value equation has no solution. For example, [latex]|x| = -2[\/latex] has no solution since the absolute value of a number is never negative.\r\n<h2>Solve Inequalities Containing Absolute Value<\/h2>\r\nLet us apply what you know about solving equations that contain absolute value and what you know about inequalities to solve inequalities that contain absolute value. Let us 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\u00a0[latex]4[\/latex].\u201d If you are asked to solve for <i>x<\/i>, you want to find out what values of <i>x <\/i>are\u00a0[latex]4[\/latex] units or less away from\u00a0[latex]0[\/latex] 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\n[latex]4[\/latex] and [latex]\u22124[\/latex] are both four units away from\u00a0[latex]0[\/latex], so they are solutions.\u00a0[latex]3[\/latex] and [latex]\u22123[\/latex] are also solutions because each of these values is less than [latex]4[\/latex] units away from\u00a0[latex]0[\/latex]. So are\u00a0[latex]1[\/latex] and [latex]\u22121[\/latex], [latex]0.5[\/latex] 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\u00a0[latex]4[\/latex] 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\r\n<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\" \/>\r\n\r\nThe solution can be written this way:\r\n\r\nInequality notation: [latex]-4\\leq x\\leq4[\/latex]\r\n\r\nInterval notation: [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\u00a0[latex]3[\/latex] units away from zero. This time,\u00a0[latex]3[\/latex] and [latex]\u22123[\/latex] are not included in the solution, so there are open circles on both of these values.\u00a0[latex]2[\/latex] and [latex]\u22122[\/latex] would not be solutions because they are not more than [latex]3[\/latex] units away from\u00a0[latex]0[\/latex]. But\u00a0[latex]5[\/latex] 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\u00a0[latex]3[\/latex]. The graph would look like the one below.\r\n\r\n<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\" \/>\r\n\r\nThe solution to this inequality can be written this way:\r\n\r\nInequality notation<i>:<\/i> [latex]x&lt;\u22123[\/latex] or [latex]x&gt;3[\/latex].\r\n\r\nInterval notation: [latex]\\left(-\\infty, -3\\right)\\cup\\left(3,\\infty\\right)[\/latex]\r\n\r\nFor any positive value of <i>a\u00a0<\/i>and\u00a0<em>x,<\/em>\u00a0a 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>\u00a0[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]\u00a0or [latex]{x}\\gt{ a}[\/latex]<\/td>\r\n<td>\u00a0[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 another example.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]x[\/latex]: [latex]\\left|x+3\\right|\\gt4[\/latex]\r\n\r\n[reveal-answer q=\"867809\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867809\"]\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\r\nInequality notation: [latex] \\displaystyle x&lt;-7\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,x&gt;1[\/latex]\r\n\r\nInterval notation: [latex]\\left(-\\infty, -7\\right)\\cup\\left(1,\\infty\\right)[\/latex]\r\n\r\nGraph:\r\n\r\n<img class=\"aligncenter wp-image-3966\" 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 is less than negative 7 or x is greater than 1 on a number line. An arrow begins with an open dot on negative 7 and extends to the left. An arrow begins with an open dot on 1 and extends to the right.\" width=\"551\" height=\"68\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<strong>Caution:<\/strong> One difference between solving an equation and solving an inequality, is that if we multiply or divide both sides by a negative number, the sense of the inequality is reversed. For example, [latex]-1 &lt; 3[\/latex]. Suppose we multiply each of these numbers by [latex]-2[\/latex]. [latex](-2)(-1)=2[\/latex], and [latex](-2)(3)=-6[\/latex]. Now we have [latex]2 &gt; -6[\/latex].","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Evaluate expressions that contain absolute value<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Solve absolute value equations<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Solve absolute value inequalities<\/li>\n<\/ul>\n<\/div>\n<p>We saw that numbers such as [latex]5[\/latex] and [latex]-5[\/latex] are opposites because they are the same distance from [latex]0[\/latex] on the number line. They are both five units from [latex]0[\/latex]. The distance between [latex]0[\/latex] and any number on the number line is called the <strong>absolute value<\/strong> of that number.<\/p>\n<p>Because distance is never negative, the absolute value of any number is never negative.<\/p>\n<p>The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[\/latex] is written as [latex]|5|[\/latex], and the absolute value of [latex]-5[\/latex] is written as [latex]|-5|[\/latex] as shown below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220135\/CNX_BMath_Figure_03_01_019.png\" alt=\"This figure is a number line. The points negative 5 and 5 are labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units. Also above the number line the distance from 0 to 5 is labeled as 5 units.\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Absolute Value<\/h3>\n<p>The absolute value of a number is its distance from [latex]0[\/latex] on the number line.<br \/>\nThe absolute value of a number [latex]n[\/latex] is written as [latex]|n|[\/latex].<\/p>\n<p>[latex]|n|\\ge 0[\/latex] for all numbers<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3><\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>\u00a0[latex]|3|[\/latex]<\/li>\n<li>\u00a0[latex]|-44|[\/latex]<\/li>\n<li>\u00a0[latex]|0|[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q558762\">Show Answer<\/span><\/p>\n<div id=\"q558762\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1165118025864\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]|3|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex] is [latex]3[\/latex] units from zero.<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1171842979184\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]|-44|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u221244[\/latex] is [latex]44[\/latex] units from zero.<\/td>\n<td>[latex]44[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1171842583307\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]|0|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex] is already at zero.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video below we show another example of how to find the absolute value of an integer.