{"id":16151,"date":"2019-10-01T18:18:42","date_gmt":"2019-10-01T18:18:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-solve-absolute-value-inequalities-2\/"},"modified":"2020-10-22T09:13:48","modified_gmt":"2020-10-22T09:13:48","slug":"read-solve-absolute-value-inequalities-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-solve-absolute-value-inequalities-2\/","title":{"raw":"8.2.c - Solving Absolute Value Inequalities","rendered":"8.2.c &#8211; Solving Absolute Value Inequalities"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Express solutions to inequalities containing absolute values<\/li>\r\n \t<li>Identify inequalities containing absolute value that have no solutions<\/li>\r\n<\/ul>\r\n<\/div>\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\nIn the following video, you will see examples of how to solve and express the solution\u00a0to absolute value inequalities involving both <em>and<\/em> and <em>or<\/em>.\r\n\r\nhttps:\/\/youtu.be\/0cXxATY2S-k\r\n<h3>Writing Solutions to Absolute Value Inequalities<\/h3>\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 us look at a few more examples of inequalities containing absolute value.\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=\"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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]y[\/latex]: [latex] \\displaystyle \\mathsf{3}\\left| \\mathsf{2}\\mathrm{y}\\mathsf{+6} \\right|-\\mathsf{9&lt;27}[\/latex]\r\n\r\n[reveal-answer q=\"632256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"632256\"]\r\n\r\nBegin isolating 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.\u00a0Subtract 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\u00a0[latex]2[\/latex] 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 notation: [latex] \\displaystyle -9&lt;\\,\\,y\\,\\,&lt;3[\/latex]\r\n\r\nInterval notation: [latex]\\left(-9,3\\right)[\/latex]\r\n\r\nGraph:\r\n\r\n<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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see an example of solving multi-step absolute value inequalities involving an <em>or<\/em> situation.\r\n\r\n[embed]https:\/\/youtu.be\/d-hUviSkmqE[\/embed]\r\n\r\nIn the following video you will see an example of solving multi-step absolute value inequalities involving an <em>and<\/em> situation.\r\n\r\n[embed]https:\/\/youtu.be\/ttUaRf-GzpM[\/embed]\r\n\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\r\n[embed]https:\/\/youtu.be\/5jRUuiMUxWQ[\/embed]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]141784[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Identify Cases of Inequalities Containing Absolute Value That Have No Solutions<\/h3>\r\nAs with equations, there may be instances where there is no solution to an inequality.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]x[\/latex]: [latex]\\left|2x+3\\right|+9\\leq 7[\/latex]\r\n\r\n[reveal-answer q=\"931656\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"931656\"]\r\n\r\nIsolate the absolute value by subtracting\u00a0[latex]9[\/latex] 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\r\nThere is no solution[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]20573[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nInequalities containing absolute value 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 <em>or<\/em> inequality or an <em>and<\/em> inequality. If the inequality is greater than a number, we will use <em>or<\/em>. If the inequality is less than a number, we will use <em>and<\/em>. Remember that if we end up with an absolute value greater than or less than a negative number, there is no solution.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Express solutions to inequalities containing absolute values<\/li>\n<li>Identify inequalities containing absolute value that have no solutions<\/li>\n<\/ul>\n<\/div>\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>In the following video, you will see examples of how to solve and express the solution\u00a0to absolute value inequalities involving both <em>and<\/em> and <em>or<\/em>.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 1:  Solve and Graph Basic Absolute Value inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/0cXxATY2S-k?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Writing Solutions to Absolute Value Inequalities<\/h3>\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 us look at a few more examples of inequalities containing absolute value.<\/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=\"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\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]y[\/latex]: [latex]\\displaystyle \\mathsf{3}\\left| \\mathsf{2}\\mathrm{y}\\mathsf{+6} \\right|-\\mathsf{9<27}[\/latex]\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><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 isolating 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.\u00a0Subtract 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\u00a0[latex]2[\/latex] 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 notation: [latex]\\displaystyle -9<\\,\\,y\\,\\,<3[\/latex]\n\nInterval notation: [latex]\\left(-9,3\\right)[\/latex]\n\nGraph:\n\n<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\" \/><\/p>\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 <em>or<\/em> situation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 2:  Solve and Graph Absolute Value inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/d-hUviSkmqE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the following video you will see an example of solving multi-step absolute value inequalities involving an <em>and<\/em> situation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 3:  Solve and Graph Absolute Value inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ttUaRf-GzpM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In 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.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 4:  Solve and Graph Absolute Value inequalities  (Requires Isolating Abs. Value)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5jRUuiMUxWQ?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=\"ohm141784\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141784&theme=oea&iframe_resize_id=ohm141784&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>Identify Cases of Inequalities Containing Absolute Value That Have No Solutions<\/h3>\n<p>As with equations, there may be instances where there is no solution to an inequality.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]x[\/latex]: [latex]\\left|2x+3\\right|+9\\leq 7[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><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\u00a0[latex]9[\/latex] 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<p>There is no solution<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm20573\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=20573&theme=oea&iframe_resize_id=ohm20573&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Summary<\/h2>\n<p>Inequalities containing absolute value 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 <em>or<\/em> inequality or an <em>and<\/em> inequality. If the inequality is greater than a number, we will use <em>or<\/em>. If the inequality is less than a number, we will use <em>and<\/em>. Remember that if we end up with an absolute value greater than or less than a negative number, there is no solution.<\/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-16151\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Solve and Graph Basic Absolute Value inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/0cXxATY2S-k\">https:\/\/youtu.be\/0cXxATY2S-k<\/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 2: Solve and Graph Absolute Value inequalities Mathispower4u . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/d-hUviSkmqE\">https:\/\/youtu.be\/d-hUviSkmqE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 3: Solve and Graph Absolute Value inequalitie. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ttUaRf-GzpM\">https:\/\/youtu.be\/ttUaRf-GzpM<\/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 4: Solve and Graph Absolute Value inequalities (Requires Isolating Abs. Value). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/5jRUuiMUxWQ\">https:\/\/youtu.be\/5jRUuiMUxWQ<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Solve and Graph Basic Absolute Value inequalities\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/0cXxATY2S-k\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2: Solve and Graph Absolute Value inequalities Mathispower4u \",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/d-hUviSkmqE\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 3: Solve and Graph Absolute Value inequalitie\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ttUaRf-GzpM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 4: Solve and Graph Absolute Value inequalities (Requires Isolating Abs. 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