{"id":112,"date":"2023-06-21T13:22:34","date_gmt":"2023-06-21T13:22:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/compound-and-absolute-value-inequalities\/"},"modified":"2023-09-07T16:41:20","modified_gmt":"2023-09-07T16:41:20","slug":"compound-and-absolute-value-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/compound-and-absolute-value-inequalities\/","title":{"raw":"\u25aa   Compound and Absolute Value Inequalities","rendered":"\u25aa   Compound and Absolute Value Inequalities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve compound inequalities.<\/li>\r\n \t<li>Solve absolute value inequalities.<\/li>\r\n<\/ul>\r\n<\/div>\r\nA <strong>compound inequality<\/strong> includes two inequalities in one statement. A statement such as [latex]4&lt;x\\le 6[\/latex] means [latex]4&lt;x[\/latex] and [latex]x\\le 6[\/latex]. There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a Compound Inequality<\/h3>\r\nSolve the compound inequality: [latex]3\\le 2x+2&lt;6[\/latex].\r\n\r\n[reveal-answer q=\"497940\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"497940\"]\r\n\r\nThe first method is to write two separate inequalities: [latex]3\\le 2x+2[\/latex] and [latex]2x+2&lt;6[\/latex]. We solve them independently.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}3\\le 2x+2\\hfill &amp; \\text{and}\\hfill &amp; 2x+2&lt;6\\hfill \\\\ 1\\le 2x\\hfill &amp; \\hfill &amp; 2x&lt;4\\hfill \\\\ \\frac{1}{2}\\le x\\hfill &amp; \\hfill &amp; x&lt;2\\hfill \\end{array}[\/latex]<\/div>\r\nThen, we can rewrite the solution as a compound inequality, the same way the problem began.\r\n<div style=\"text-align: center;\">[latex]\\frac{1}{2}\\le x&lt;2[\/latex]<\/div>\r\nIn interval notation, the solution is written as [latex]\\left[\\frac{1}{2},2\\right)[\/latex].\r\n\r\nThe second method is to leave the compound inequality intact and perform solving procedures on the three parts at the same time.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}3\\le 2x+2&lt;6\\hfill &amp; \\hfill \\\\ 1\\le 2x&lt;4\\hfill &amp; \\text{Isolate the variable term and subtract 2 from all three parts}.\\hfill \\\\ \\frac{1}{2}\\le x&lt;2\\hfill &amp; \\text{Divide through all three parts by 2}.\\hfill \\end{array}[\/latex]<\/div>\r\nWe get the same solution: [latex]\\left[\\frac{1}{2},2\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the compound inequality [latex]4&lt;2x - 8\\le 10[\/latex].\r\n\r\n[reveal-answer q=\"265531\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"265531\"]\r\n\r\n[latex]6&lt;x\\le 9\\text{ }\\text{ }\\text{or}\\left(6,9\\right][\/latex][\/hidden-answer]\r\n\r\n[ohm_question height=\"170\"]3816[\/ohm_question]\r\n\r\n[ohm_question height=\"230\"]92608[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a Compound Inequality with the Variable in All Three Parts<\/h3>\r\nSolve the compound inequality with variables in all three parts: [latex]3+x&gt;7x - 2&gt;5x - 10[\/latex].\r\n\r\n[reveal-answer q=\"658677\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"658677\"]\r\n\r\nLet's try the first method. Write two inequalities<strong>:<\/strong>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}3+x&gt; 7x - 2\\hfill &amp; \\text{and}\\hfill &amp; 7x - 2&gt; 5x - 10\\hfill \\\\ 3&gt; 6x - 2\\hfill &amp; \\hfill &amp; 2x - 2&gt; -10\\hfill \\\\ 5&gt; 6x\\hfill &amp; \\hfill &amp; 2x&gt; -8\\hfill \\\\ \\frac{5}{6}&gt; x\\hfill &amp; \\hfill &amp; x&gt; -4\\hfill \\\\ x&lt; \\frac{5}{6}\\hfill &amp; \\hfill &amp; -4&lt; x\\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is [latex]-4&lt;x&lt;\\frac{5}{6}[\/latex] or in interval notation [latex]\\left(-4,\\frac{5}{6}\\right)[\/latex]. Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right as they appear on a number line.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225901\/CNX_CAT_Figure_02_07_003.jpg\" alt=\"A number line with the points -4 and 5\/6 labeled. Dots appear at these points and a line connects these two dots.\" width=\"487\" height=\"69\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the compound inequality: [latex]3y&lt;4 - 5y&lt;5+3y[\/latex].\r\n\r\n[reveal-answer q=\"661493\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"661493\"]\r\n\r\n[latex]\\left(-\\frac{1}{8},\\frac{1}{2}\\right)[\/latex][\/hidden-answer]\r\n\r\n[ohm_question height=\"220\"]13990[\/ohm_question]\r\n\r\n[ohm_question height=\"300\"]129529[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Solving Absolute Value Inequalities<\/h2>\r\nAs we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at [latex]\\left(-x,0\\right)[\/latex] has an absolute value of [latex]x[\/latex] as it is <em>x <\/em>units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.