{"id":13736,"date":"2018-08-24T18:09:44","date_gmt":"2018-08-24T18:09:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalcone\/?post_type=chapter&#038;p=13736"},"modified":"2025-02-05T05:18:04","modified_gmt":"2025-02-05T05:18:04","slug":"absolute-value-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/absolute-value-functions\/","title":{"raw":"Absolute Value Functions","rendered":"Absolute Value Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph an absolute value function.<\/li>\r\n \t<li>Solve an absolute value equation.<\/li>\r\n \t<li>Solve an absolute value inequality.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<figure id=\"Figure_01_06_001\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010617\/CNX_Precalc_Figure_01_06_001n2.jpg\" alt=\"The Milky Way.\" width=\"488\" height=\"338\" \/> <b>Figure 1.<\/b> Distances in deep space can be measured in all directions. As such, it is useful to consider distance in terms of absolute values. (credit: \"s58y\"\/Flickr)[\/caption]<\/figure>\r\n<p id=\"fs-id1165137475222\">Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate <strong>absolute value functions<\/strong>.<\/p>\r\n\r\n<h2>Understanding Absolute Value<\/h2>\r\n<p id=\"fs-id1165135449691\">Recall that in its basic form [latex]\\displaystyle{f}\\left({x}\\right)={|x|}[\/latex], the absolute value function, is one of our toolkit functions. The <strong>absolute value<\/strong> function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.<\/p>\r\n\r\n<div id=\"fs-id1165135404116\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Absolute Value Function<\/h3>\r\n<p id=\"fs-id1165137832269\">The absolute value function can be defined as a piecewise function<\/p>\r\n<p style=\"text-align: center;\">$latex f(x) = \\begin{cases} x ,\\ x \\geq 0 \\\\ -x , x &lt; 0 \\end{cases} $<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_01_06_01\" class=\"example\">\r\n<div id=\"fs-id1165137437173\" class=\"exercise\">\r\n<div id=\"fs-id1165137618976\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Determine a Number within a Prescribed Distance<\/h3>\r\n<p id=\"fs-id1165137761508\">Describe all values [latex]x[\/latex] within or including a distance of 4 from the number 5.<\/p>\r\n[reveal-answer q=\"617814\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617814\"]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010617\/CNX_Precalc_Figure_01_06_0022.jpg\" alt=\"Number line describing the difference of the distance of 4 away from 5.\" width=\"487\" height=\"81\" \/> <b>Figure 2<\/b>[\/caption]\r\n<p id=\"fs-id1165135424674\">We want the distance between [latex]x[\/latex] and 5 to be less than or equal to 4. We can draw a number line\u00a0to represent the condition to be satisfied.<span id=\"fs-id1165137761581\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165137772130\">The distance from [latex]x[\/latex] to 5 can be represented using the absolute value as [latex]|x - 5|[\/latex]. We want the values of [latex]x[\/latex] that satisfy the condition [latex]|x - 5|\\le 4[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165135161478\">Note that<\/p>\r\n\r\n<div id=\"fs-id1165134394601\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{-4}\\le{x - 5}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{1}\\le{x}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\">And:<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{x-5}\\le{4}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{x}\\le{9}[\/latex]<\/div>\r\n<p id=\"fs-id1165137569650\">So [latex]|x - 5|\\le 4[\/latex] is equivalent to [latex]1\\le x\\le 9[\/latex].<\/p>\r\n<p id=\"fs-id1165137539782\">However, mathematicians generally prefer absolute value notation.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135394310\">Describe all values [latex]x[\/latex] within a distance of 3 from the number 2.<\/p>\r\n[reveal-answer q=\"308596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"308596\"]\r\n\r\n<span class=\"s1\">[latex]|x - 2|\\le 3[\/latex]<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_01_06_02\" class=\"example\">\r\n<div id=\"fs-id1165137657277\" class=\"exercise\">\r\n<div id=\"fs-id1165137579723\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Resistance of a Resistor<\/h3>\r\n<p id=\"fs-id1165135203760\">Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often [latex]\\displaystyle\\pm\\text{1%,}\\pm\\text{5%,}[\/latex] or [latex]\\displaystyle\\pm\\text{10%}[\/latex].<\/p>\r\n<p id=\"fs-id1165135175007\">Suppose we have a resistor rated at 680 ohms, [latex]\\pm 5%[\/latex]. Use the absolute value function to express the range of possible values of the actual resistance.<\/p>\r\n[reveal-answer q=\"430965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"430965\"]\r\n<p id=\"fs-id1165137600783\">5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance [latex]R[\/latex] in ohms,<\/p>\r\n\r\n<div id=\"fs-id1165135176481\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|R - 680|\\le 34[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137828266\">Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.<\/p>\r\n[reveal-answer q=\"365106\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"365106\"]\r\n\r\n<span class=\"s1\">Using the variable [latex]p[\/latex] for passing, [latex]|p - 80|\\le 20[\/latex].<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Graphing an Absolute Value Function<\/h2>\r\n<p id=\"fs-id1165135570012\">The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the <strong>origin<\/strong>.<span id=\"fs-id1165137530693\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010618\/CNX_Precalc_Figure_01_06_0032.jpg\" alt=\"Graph of an absolute function\" width=\"487\" height=\"251\" \/> <b>Figure 3<\/b>[\/caption]\r\n\r\nFigure 4 shows how to find the graph of [latex]y=2\\left|x - 3\\right|+4[\/latex]. The graph of [latex]y=|x|[\/latex] has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at [latex]\\left(3,4\\right)[\/latex] for this transformed function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010618\/CNX_Precalc_Figure_01_06_0042.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"487\" height=\"486\" \/> <b>Figure 4<\/b>[\/caption]\r\n\r\n<div id=\"Example_01_06_03\" class=\"example\">\r\n<div id=\"fs-id1165135187768\" class=\"exercise\">\r\n<div id=\"fs-id1165137741094\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Writing an Equation for an Absolute Value Function<\/h3>\r\nWrite an equation for the function graphed in Figure 5.