<\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7115017&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=I8bTqGmkqGI&amp;video_target=tpm-plugin-24gvgwzn-I8bTqGmkqGI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<h2>Solve Equations Containing Absolute Value<\/h2>\n<p>When solving absolute value equations, we are asking about when the distance of an expression from [latex]0[\/latex] is a particular value. 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>Consider the equation [latex]|x|=5[\/latex]. This equation asks you to find all numbers whose distance is [latex]5[\/latex] units from [latex]0[\/latex]. Its solutions are [latex]5[\/latex] and [latex]-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 [latex]15[\/latex] 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 [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 style=\"text-align: center;\">The following video provides worked examples of solving linear equations with absolute value terms.<\/p>\n<p><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<p>Now we will see how to solve equations with absolute value that include 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<p style=\"text-align: center;\">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<p>Sometimes an absolute value equation has no solution. For example, [latex]|x| = -2[\/latex] has no solution since the absolute value of a number is never negative.<\/p>\n<h2>Solve Inequalities Containing Absolute Value<\/h2>\n<p>Let us apply what you know about solving equations that contain absolute value and what you know about inequalities to solve inequalities that contain absolute value. Let us 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\u00a0[latex]4[\/latex].\u201d If you are asked to solve for <i>x<\/i>, you want to find out what values of <i>x <\/i>are\u00a0[latex]4[\/latex] units or less away from\u00a0[latex]0[\/latex] 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>[latex]4[\/latex] and [latex]\u22124[\/latex] are both four units away from\u00a0[latex]0[\/latex], so they are solutions.\u00a0[latex]3[\/latex] and [latex]\u22123[\/latex] are also solutions because each of these values is less than [latex]4[\/latex] units away from\u00a0[latex]0[\/latex]. So are\u00a0[latex]1[\/latex] and [latex]\u22121[\/latex], [latex]0.5[\/latex] 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\u00a0[latex]4[\/latex] 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<p><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\" \/><\/p>\n<p>The solution can be written this way:<\/p>\n<p>Inequality notation: [latex]-4\\leq x\\leq4[\/latex]<\/p>\n<p>Interval notation: [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\u00a0[latex]3[\/latex] units away from zero. This time,\u00a0[latex]3[\/latex] and [latex]\u22123[\/latex] are not included in the solution, so there are open circles on both of these values.\u00a0[latex]2[\/latex] and [latex]\u22122[\/latex] would not be solutions because they are not more than [latex]3[\/latex] units away from\u00a0[latex]0[\/latex]. But\u00a0[latex]5[\/latex] 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\u00a0[latex]3[\/latex]. The graph would look like the one below.<\/p>\n<p><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\" \/><\/p>\n<p>The solution to this inequality can be written this way:<\/p>\n<p>Inequality notation<i>:<\/i> [latex]x<\u22123[\/latex] or [latex]x>3[\/latex].<\/p>\n<p>Interval notation: [latex]\\left(-\\infty, -3\\right)\\cup\\left(3,\\infty\\right)[\/latex]<\/p>\n<p>For any positive value of <i>a\u00a0<\/i>and\u00a0<em>x,<\/em>\u00a0a 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>\u00a0[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]\u00a0or [latex]{x}\\gt{ a}[\/latex]<\/td>\n<td>\u00a0[latex]\\left(-\\infty,-a\\right)\\cup\\left(a,\\infty\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let\u2019s look at another example.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]x[\/latex]: [latex]\\left|x+3\\right|\\gt4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><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<p>Inequality notation: [latex]\\displaystyle x<-7\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,x>1[\/latex]<\/p>\n<p>Interval notation: [latex]\\left(-\\infty, -7\\right)\\cup\\left(1,\\infty\\right)[\/latex]<\/p>\n<p>Graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3966\" 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 is less than negative 7 or x is greater than 1 on a number line. An arrow begins with an open dot on negative 7 and extends to the left. An arrow begins with an open dot on 1 and extends to the right.\" width=\"551\" height=\"68\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><strong>Caution:<\/strong> One difference between solving an equation and solving an inequality, is that if we multiply or divide both sides by a negative number, the sense of the inequality is reversed. For example, [latex]-1 < 3[\/latex]. Suppose we multiply each of these numbers by [latex]-2[\/latex]. [latex](-2)(-1)=2[\/latex], and [latex](-2)(3)=-6[\/latex]. Now we have [latex]2 > -6[\/latex].<\/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-1714\">\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><strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Determine the Absolute Value of an Integer. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/I8bTqGmkqGI\">https:\/\/youtu.be\/I8bTqGmkqGI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 1: Solving Absolute Value Equations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/U-7fF-W8_xE\">https:\/\/youtu.be\/U-7fF-W8_xE<\/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>Solving Absolute Value Equation Using Multiplication and Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/CTLnJ955xzc\">https:\/\/youtu.be\/CTLnJ955xzc<\/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>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\">https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction<\/a>. <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>: Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction<\/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":169134,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Determine the Absolute Value of an Integer\",\"author\":\"James Sousa 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