\r\n\r\nAn <strong>absolute value inequality<\/strong> is an equation of the form\r\n<div style=\"text-align: center;\">[latex]|A|&lt;B,|A|\\le B,|A|&gt;B,\\text{or }|A|\\ge B[\/latex],<\/div>\r\nwhere <em>A<\/em>, and sometimes <em>B<\/em>, represents an algebraic expression dependent on a variable <em>x. <\/em>Solving the inequality means finding the set of all [latex]x[\/latex] <em>-<\/em>values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.\r\n\r\nThere are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.\r\n\r\nSuppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of <em>x-<\/em>values such that the distance between [latex]x[\/latex] and 600 is less than 200. We represent the distance between [latex]x[\/latex] and 600 as [latex]|x - 600|[\/latex], and therefore, [latex]|x - 600|\\le 200[\/latex] or\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{c}-200\\le x - 600\\le 200\\\\ -200+600\\le x - 600+600\\le 200+600\\\\ 400\\le x\\le 800\\end{array}[\/latex]<\/div>\r\nThis means our returns would be between $400 and $800.\r\n\r\nTo solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Absolute Value Inequalities<\/h3>\r\nFor an algebraic expression <em>X\u00a0<\/em>and [latex]k&gt;0[\/latex], an <strong>absolute value inequality<\/strong> is an inequality of the form:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}|X|&lt; k\\text{ which is equivalent to }-k&lt; X&lt; k\\hfill \\text{ or }\\ |X|&gt; k\\text{ which is equivalent to }X&lt; -k\\text{ or }X&gt; k\\hfill \\end{array}[\/latex]<\/div>\r\nThese statements also apply to [latex]|X|\\le k[\/latex] and [latex]|X|\\ge k[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining a Number within a Prescribed Distance<\/h3>\r\nDescribe all values [latex]x[\/latex] within a distance of 4 from the number 5.\r\n\r\n[reveal-answer q=\"746497\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"746497\"]\r\n\r\nWe want the distance between [latex]x[\/latex] and 5 to be less than or equal to 4. We can draw a number line to represent the condition to be satisfied.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225903\/CNX_Precalc_Figure_01_06_002.jpg\" alt=\"A number line with one tick mark in the center labeled: 5. The tick marks on either side of the center one are not marked. Arrows extend from the center tick mark to the outer tick marks, both are labeled 4.\" width=\"487\" height=\"81\" \/>\r\n\r\nThe distance from [latex]x[\/latex] to 5 can be represented using an absolute value symbol, [latex]|x - 5|[\/latex]. Write the values of [latex]x[\/latex] that satisfy the condition as an absolute value inequality.\r\n<div style=\"text-align: center;\">[latex]|x - 5|\\le 4[\/latex]<\/div>\r\nWe need to write two inequalities as there are always two solutions to an absolute value equation.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}x - 5\\le 4\\hfill &amp; \\text{and}\\hfill &amp; x - 5\\ge -4\\hfill \\\\ x\\le 9\\hfill &amp; \\hfill &amp; x\\ge 1\\hfill \\end{array}[\/latex]<\/div>\r\nIf the solution set is [latex]x\\le 9[\/latex] and [latex]x\\ge 1[\/latex], then the solution set is an interval including all real numbers between and including 1 and 9.\r\n\r\nSo [latex]|x - 5|\\le 4[\/latex] is equivalent to [latex]\\left[1,9\\right][\/latex] in interval notation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDescribe all <em>x-<\/em>values within a distance of 3 from the number 2.\r\n\r\n[reveal-answer q=\"7507\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"7507\"]\r\n\r\n[latex]|x - 2|\\le 3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving an Absolute Value Inequality<\/h3>\r\nSolve [latex]|x - 1|\\le 3[\/latex].\r\n\r\n[reveal-answer q=\"4865\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"4865\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}|x - 1|\\le 3\\hfill \\\\ \\hfill \\\\ -3\\le x - 1\\le 3\\hfill \\\\ \\hfill \\\\ -2\\le x\\le 4\\hfill \\\\ \\hfill \\\\ \\left[-2,4\\right]\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using a Graphical Approach to Solve Absolute Value Inequalities<\/h3>\r\nGiven the equation [latex]y=-\\frac{1}{2}|4x - 5|+3[\/latex], determine the <em>x<\/em>-values for which the <em>y<\/em>-values are negative.