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010618\/CNX_Precalc_Figure_01_06_0052.jpg\" alt=\"Graph of an absolute function. Two rays stem from the point 3, negative 2. One ray crosses the point 0, 4. The other ray crosses the point 5, 2.\" width=\"487\" height=\"363\" \/> <b>Figure 5<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"729255\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"729255\"]\r\n\r\nThe basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010618\/CNX_Precalc_Figure_01_06_0062.jpg\" alt=\"Graph of two transformations for an absolute function at (3, -2).\" width=\"487\" height=\"363\" \/> <b>Figure 6<\/b>[\/caption]\r\n<p id=\"fs-id1165137680556\"><span id=\"fs-id1165137901124\">We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance.\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010618\/CNX_Precalc_Figure_01_06_0072.jpg\" alt=\"Graph of two transformations for an absolute function at (3, -2) and describes the ratios between the two different transformations.\" width=\"487\" height=\"363\" \/> <b>Figure 7<\/b>[\/caption]\r\n<p id=\"fs-id1165137732766\">From this information we can write the equation<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;f\\left(x\\right)=2\\left|x - 3\\right|-2, &amp;&amp; \\text{treating the stretch as a vertical stretch,} \\\\[2mm] \\text{or } &amp;f\\left(x\\right)=\\left|2\\left(x - 3\\right)\\right|-2, &amp;&amp; \\text{treating the stretch as a horizontal compression}. \\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137591631\">Note that these equations are algebraically equivalent\u2014the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134377948\" class=\"note precalculus qa textbox\">\r\n<p id=\"fs-id1165135245777\"><strong>Q &amp; A<\/strong><\/p>\r\n<strong>If we couldn\u2019t observe the stretch of the function from the graphs, could we algebraically determine it?<\/strong>\r\n<p id=\"fs-id1165137473393\"><em>Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for [latex]x[\/latex] and [latex]f\\left(x\\right)[\/latex].<\/em><\/p>\r\n\r\n<div id=\"fs-id1165135514699\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=a|x - 3|-2[\/latex]<\/div>\r\n<p id=\"fs-id1165137694034\"><em>Now substituting in the point <\/em>(1, 2)<\/p>\r\n\r\n<div id=\"fs-id1165135173265\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}&amp;2=a|1 - 3|-2 \\\\ &amp;4=2a \\\\ &amp;a=2 \\end{align}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135497155\">Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.<\/p>\r\n[reveal-answer q=\"637365\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"637365\"]\r\n\r\n<span class=\"s1\">[latex]f\\left(x\\right)=-|x+2|+3[\/latex]<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135203778\" class=\"note precalculus qa textbox\">\r\n<p id=\"fs-id1165137527840\"><strong>Q &amp; A<\/strong><\/p>\r\n<strong>Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?\r\n<\/strong>\r\n<p id=\"fs-id1165137581861\"><em>Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.\r\n<\/em><\/p>\r\n<p id=\"fs-id1165137444543\"><em>No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points.<\/em><\/p>\r\n\r\n<\/div>\r\n<figure id=\"Figure_01_06_008\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010619\/CNX_Precalc_Figure_01_06_008abc2.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"975\" height=\"415\" \/> <b>Figure 8.<\/b> (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.[\/caption]<\/figure>\r\n<h2>Solving an Absolute Value Equation<\/h2>\r\n<p id=\"fs-id1165137401775\">Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as [latex]{8}=\\left|{2}x - {6}\\right|[\/latex], we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.<\/p>\r\n\r\n<div id=\"fs-id1165137583696\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}2x - 6&amp;=8 &amp; \\text{or} &amp;&amp; 2x - 6&amp;=-8 \\\\ 2x&amp;=14 &amp;&amp;&amp; 2x&amp;=-2 \\\\ x&amp;=7 &amp;&amp;&amp; x&amp;=-1 \\\\ \\text{ } \\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165137641126\">Knowing how to solve problems involving <strong>absolute value functions<\/strong> is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\r\n<p id=\"fs-id1165137937577\">An <strong>absolute value equation<\/strong> is an equation in which the unknown variable appears in absolute value bars. For example,<\/p>\r\n\r\n<div id=\"fs-id1165137646929\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|x|=4[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|2x - 1|=3[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|5x+2|-4=9[\/latex]<\/div>\r\n<div id=\"fs-id1165137692078\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Solutions to Absolute Value Equations<\/h3>\r\n<p id=\"fs-id1165137809877\">For real numbers [latex]A[\/latex] and [latex]B[\/latex], an equation of the form [latex]|A|=B[\/latex], with [latex]B\\ge 0[\/latex], will have solutions when [latex]A=B[\/latex] or [latex]A=-B[\/latex]. If [latex]B&lt;0[\/latex], the equation [latex]|A|=B[\/latex] has no solution.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135160087\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135593248\">How To: Given the formula for an absolute value function, find the horizontal intercepts of its graph.<\/h3>\r\n<ol id=\"fs-id1165131968095\">\r\n \t<li>Set the function equal to [latex]0[\\latex].<\/li>\r\n \t<li>Isolate the absolute value term.<\/li>\r\n \t<li>Use [latex]|A|=B[\/latex] to write [latex]A=B[\/latex] or [latex]\\mathrm{-A}=B[\/latex], assuming [latex]B&gt;0[\/latex].<\/li>\r\n \t<li>Solve for [latex]x[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_06_04\" class=\"example\">\r\n<div id=\"fs-id1165137619575\" class=\"exercise\">\r\n<div id=\"fs-id1165135309797\" class=\"problem textbox shaded\">\r\n<h3>Example 4: Finding the Zeros of an Absolute Value Function<\/h3>\r\n<p id=\"fs-id1165137527684\">For the function [latex]f\\left(x\\right)=|4x+1|-7[\/latex] , find the values of\u00a0[latex]x[\/latex] such that\u00a0[latex]\\text{ }f\\left(x\\right)=0[\/latex] .<\/p>\r\n[reveal-answer q=\"621359\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"621359\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;0=|4x+1|-7 &amp;&amp;&amp;&amp;&amp;&amp; \\text{Substitute 0 for }f\\left(x\\right). \\\\ &amp;7=|4x+1| &amp;&amp;&amp;&amp;&amp;&amp; \\text{Isolate the absolute value on one side of the equation}.\\\\ &amp; \\\\ &amp;7=4x+1 &amp; \\text{or} &amp;&amp;&amp; -7=4x+1 &amp;&amp; \\text{Break into two separate equations and solve}. \\\\ &amp;6=4x &amp;&amp;&amp;&amp; -8=4x \\\\ &amp; \\\\ &amp;x=\\frac{6}{4}=\\frac{3}{2}=1.5 &amp;&amp;&amp;&amp; \\text{ }x=\\frac{-8}{4}=-2 \\end{align}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010619\/CNX_Precalc_Figure_01_06_011F2.jpg\" alt=\"Graph an absolute function with x-intercepts at -2 and 1.5.\" width=\"731\" height=\"476\" \/> <b>Figure 9<\/b>[\/caption]\r\n<p id=\"fs-id1165137870931\">The function outputs 0 when [latex]x=1.5[\/latex] or [latex]x=-2[\/latex].<\/p>\r\n[\/hidden-answer]<span id=\"fs-id1165137662351\">\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137843093\">For the function [latex]f\\left(x\\right)=|2x - 1|-3[\/latex], find the values of [latex]x[\/latex] such that [latex]f\\left(x\\right)=0[\/latex].<\/p>\r\n[reveal-answer q=\"796930\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"796930\"]\r\n\r\n<span class=\"s1\">[latex]x=-1[\/latex] or [latex]x=2[\/latex]<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]165746[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135175321\" class=\"note precalculus qa textbox\">\r\n<p id=\"fs-id1165135606935\"><strong>Q &amp; A<\/strong><\/p>\r\n<strong>Should we always expect two answers when solving [latex]|A|=B?[\/latex]<\/strong>\r\n<p id=\"fs-id1165137755892\"><em>No. We may find one, two, or even no answers. For example, there is no solution to\u00a0<\/em>[latex]2+|3x - 5|=1[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137911662\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137647413\">How To: Given an absolute value equation, solve it.<\/h3>\r\n<ol id=\"fs-id1165137589466\">\r\n \t<li>Isolate the absolute value term.<\/li>\r\n \t<li>Use [latex]|A|=B[\/latex] to write [latex]A=B[\/latex] or [latex]A=\\mathrm{-B}[\/latex].<\/li>\r\n \t<li>Solve for [latex]x[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_06_05\" class=\"example\">\r\n<div id=\"fs-id1165137727865\" class=\"exercise\">\r\n<div id=\"fs-id1165135195112\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Solving an Absolute Value Equation<\/h3>\r\n<p id=\"fs-id1165137695200\">Solve [latex]1=4|x - 2|+2[\/latex].<\/p>\r\n[reveal-answer q=\"134452\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"134452\"]\r\n<p id=\"fs-id1165135210177\">First we isolate the absolute value expression on one side of the equation.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}1&amp;=4|x - 2|+2 \\\\ -1&amp;=4|x - 2| \\\\ -\\frac{1}{4}&amp;=|x - 2| \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137611734\">The absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. At this point, we notice that this equation has no solutions.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]152838[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137465993\" class=\"note precalculus qa textbox\">\r\n<p id=\"fs-id1165137573052\"><strong>Q &amp; A<\/strong><\/p>\r\n<strong>In Example 5, if the functions [latex]f\\left(x\\right)=1[\/latex] and [latex]g\\left(x\\right)=4|x - 2|+2[\/latex] were graphed on the same set of axes, would the graphs intersect?<\/strong>\r\n<p id=\"fs-id1165137602208\"><em>No. The graphs of [latex]f[\/latex] and [latex]g[\/latex] would not intersect. This confirms, graphically, that the equation [latex]1=4|x - 2|+2[\/latex] has no solution.<\/em><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010619\/CNX_Precalc_Figure_01_06_0122.jpg\" alt=\"Graph of g(x)=4|x-2|+2 and f(x)=1.\" width=\"487\" height=\"476\" \/> <b>Figure 10<\/b>[\/caption]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137735930\">Find where the graph of the function [latex]f\\left(x\\right)=-|x+2|+3[\/latex] intersects the horizontal and vertical axes.<\/p>\r\n[reveal-answer q=\"960627\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"960627\"]\r\n\r\n<span class=\"s1\">[latex]f\\left(0\\right)=1[\/latex], so the graph intersects the vertical axis at [latex]\\left(0,1\\right)[\/latex]. [latex]f\\left(x\\right)=0[\/latex] when [latex]x=-5[\/latex] and [latex]x=1[\/latex] so the graph intersects the horizontal axis at [latex]\\left(-5,0\\right)[\/latex] and [latex]\\left(1,0\\right)[\/latex].<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Solving an Absolute Value Inequality<\/h2>\r\n<p id=\"fs-id1165137583863\">Absolute value expressions may not always involve equations. Instead we may need to solve where an expression is within a range of values. We would use an absolute value inequality to solve such an equation. An <strong>absolute value inequality<\/strong> is an inequality of the form<\/p>\r\n\r\n<div id=\"fs-id1165134065110\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|{A}|&lt;{ B },|{ A }|\\le{ B },|{ A }|&gt;{ B },\\text{ or } |{ A }|\\ge { B }[\/latex],<\/div>\r\n<p id=\"fs-id1165135154162\">where an expression [latex]A[\/latex] (and possibly but not usually [latex]B[\/latex] ) depends on a variable [latex]x[\/latex]. Solving the inequality means finding the set of all [latex]x[\/latex] that satisfy the inequality. Usually this set will be an interval or the union of two intervals.<\/p>\r\n<p id=\"fs-id1165137580992\">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 functions. The advantage of the algebraic approach is it yields solutions that may be difficult to read from the graph.<\/p>\r\n<p id=\"fs-id1165137557647\">For example, we know that all numbers within 200 units of 0 may be expressed as<\/p>\r\n\r\n<div id=\"fs-id1165137543814\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|x|&lt;{ 200 }\\text{ or }{ -200 }&lt;{ x }&lt;{ 200 }\\text{ }[\/latex]<\/div>\r\n<p id=\"fs-id1165137610749\">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 values [latex]x[\/latex]\u00a0such that the distance between [latex]x[\/latex] and 600 is less than 200. We represent the distance between [latex]x[\/latex]\u00a0and 600 as [latex]|{ x } - {600 }|[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165137755666\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|{ x } -{ 600 }|&lt;{ 200 }[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">OR<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{ -200 }&lt;{ x } - { 600 }&lt;{ 200 }[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{-200 }+{ 600 }&lt;{ x } - {600 }+{ 600 }&lt;{ 200 }+{ 600 }[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{ 400 }&lt;{ x }&lt;{ 800 }[\/latex]<\/div>\r\n<p id=\"fs-id1165137804310\">This means our returns would be between $400 and $800.