\r\n\r\n[reveal-answer q=\"624558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624558\"]\r\n\r\nWe are trying to determine where [latex]y&lt;0[\/latex] which is when [latex]-\\frac{1}{2}|4x - 5|+3&lt;0[\/latex]. We begin by isolating the absolute value.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}-\\frac{1}{2}|4x - 5|&lt; -3\\hfill &amp; \\text{Multiply both sides by -2, and reverse the inequality}.\\hfill \\\\ |4x - 5|&gt; 6\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\r\nNext, we solve [latex]|4x - 5|=6[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}4x - 5=6\\hfill &amp; \\hfill &amp; 4x - 5=-6\\hfill \\\\ 4x=11\\hfill &amp; \\text{or}\\hfill &amp; 4x=-1\\hfill \\\\ x=\\frac{11}{4}\\hfill &amp; \\hfill &amp; x=-\\frac{1}{4}\\hfill \\end{array}[\/latex]<\/div>\r\nNow, we can examine the graph to observe where the <em>y-<\/em>values are negative. We observe where the branches are below the <em>x-<\/em>axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at [latex]x=-\\frac{1}{4}[\/latex] and [latex]x=\\frac{11}{4}[\/latex] and that the graph opens downward.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225906\/CNX_CAT_Figure_02_07_006.jpg\" alt=\"A coordinate plan with the x-axis ranging from -5 to 5 and the y-axis ranging from -4 to 4. The function y = -1\/2|4x \u2013 5| + 3 is graphed. An open circle appears at the point -0.25 and an arrow\" width=\"487\" height=\"363\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve [latex]-2|k - 4|\\le -6[\/latex].\r\n\r\n[reveal-answer q=\"96760\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96760\"]\r\n\r\n[latex]k\\le 1[\/latex] or [latex]k\\ge 7[\/latex]; in interval notation, this would be [latex]\\left(-\\infty ,1\\right]\\cup \\left[7,\\infty \\right)[\/latex].\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200413\/CNX_CAT_Figure_02_07_007.jpg\" alt=\"A coordinate plane with the x-axis ranging from -1 to 9 and the y-axis ranging from -3 to 8. The function y = -2|k 4| + 6 is graphed and everything above the function is shaded in.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=89935&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\r\n<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=15505&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p style=\"text-align: left;\">Sometimes a picture is worth a thousand words. You can turn a single variable inequality into a two variable inequality and make a graph. The x-intercepts of the graph will correspond with the solution to the inequality you can find by hand.\r\nLet's use the last example to try it. We will change the variable to x to make it easier to enter in an online graphing calculator.\r\nTo turn [latex]-2|x - 4|\\le -6[\/latex] into a two variable equation, move everything to one side, and place the variable y on the other side like this:<\/p>\r\n<p style=\"text-align: center;\">[latex]-2|x - 4|\\le -6[\/latex]\r\n[latex]-2|x - 4|+6\\le y[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Now enter this inequality in an online graphing calculator and hover over the x-intercepts.<\/p>\r\nIf you need instruction on how to enter inequalities in an online graphing calculator, watch this tutorial for the process within Desmos. Other calculators will behave slightly differently.\r\n\r\nhttps:\/\/youtu.be\/2H3cAYmBdyI\r\n\r\nAre the x-values of the intercepts the same values as the solution we found above?\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n<p style=\"text-align: left;\">Now you try turning this single variable inequality into a two variable inequality:<\/p>\r\n<p style=\"text-align: center;\">[latex]5|9-2x|\\ge10[\/latex]<\/p>\r\nGraph your inequality with an online graphing calculator, and write the solution interval.\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve compound inequalities.<\/li>\n<li>Solve absolute value inequalities.<\/li>\n<\/ul>\n<\/div>\n<p>A <strong>compound inequality<\/strong> includes two inequalities in one statement. A statement such as [latex]4<x\\le 6[\/latex] means [latex]4<x[\/latex] and [latex]x\\le 6[\/latex]. There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.\n\n\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Compound Inequality<\/h3>\n<p>Solve the compound inequality: [latex]3\\le 2x+2<6[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q497940\">Show Solution<\/span><\/p>\n<div id=\"q497940\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first method is to write two separate inequalities: [latex]3\\le 2x+2[\/latex] and [latex]2x+2<6[\/latex]. We solve them independently.