<\/p>\r\n<p id=\"fs-id1165137507358\">Sometimes an absolute value inequality problem will be presented to us in terms of a shifted and\/or stretched or compressed absolute value function, where we must determine for which values of the input the function\u2019s output will be negative or positive.<\/p>\r\n\r\n<div id=\"fs-id1165137667916\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137502428\">How To: Given an absolute value inequality of the form [latex]|x-A|\\le B[\/latex] for real numbers [latex]a[\/latex] and [latex]b[\/latex] where [latex]b[\/latex] is positive, solve the absolute value inequality algebraically.<\/h3>\r\n<ol id=\"fs-id1165137563287\">\r\n \t<li>Find boundary points by solving [latex]|x-A|=B[\/latex].<\/li>\r\n \t<li>Test intervals created by the boundary points to determine where [latex]|x-A|\\le B[\/latex].<\/li>\r\n \t<li>Write the interval or union of intervals satisfying the inequality in interval, inequality, or set-builder notation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_06_06\" class=\"example\">\r\n<div id=\"fs-id1165135704112\" class=\"exercise\">\r\n<div id=\"fs-id1165137401703\" class=\"problem textbox shaded\">\r\n<h3>Example 6: Solving an Absolute Value Inequality<\/h3>\r\n<p id=\"fs-id1165135342955\">Solve [latex]|x - 5|\\le 4[\/latex].<\/p>\r\n[reveal-answer q=\"183338\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"183338\"]\r\n<p id=\"fs-id1165137645044\">With both approaches, we will need to know first where the corresponding equality is true. In this case we first will find where [latex]|x - 5|=4[\/latex]. We do this because the absolute value is a function with no breaks, so the only way the function values can switch from being less than 4 to being greater than 4 is by passing through where the values equal 4. Solve [latex]|x - 5|=4[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x - 5&amp;=4 &amp; \\text{ or } &amp;&amp; {x - 5 }&amp;={ -4 }\\\\ {x }&amp;= {9} &amp;\\text{ or } &amp;&amp; { x }&amp;={ 1 } \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137665217\">After determining that the absolute value is equal to 4 at [latex]x=1[\/latex] and [latex]x=9[\/latex], we know the graph can change only from being less than 4 to greater than 4 at these values. This divides the number line up into three intervals:<\/p>\r\n<p style=\"text-align: center;\">[latex]{ x }&lt;{ 1 },\\text{ }{ 1 }&lt;{ x }&lt;{ 9 },\\text{ and }{ x }&gt;{ 9 }[\/latex].<\/p>\r\n<p id=\"fs-id1165137422669\">To determine when the function is less than 4, we could choose a value in each interval and see if the output is less than or greater than 4, as shown in the table below.<\/p>\r\n\r\n<table id=\"Table_01_06_01\" style=\"border: 1px dashed #bbbbbb;\" summary=\"Table describing the interval test for certain inequalities for x. So if x&lt;1 and f(x)=0, then |0-5|&gt;4. If1&lt; x&lt;9 and f(x)=6, then |6-5|&lt;4. If x&lt;9 and f(x)=11, then |11-5|&gt;4.\"><colgroup> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th>Interval test [latex]x[\/latex]<\/th>\r\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\r\n<th colspan=\"2\">[latex]&lt;4[\/latex] or [latex]&gt;4?[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]{ x }&lt;{ 1 }[\/latex]<\/td>\r\n<td>0<\/td>\r\n<td>[latex]|0 - 5|=5[\/latex]<\/td>\r\n<td>Greater than<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{ 1 }&lt;{ x }&lt;{ 9 }[\/latex]<\/td>\r\n<td>6<\/td>\r\n<td>[latex]|6 - 5|=1[\/latex]<\/td>\r\n<td>Less than<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{ x }&gt;{ 9 }[\/latex]<\/td>\r\n<td>11<\/td>\r\n<td>[latex]|11 - 5|=6[\/latex]<\/td>\r\n<td>Greater than<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137558949\">Because [latex]1\\le x\\le 9[\/latex] is the only interval in which the output at the test value is less than 4, we can conclude that the solution to [latex]|x - 5|\\le 4[\/latex] is [latex]1\\le x\\le 9[\/latex], or [latex]\\left[1,9\\right][\/latex].<\/p>\r\nTo use a graph, we can sketch the function [latex]f\\left(x\\right)=|x - 5|[\/latex]. To help us see where the outputs are 4, the line [latex]g\\left(x\\right)=4[\/latex] could also be sketched.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010620\/CNX_Precalc_Figure_01_06_0132.jpg\" alt=\"Graph of an absolute function and a vertical line, demonstrating how to see what outputs are less than the vertical line.\" width=\"487\" height=\"288\" \/> <b>Figure 11.<\/b> Graph to find the points satisfying an absolute value inequality.[\/caption]\r\n<p id=\"fs-id1165137874583\">We can see the following:<\/p>\r\n\r\n<ul id=\"fs-id1165134148370\">\r\n \t<li>The output values of the absolute value are equal to 4 at [latex]x=1[\/latex] and [latex]x=9[\/latex].<\/li>\r\n \t<li>The graph of [latex]f[\/latex] is below the graph of [latex]g[\/latex] on [latex]1&lt;x&lt;9[\/latex]. This means the output values of [latex]f\\left(x\\right)[\/latex] are less than the output values of [latex]g\\left(x\\right)[\/latex].<\/li>\r\n \t<li>The absolute value is less than or equal to 4 between these two points, when [latex]1\\le x\\le 9[\/latex]. In interval notation, this would be the interval [latex]\\left[1,9\\right][\/latex].<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135381301\" class=\"commentary\">\r\n<h3>Analysis of the Solution<\/h3>\r\n<p id=\"fs-id1165135689465\">For absolute value inequalities,<\/p>\r\n\r\n<div id=\"fs-id1165135650752\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]|x-A|&lt;C[latex] can be rewritten [latex]-C&lt;x-A&lt;C[\/latex] and [latex] |x-A| &gt; C[\/latex] can be rewritten [latex] x-A &lt; -C \\text{ or } x-A &gt; C [\/latex].<\/div>\r\n<p id=\"fs-id1165135195336\">The [latex]&lt;[\/latex] or [latex]&gt;[\/latex] symbol may be replaced by [latex]\\le \\text{ or }\\ge [\/latex].<\/p>\r\n<p id=\"fs-id1165135524557\">So, for this example, we could use this alternative approach.<\/p>\r\n\r\n<div id=\"fs-id1165134226778\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{gathered}|x - 5|\\le 4 \\\\ -4\\le x - 5\\le 4 \\\\ -4+5\\le x - 5+5\\le 4+5 \\\\ 1\\le x\\le 9 \\end{gathered}[\/latex] [latex]\\begin{align} &amp;\\\\&amp;&amp;&amp; \\text{Rewrite by removing the absolute value bars}. \\\\ &amp;&amp;&amp; \\text{Isolate the }x. \\\\&amp; \\end{align}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137450875\">Solve [latex]|x+2|\\le 6[\/latex].<\/p>\r\n[reveal-answer q=\"606238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"606238\"]\r\n\r\n<span class=\"s1\">[latex]4\\le x\\le 8[\/latex]<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137530158\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135445865\">How To: Given an absolute value function, solve for the set of inputs where the output is positive (or negative).<strong>\r\n<\/strong><\/h3>\r\n<ol id=\"fs-id1165137572514\">\r\n \t<li>Set the function equal to zero and solve for the boundary points of the solution set.<\/li>\r\n \t<li>Use test points or a graph to determine where the function\u2019s output is positive or negative.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_06_07\" class=\"example\">\r\n<div id=\"fs-id1165137409791\" class=\"exercise\">\r\n<div id=\"fs-id1165137838822\" class=\"problem textbox shaded\">\r\n<h3>Example 7: Using a Graphical Approach to Solve Absolute Value Inequalities<\/h3>\r\n<p id=\"fs-id1165137933778\">Given the function [latex]f\\left(x\\right)=-\\frac{1}{2}|4x - 5|+3[\/latex], determine the [latex]x\\text{-}[\/latex] values for which the function values are negative.<\/p>\r\n[reveal-answer q=\"767825\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"767825\"]\r\n<p id=\"fs-id1165137431799\">We are trying to determine where [latex]f\\left(x\\right)&lt;0[\/latex], which is when [latex]-\\frac{1}{2}|4x - 5|+3&lt;0[\/latex]. We begin by isolating the absolute value.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}-\\frac{1}{2}|4x - 5|&amp;&lt;-3 \\hfill &amp;&amp; \\text{Multiply both sides by -2, and reverse the inequality}. \\\\ |4x - 5|&amp;&gt;6 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135317447\">Next we solve for the equality [latex]|4x - 5|=6[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}4x - 5&amp;=6 &amp; \\text{or} &amp;&amp; 4x - 5&amp;=-6 \\\\ 4x - 5&amp;=6 &amp;&amp;&amp; 4x&amp;=-1 \\\\ x&amp;=\\frac{11}{4} &amp;&amp;&amp; x&amp;=-\\frac{1}{4} \\end{align}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010620\/CNX_Precalc_Figure_01_06_0142.jpg\" alt=\"Graph of an absolute function with x-intercepts at -0.25 and 2.75.\" width=\"487\" height=\"365\" \/> <b>Figure 12<\/b>[\/caption]\r\n<p id=\"fs-id1165135344887\">Now, we can examine the graph of [latex]f[\/latex] to observe where the output is negative. We will observe where the branches are below the <em>x<\/em>-axis. Notice that it is not even 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 has been reflected vertically.<span id=\"fs-id1165137728256\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165137725461\">We observe that the graph of the function is below the <em>x<\/em>-axis left of [latex]x=-\\frac{1}{4}[\/latex] and right of [latex]x=\\frac{11}{4}[\/latex]. This means the function values are negative to the left of the first horizontal intercept at [latex]x=-\\frac{1}{4}[\/latex], and negative to the right of the second intercept at [latex]x=\\frac{11}{4}[\/latex]. This gives us the solution to the inequality.<\/p>\r\n<p style=\"text-align: center;\">[latex]x&lt;-\\frac{1}{4}\\text{ }\\text{or}\\text{ }x&gt;\\frac{11}{4}[\/latex]<\/p>\r\n<p id=\"fs-id1165135502945\">In interval notation, this would be [latex]\\left(-\\infty ,-0.25\\right)\\cup \\left(2.75,\\infty \\right)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137784438\">Solve [latex]-2|k - 4|\\le -6[\/latex].<\/p>\r\n[reveal-answer q=\"416287\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"416287\"]\r\n\r\n<span class=\"s1\">[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]<\/span>\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\n[ohm_question hide_question_numbers=1]86181[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135332513\">\r\n \t<li>The absolute value function is commonly used to measure distances between points.<\/li>\r\n \t<li>Applied problems, such as ranges of possible values, can also be solved using the absolute value function.<\/li>\r\n \t<li>The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction.<\/li>\r\n \t<li>In an absolute value equation, an unknown variable is the input of an absolute value function.<\/li>\r\n \t<li>If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable.<\/li>\r\n \t<li>An absolute value equation may have one solution, two solutions, or no solutions.<\/li>\r\n \t<li>An absolute value inequality is similar to an absolute value equation but takes the form [latex]|A|&lt;B,|A|\\le B,|A|&gt;B,\\text{ or }|A|\\ge B[\/latex]. It can be solved by determining the boundaries of the solution set and then testing which segments are in the set.<\/li>\r\n \t<li>Absolute value inequalities can also be solved graphically.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135191341\" class=\"definition\">\r\n \t<dt>absolute value equation<\/dt>\r\n \t<dd id=\"fs-id1165137627032\">an equation of the form [latex]|A|=B[\/latex], with [latex]B\\ge 0[\/latex]; it will have solutions when [latex]A=B[\/latex] or [latex]A=-B[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137560214\" class=\"definition\">\r\n \t<dt>absolute value inequality<\/dt>\r\n \t<dd id=\"fs-id1165135173524\">a relationship in the form [latex]|{ A }|&lt;{ B },|{ A }|\\le { B },|{ A }|&gt;{ B },\\text{or }|{ A }|\\ge{ B }[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph an absolute value function.<\/li>\n<li>Solve an absolute value equation.<\/li>\n<li>Solve an absolute value inequality.<\/li>\n<\/ul>\n<\/div>\n<figure id=\"Figure_01_06_001\" class=\"medium\">\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010617\/CNX_Precalc_Figure_01_06_001n2.jpg\" alt=\"The Milky Way.\" width=\"488\" height=\"338\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> Distances in deep space can be measured in all directions. As such, it is useful to consider distance in terms of absolute values. (credit: &#8220;s58y&#8221;\/Flickr)<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1165137475222\">Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate <strong>absolute value functions<\/strong>.<\/p>\n<h2>Understanding Absolute Value<\/h2>\n<p id=\"fs-id1165135449691\">Recall that in its basic form [latex]\\displaystyle{f}\\left({x}\\right)={|x|}[\/latex], the absolute value function, is one of our toolkit functions. The <strong>absolute value<\/strong> function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.