\n\n\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}3\\le 2x+2\\hfill & \\text{and}\\hfill & 2x+2<6\\hfill \\\\ 1\\le 2x\\hfill & \\hfill & 2x<4\\hfill \\\\ \\frac{1}{2}\\le x\\hfill & \\hfill & x<2\\hfill \\end{array}[\/latex]<\/div>\n<p>Then, we can rewrite the solution as a compound inequality, the same way the problem began.<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{1}{2}\\le x<2[\/latex]<\/div>\n<p>In interval notation, the solution is written as [latex]\\left[\\frac{1}{2},2\\right)[\/latex].<\/p>\n<p>The second method is to leave the compound inequality intact and perform solving procedures on the three parts at the same time.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}3\\le 2x+2<6\\hfill & \\hfill \\\\ 1\\le 2x<4\\hfill & \\text{Isolate the variable term and subtract 2 from all three parts}.\\hfill \\\\ \\frac{1}{2}\\le x<2\\hfill & \\text{Divide through all three parts by 2}.\\hfill \\end{array}[\/latex]<\/div>\n<p>We get the same solution: [latex]\\left[\\frac{1}{2},2\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the compound inequality [latex]4<2x - 8\\le 10[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q265531\">Show Solution<\/span><\/p>\n<div id=\"q265531\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]6<x\\le 9\\text{ }\\text{ }\\text{or}\\left(6,9\\right][\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm3816\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3816&theme=oea&iframe_resize_id=ohm3816&show_question_numbers\" width=\"100%\" height=\"170\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm92608\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92608&theme=oea&iframe_resize_id=ohm92608&show_question_numbers\" width=\"100%\" height=\"230\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Compound Inequality with the Variable in All Three Parts<\/h3>\n<p>Solve the compound inequality with variables in all three parts: [latex]3+x>7x - 2>5x - 10[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q658677\">Show Solution<\/span><\/p>\n<div id=\"q658677\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let&#8217;s try the first method. Write two inequalities<strong>:<\/strong><\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}3+x> 7x - 2\\hfill & \\text{and}\\hfill & 7x - 2> 5x - 10\\hfill \\\\ 3> 6x - 2\\hfill & \\hfill & 2x - 2> -10\\hfill \\\\ 5> 6x\\hfill & \\hfill & 2x> -8\\hfill \\\\ \\frac{5}{6}> x\\hfill & \\hfill & x> -4\\hfill \\\\ x< \\frac{5}{6}\\hfill & \\hfill & -4< x\\hfill \\end{array}[\/latex]<\/div>\n<p>The solution set is [latex]-4<x<\\frac{5}{6}[\/latex] or in interval notation [latex]\\left(-4,\\frac{5}{6}\\right)[\/latex]. Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right as they appear on a number line.\n\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225901\/CNX_CAT_Figure_02_07_003.jpg\" alt=\"A number line with the points -4 and 5\/6 labeled. Dots appear at these points and a line connects these two dots.\" width=\"487\" height=\"69\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the compound inequality: [latex]3y<4 - 5y<5+3y[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q661493\">Show Solution<\/span><\/p>\n<div id=\"q661493\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\frac{1}{8},\\frac{1}{2}\\right)[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm13990\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=13990&theme=oea&iframe_resize_id=ohm13990&show_question_numbers\" width=\"100%\" height=\"220\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm129529\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=129529&theme=oea&iframe_resize_id=ohm129529&show_question_numbers\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<h2>Solving Absolute Value Inequalities<\/h2>\n<p>As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at [latex]\\left(-x,0\\right)[\/latex] has an absolute value of [latex]x[\/latex] as it is <em>x <\/em>units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.<\/p>\n<p>An <strong>absolute value inequality<\/strong> is an equation of the form<\/p>\n<div style=\"text-align: center;\">[latex]|A|<B,|A|\\le B,|A|>B,\\text{or }|A|\\ge B[\/latex],<\/div>\n<p>where <em>A<\/em>, and sometimes <em>B<\/em>, represents an algebraic expression dependent on a variable <em>x. <\/em>Solving the inequality means finding the set of all [latex]x[\/latex] <em>&#8211;<\/em>values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.