<\/p>\n<div id=\"fs-id1165135404116\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Absolute Value Function<\/h3>\n<p id=\"fs-id1165137832269\">The absolute value function can be defined as a piecewise function<\/p>\n<p style=\"text-align: center;\">[latex]f(x) = \\begin{cases} x ,\\ x \\geq 0 \\\\ -x , x < 0 \\end{cases}[\/latex]<\/p>\n<\/div>\n<div id=\"Example_01_06_01\" class=\"example\">\n<div id=\"fs-id1165137437173\" class=\"exercise\">\n<div id=\"fs-id1165137618976\" class=\"problem textbox shaded\">\n<h3>Example 1: Determine a Number within a Prescribed Distance<\/h3>\n<p id=\"fs-id1165137761508\">Describe all values [latex]x[\/latex] within or including 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=\"q617814\">Show Solution<\/span><\/p>\n<div id=\"q617814\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010617\/CNX_Precalc_Figure_01_06_0022.jpg\" alt=\"Number line describing the difference of the distance of 4 away from 5.\" width=\"487\" height=\"81\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135424674\">We want the distance between [latex]x[\/latex] and 5 to be less than or equal to 4. We can draw a number line\u00a0to represent the condition to be satisfied.<span id=\"fs-id1165137761581\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137772130\">The distance from [latex]x[\/latex] to 5 can be represented using the absolute value as [latex]|x - 5|[\/latex]. We want the values of [latex]x[\/latex] that satisfy the condition [latex]|x - 5|\\le 4[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165135161478\">Note that<\/p>\n<div id=\"fs-id1165134394601\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{-4}\\le{x - 5}[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{1}\\le{x}[\/latex]<\/div>\n<div class=\"equation unnumbered\">And:<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{x-5}\\le{4}[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{x}\\le{9}[\/latex]<\/div>\n<p id=\"fs-id1165137569650\">So [latex]|x - 5|\\le 4[\/latex] is equivalent to [latex]1\\le x\\le 9[\/latex].<\/p>\n<p id=\"fs-id1165137539782\">However, mathematicians generally prefer absolute value notation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135394310\">Describe all values [latex]x[\/latex] 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=\"q308596\">Show Solution<\/span><\/p>\n<div id=\"q308596\" class=\"hidden-answer\" style=\"display: none\">\n<p><span class=\"s1\">[latex]|x - 2|\\le 3[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_06_02\" class=\"example\">\n<div id=\"fs-id1165137657277\" class=\"exercise\">\n<div id=\"fs-id1165137579723\" class=\"problem textbox shaded\">\n<h3>Example 2: Resistance of a Resistor<\/h3>\n<p id=\"fs-id1165135203760\">Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often [latex]\\displaystyle\\pm\\text{1%,}\\pm\\text{5%,}[\/latex] or [latex]\\displaystyle\\pm\\text{10%}[\/latex].<\/p>\n<p id=\"fs-id1165135175007\">Suppose we have a resistor rated at 680 ohms, [latex]\\pm 5%[\/latex]. Use the absolute value function to express the range of possible values of the actual resistance.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q430965\">Show Solution<\/span><\/p>\n<div id=\"q430965\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137600783\">5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance [latex]R[\/latex] in ohms,<\/p>\n<div id=\"fs-id1165135176481\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|R - 680|\\le 34[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137828266\">Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q365106\">Show Solution<\/span><\/p>\n<div id=\"q365106\" class=\"hidden-answer\" style=\"display: none\">\n<p><span class=\"s1\">Using the variable [latex]p[\/latex] for passing, [latex]|p - 80|\\le 20[\/latex].<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Graphing an Absolute Value Function<\/h2>\n<p id=\"fs-id1165135570012\">The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the <strong>origin<\/strong>.<span id=\"fs-id1165137530693\"><br \/>\n<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010618\/CNX_Precalc_Figure_01_06_0032.jpg\" alt=\"Graph of an absolute function\" width=\"487\" height=\"251\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<p>Figure 4 shows how to find the graph of [latex]y=2\\left|x - 3\\right|+4[\/latex]. The graph of [latex]y=|x|[\/latex] has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at [latex]\\left(3,4\\right)[\/latex] for this transformed function.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010618\/CNX_Precalc_Figure_01_06_0042.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"487\" height=\"486\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<div id=\"Example_01_06_03\" class=\"example\">\n<div id=\"fs-id1165135187768\" class=\"exercise\">\n<div id=\"fs-id1165137741094\" class=\"problem textbox shaded\">\n<h3>Example 3: Writing an Equation for an Absolute Value Function<\/h3>\n<p>Write an equation for the function graphed in Figure 5.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010618\/CNX_Precalc_Figure_01_06_0052.jpg\" alt=\"Graph of an absolute function. Two rays stem from the point 3, negative 2. One ray crosses the point 0, 4. The other ray crosses the point 5, 2.\" width=\"487\" height=\"363\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q729255\">Show Solution<\/span><\/p>\n<div id=\"q729255\" class=\"hidden-answer\" style=\"display: none\">\n<p>The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010618\/CNX_Precalc_Figure_01_06_0062.jpg\" alt=\"Graph of two transformations for an absolute function at (3, -2).\" width=\"487\" height=\"363\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137680556\"><span id=\"fs-id1165137901124\">We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance.<br \/>\n<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010618\/CNX_Precalc_Figure_01_06_0072.jpg\" alt=\"Graph of two transformations for an absolute function at (3, -2) and describes the ratios between the two different transformations.\" width=\"487\" height=\"363\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137732766\">From this information we can write the equation<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&f\\left(x\\right)=2\\left|x - 3\\right|-2, && \\text{treating the stretch as a vertical stretch,} \\\\[2mm] \\text{or } &f\\left(x\\right)=\\left|2\\left(x - 3\\right)\\right|-2, && \\text{treating the stretch as a horizontal compression}. \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137591631\">Note that these equations are algebraically equivalent\u2014the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134377948\" class=\"note precalculus qa textbox\">\n<p id=\"fs-id1165135245777\"><strong>Q &amp; A<\/strong><\/p>\n<p><strong>If we couldn\u2019t observe the stretch of the function from the graphs, could we algebraically determine it?