<\/p>\n<p>There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.<\/p>\n<p>Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of <em>x-<\/em>values such that the distance between [latex]x[\/latex] and 600 is less than 200. We represent the distance between [latex]x[\/latex] and 600 as [latex]|x - 600|[\/latex], and therefore, [latex]|x - 600|\\le 200[\/latex] or<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{c}-200\\le x - 600\\le 200\\\\ -200+600\\le x - 600+600\\le 200+600\\\\ 400\\le x\\le 800\\end{array}[\/latex]<\/div>\n<p>This means our returns would be between $400 and $800.<\/p>\n<p>To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Absolute Value Inequalities<\/h3>\n<p>For an algebraic expression <em>X\u00a0<\/em>and [latex]k>0[\/latex], an <strong>absolute value inequality<\/strong> is an inequality of the form:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}|X|< k\\text{ which is equivalent to }-k< X< k\\hfill \\text{ or }\\ |X|> k\\text{ which is equivalent to }X< -k\\text{ or }X> k\\hfill \\end{array}[\/latex]<\/div>\n<p>These statements also apply to [latex]|X|\\le k[\/latex] and [latex]|X|\\ge k[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determining a Number within a Prescribed Distance<\/h3>\n<p>Describe all values [latex]x[\/latex] within a distance of 4 from the number 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q746497\">Show Solution<\/span><\/p>\n<div id=\"q746497\" class=\"hidden-answer\" style=\"display: none\">\n<p>We want the distance between [latex]x[\/latex] and 5 to be less than or equal to 4. We can draw a number line to represent the condition to be satisfied.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225903\/CNX_Precalc_Figure_01_06_002.jpg\" alt=\"A number line with one tick mark in the center labeled: 5. The tick marks on either side of the center one are not marked. Arrows extend from the center tick mark to the outer tick marks, both are labeled 4.\" width=\"487\" height=\"81\" \/><\/p>\n<p>The distance from [latex]x[\/latex] to 5 can be represented using an absolute value symbol, [latex]|x - 5|[\/latex]. Write the values of [latex]x[\/latex] that satisfy the condition as an absolute value inequality.<\/p>\n<div style=\"text-align: center;\">[latex]|x - 5|\\le 4[\/latex]<\/div>\n<p>We need to write two inequalities as there are always two solutions to an absolute value equation.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}x - 5\\le 4\\hfill & \\text{and}\\hfill & x - 5\\ge -4\\hfill \\\\ x\\le 9\\hfill & \\hfill & x\\ge 1\\hfill \\end{array}[\/latex]<\/div>\n<p>If the solution set is [latex]x\\le 9[\/latex] and [latex]x\\ge 1[\/latex], then the solution set is an interval including all real numbers between and including 1 and 9.<\/p>\n<p>So [latex]|x - 5|\\le 4[\/latex] is equivalent to [latex]\\left[1,9\\right][\/latex] in interval notation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Describe all <em>x-<\/em>values within a distance of 3 from the number 2.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q7507\">Show Solution<\/span><\/p>\n<div id=\"q7507\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]|x - 2|\\le 3[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Absolute Value Inequality<\/h3>\n<p>Solve [latex]|x - 1|\\le 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q4865\">Show Solution<\/span><\/p>\n<div id=\"q4865\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}|x - 1|\\le 3\\hfill \\\\ \\hfill \\\\ -3\\le x - 1\\le 3\\hfill \\\\ \\hfill \\\\ -2\\le x\\le 4\\hfill \\\\ \\hfill \\\\ \\left[-2,4\\right]\\hfill \\end{array}[\/latex]<\/p>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using a Graphical Approach to Solve Absolute Value Inequalities<\/h3>\n<p>Given the equation [latex]y=-\\frac{1}{2}|4x - 5|+3[\/latex], determine the <em>x<\/em>-values for which the <em>y<\/em>-values are negative.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q624558\">Show Solution<\/span><\/p>\n<div id=\"q624558\" class=\"hidden-answer\" style=\"display: none\">\n<p>We are trying to determine where [latex]y<0[\/latex] which is when [latex]-\\frac{1}{2}|4x - 5|+3<0[\/latex]. We begin by isolating the absolute value.\n\n\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}-\\frac{1}{2}|4x - 5|< -3\\hfill & \\text{Multiply both sides by -2, and reverse the inequality}.\\hfill \\\\ |4x - 5|> 6\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<p>Next, we solve [latex]|4x - 5|=6[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}4x - 5=6\\hfill & \\hfill & 4x - 5=-6\\hfill \\\\ 4x=11\\hfill & \\text{or}\\hfill & 4x=-1\\hfill \\\\ x=\\frac{11}{4}\\hfill & \\hfill & x=-\\frac{1}{4}\\hfill \\end{array}[\/latex]<\/div>\n<p>Now, we can examine the graph to observe where the <em>y-<\/em>values are negative. We observe where the branches are below the <em>x-<\/em>axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at [latex]x=-\\frac{1}{4}[\/latex] and [latex]x=\\frac{11}{4}[\/latex] and that the graph opens downward.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225906\/CNX_CAT_Figure_02_07_006.jpg\" alt=\"A coordinate plan with the x-axis ranging from -5 to 5 and the y-axis ranging from -4 to 4. The function y = -1\/2|4x \u2013 5| + 3 is graphed. An open circle appears at the point -0.25 and an arrow\" width=\"487\" height=\"363\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve [latex]-2|k - 4|\\le -6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q96760\">Show Solution<\/span><\/p>\n<div id=\"q96760\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]k\\le 1[\/latex] or [latex]k\\ge 7[\/latex]; in interval notation, this would be [latex]\\left(-\\infty ,1\\right]\\cup \\left[7,\\infty \\right)[\/latex].<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200413\/CNX_CAT_Figure_02_07_007.jpg\" alt=\"A coordinate plane with the x-axis ranging from -1 to 9 and the y-axis ranging from -3 to 8. The function y = -2|k 4| + 6 is graphed and everything above the function is shaded in.\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=89935&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=15505&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p style=\"text-align: left;\">Sometimes a picture is worth a thousand words. You can turn a single variable inequality into a two variable inequality and make a graph. The x-intercepts of the graph will correspond with the solution to the inequality you can find by hand.<br \/>\nLet&#8217;s use the last example to try it. We will change the variable to x to make it easier to enter in an online graphing calculator.<br \/>\nTo turn [latex]-2|x - 4|\\le -6[\/latex] into a two variable equation, move everything to one side, and place the variable y on the other side like this:<\/p>\n<p style=\"text-align: center;\">[latex]-2|x - 4|\\le -6[\/latex]<br \/>\n[latex]-2|x - 4|+6\\le y[\/latex]<\/p>\n<p style=\"text-align: left;\">Now enter this inequality in an online graphing calculator and hover over the x-intercepts.<\/p>\n<p>If you need instruction on how to enter inequalities in an online graphing calculator, watch this tutorial for the process within Desmos. Other calculators will behave slightly differently.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Learn Desmos: Inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/2H3cAYmBdyI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Are the x-values of the intercepts the same values as the solution we found above?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p style=\"text-align: left;\">Now you try turning this single variable inequality into a two variable inequality:<\/p>\n<p style=\"text-align: center;\">[latex]5|9-2x|\\ge10[\/latex]<\/p>\n<p>Graph your inequality with an online graphing calculator, and write the solution interval.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<\/div>\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-112\">\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>Revision and Adaptation. <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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 92608, 92609. <strong>Authored by<\/strong>: Michael Jenck. <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>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 89935. <strong>Authored by<\/strong>: Krystal Meier. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 15505. <strong>Authored by<\/strong>: Tophe Anderson. <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>: IMathAS Community License CC- BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Learn Desmos: Inequalities. <strong>Authored by<\/strong>: Desmos. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/2H3cAYmBdyI\">https:\/\/youtu.be\/2H3cAYmBdyI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/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":395986,"menu_order":27,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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