<\/strong><\/p>\n<p id=\"fs-id1165137473393\"><em>Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for [latex]x[\/latex] and [latex]f\\left(x\\right)[\/latex].<\/em><\/p>\n<div id=\"fs-id1165135514699\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=a|x - 3|-2[\/latex]<\/div>\n<p id=\"fs-id1165137694034\"><em>Now substituting in the point <\/em>(1, 2)<\/p>\n<div id=\"fs-id1165135173265\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}&2=a|1 - 3|-2 \\\\ &4=2a \\\\ &a=2 \\end{align}[\/latex]<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135497155\">Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q637365\">Show Solution<\/span><\/p>\n<div id=\"q637365\" class=\"hidden-answer\" style=\"display: none\">\n<p><span class=\"s1\">[latex]f\\left(x\\right)=-|x+2|+3[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203778\" class=\"note precalculus qa textbox\">\n<p id=\"fs-id1165137527840\"><strong>Q &amp; A<\/strong><\/p>\n<p><strong>Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?<br \/>\n<\/strong><\/p>\n<p id=\"fs-id1165137581861\"><em>Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.<br \/>\n<\/em><\/p>\n<p id=\"fs-id1165137444543\"><em>No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points.<\/em><\/p>\n<\/div>\n<figure id=\"Figure_01_06_008\">\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010619\/CNX_Precalc_Figure_01_06_008abc2.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"975\" height=\"415\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8.<\/b> (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.<\/p>\n<\/div>\n<\/figure>\n<h2>Solving an Absolute Value Equation<\/h2>\n<p id=\"fs-id1165137401775\">Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as [latex]{8}=\\left|{2}x - {6}\\right|[\/latex], we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.<\/p>\n<div id=\"fs-id1165137583696\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}2x - 6&=8 & \\text{or} && 2x - 6&=-8 \\\\ 2x&=14 &&& 2x&=-2 \\\\ x&=7 &&& x&=-1 \\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165137641126\">Knowing how to solve problems involving <strong>absolute value functions<\/strong> is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\n<p id=\"fs-id1165137937577\">An <strong>absolute value equation<\/strong> is an equation in which the unknown variable appears in absolute value bars. For example,<\/p>\n<div id=\"fs-id1165137646929\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|x|=4[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|2x - 1|=3[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|5x+2|-4=9[\/latex]<\/div>\n<div id=\"fs-id1165137692078\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Solutions to Absolute Value Equations<\/h3>\n<p id=\"fs-id1165137809877\">For real numbers [latex]A[\/latex] and [latex]B[\/latex], an equation of the form [latex]|A|=B[\/latex], with [latex]B\\ge 0[\/latex], will have solutions when [latex]A=B[\/latex] or [latex]A=-B[\/latex]. If [latex]B<0[\/latex], the equation [latex]|A|=B[\/latex] has no solution.<\/p>\n<\/div>\n<div id=\"fs-id1165135160087\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135593248\">How To: Given the formula for an absolute value function, find the horizontal intercepts of its graph.<\/h3>\n<ol id=\"fs-id1165131968095\">\n<li>Set the function equal to [latex]0[\\latex].<\/li>\n<li>Isolate the absolute value term.<\/li>\n<li>Use [latex]|A|=B[\/latex] to write [latex]A=B[\/latex] or [latex]\\mathrm{-A}=B[\/latex], assuming [latex]B>0[\/latex].<\/li>\n<li>Solve for [latex]x[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_04\" class=\"example\">\n<div id=\"fs-id1165137619575\" class=\"exercise\">\n<div id=\"fs-id1165135309797\" class=\"problem textbox shaded\">\n<h3>Example 4: Finding the Zeros of an Absolute Value Function<\/h3>\n<p id=\"fs-id1165137527684\">For the function [latex]f\\left(x\\right)=|4x+1|-7[\/latex] , find the values of\u00a0[latex]x[\/latex] such that\u00a0[latex]\\text{ }f\\left(x\\right)=0[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q621359\">Show Solution<\/span><\/p>\n<div id=\"q621359\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}&0=|4x+1|-7 &&&&&& \\text{Substitute 0 for }f\\left(x\\right). \\\\ &7=|4x+1| &&&&&& \\text{Isolate the absolute value on one side of the equation}.\\\\ & \\\\ &7=4x+1 & \\text{or} &&& -7=4x+1 && \\text{Break into two separate equations and solve}. \\\\ &6=4x &&&& -8=4x \\\\ & \\\\ &x=\\frac{6}{4}=\\frac{3}{2}=1.5 &&&& \\text{ }x=\\frac{-8}{4}=-2 \\end{align}[\/latex]<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010619\/CNX_Precalc_Figure_01_06_011F2.jpg\" alt=\"Graph an absolute function with x-intercepts at -2 and 1.5.\" width=\"731\" height=\"476\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137870931\">The function outputs 0 when [latex]x=1.5[\/latex] or [latex]x=-2[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><span id=\"fs-id1165137662351\"><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137843093\">For the function [latex]f\\left(x\\right)=|2x - 1|-3[\/latex], find the values of [latex]x[\/latex] such that [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q796930\">Show Solution<\/span><\/p>\n<div id=\"q796930\" class=\"hidden-answer\" style=\"display: none\">\n<p><span class=\"s1\">[latex]x=-1[\/latex] or [latex]x=2[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm165746\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=165746&theme=oea&iframe_resize_id=ohm165746\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1165135175321\" class=\"note precalculus qa textbox\">\n<p id=\"fs-id1165135606935\"><strong>Q &amp; A<\/strong><\/p>\n<p><strong>Should we always expect two answers when solving [latex]|A|=B?[\/latex]<\/strong><\/p>\n<p id=\"fs-id1165137755892\"><em>No. We may find one, two, or even no answers. For example, there is no solution to\u00a0<\/em>[latex]2+|3x - 5|=1[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165137911662\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137647413\">How To: Given an absolute value equation, solve it.<\/h3>\n<ol id=\"fs-id1165137589466\">\n<li>Isolate the absolute value term.<\/li>\n<li>Use [latex]|A|=B[\/latex] to write [latex]A=B[\/latex] or [latex]A=\\mathrm{-B}[\/latex].<\/li>\n<li>Solve for [latex]x[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_05\" class=\"example\">\n<div id=\"fs-id1165137727865\" class=\"exercise\">\n<div id=\"fs-id1165135195112\" class=\"problem textbox shaded\">\n<h3>Example 5: Solving an Absolute Value Equation<\/h3>\n<p id=\"fs-id1165137695200\">Solve [latex]1=4|x - 2|+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q134452\">Show Solution<\/span><\/p>\n<div id=\"q134452\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135210177\">First we isolate the absolute value expression on one side of the equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}1&=4|x - 2|+2 \\\\ -1&=4|x - 2| \\\\ -\\frac{1}{4}&=|x - 2| \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137611734\">The absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. At this point, we notice that this equation has no solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm152838\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=152838&theme=oea&iframe_resize_id=ohm152838\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1165137465993\" class=\"note precalculus qa textbox\">\n<p id=\"fs-id1165137573052\"><strong>Q &amp; A<\/strong><\/p>\n<p><strong>In Example 5, if the functions [latex]f\\left(x\\right)=1[\/latex] and [latex]g\\left(x\\right)=4|x - 2|+2[\/latex] were graphed on the same set of axes, would the graphs intersect?<\/strong><\/p>\n<p id=\"fs-id1165137602208\"><em>No. The graphs of [latex]f[\/latex] and [latex]g[\/latex] would not intersect. This confirms, graphically, that the equation [latex]1=4|x - 2|+2[\/latex] has no solution.<\/em><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010619\/CNX_Precalc_Figure_01_06_0122.jpg\" alt=\"Graph of g(x)=4|x-2|+2 and f(x)=1.\" width=\"487\" height=\"476\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137735930\">Find where the graph of the function [latex]f\\left(x\\right)=-|x+2|+3[\/latex] intersects the horizontal and vertical axes.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q960627\">Show Solution<\/span><\/p>\n<div id=\"q960627\" class=\"hidden-answer\" style=\"display: none\">\n<p><span class=\"s1\">[latex]f\\left(0\\right)=1[\/latex], so the graph intersects the vertical axis at [latex]\\left(0,1\\right)[\/latex]. [latex]f\\left(x\\right)=0[\/latex] when [latex]x=-5[\/latex] and [latex]x=1[\/latex] so the graph intersects the horizontal axis at [latex]\\left(-5,0\\right)[\/latex] and [latex]\\left(1,0\\right)[\/latex].<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Solving an Absolute Value Inequality<\/h2>\n<p id=\"fs-id1165137583863\">Absolute value expressions may not always involve equations. Instead we may need to solve where an expression is within a range of values. We would use an absolute value inequality to solve such an equation. An <strong>absolute value inequality<\/strong> is an inequality of the form<\/p>\n<div id=\"fs-id1165134065110\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|{A}|<{ B },|{ A }|\\le{ B },|{ A }|>{ B },\\text{ or } |{ A }|\\ge { B }[\/latex],<\/div>\n<p id=\"fs-id1165135154162\">where an expression [latex]A[\/latex] (and possibly but not usually [latex]B[\/latex] ) depends on a variable [latex]x[\/latex]. Solving the inequality means finding the set of all [latex]x[\/latex] that satisfy the inequality. Usually this set will be an interval or the union of two intervals.<\/p>\n<p id=\"fs-id1165137580992\">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 functions. The advantage of the algebraic approach is it yields solutions that may be difficult to read from the graph.<\/p>\n<p id=\"fs-id1165137557647\">For example, we know that all numbers within 200 units of 0 may be expressed as<\/p>\n<div id=\"fs-id1165137543814\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|x|<{ 200 }\\text{ or }{ -200 }<{ x }<{ 200 }\\text{ }[\/latex]<\/div>\n<p id=\"fs-id1165137610749\">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 values [latex]x[\/latex]\u00a0such that the distance between [latex]x[\/latex] and 600 is less than 200. We represent the distance between [latex]x[\/latex]\u00a0and 600 as [latex]|{ x } - {600 }|[\/latex].<\/p>\n<div id=\"fs-id1165137755666\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|{ x } -{ 600 }|<{ 200 }[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">OR<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{ -200 }<{ x } - { 600 }<{ 200 }[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{-200 }+{ 600 }<{ x } - {600 }+{ 600 }<{ 200 }+{ 600 }[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{ 400 }<{ x }<{ 800 }[\/latex]<\/div>\n<p id=\"fs-id1165137804310\">This means our returns would be between $400 and $800.<\/p>\n<p id=\"fs-id1165137507358\">Sometimes an absolute value inequality problem will be presented to us in terms of a shifted and\/or stretched or compressed absolute value function, where we must determine for which values of the input the function\u2019s output will be negative or positive.<\/p>\n<div id=\"fs-id1165137667916\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137502428\">How To: Given an absolute value inequality of the form [latex]|x-A|\\le B[\/latex] for real numbers [latex]a[\/latex] and [latex]b[\/latex] where [latex]b[\/latex] is positive, solve the absolute value inequality algebraically.<\/h3>\n<ol id=\"fs-id1165137563287\">\n<li>Find boundary points by solving [latex]|x-A|=B[\/latex].<\/li>\n<li>Test intervals created by the boundary points to determine where [latex]|x-A|\\le B[\/latex].<\/li>\n<li>Write the interval or union of intervals satisfying the inequality in interval, inequality, or set-builder notation.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_06\" class=\"example\">\n<div id=\"fs-id1165135704112\" class=\"exercise\">\n<div id=\"fs-id1165137401703\" class=\"problem textbox shaded\">\n<h3>Example 6: Solving an Absolute Value Inequality<\/h3>\n<p id=\"fs-id1165135342955\">Solve [latex]|x - 5|\\le 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q183338\">Show Solution<\/span><\/p>\n<div id=\"q183338\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137645044\">With both approaches, we will need to know first where the corresponding equality is true. In this case we first will find where [latex]|x - 5|=4[\/latex]. We do this because the absolute value is a function with no breaks, so the only way the function values can switch from being less than 4 to being greater than 4 is by passing through where the values equal 4. Solve [latex]|x - 5|=4[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x - 5&=4 & \\text{ or } && {x - 5 }&={ -4 }\\\\ {x }&= {9} &\\text{ or } && { x }&={ 1 } \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137665217\">After determining that the absolute value is equal to 4 at [latex]x=1[\/latex] and [latex]x=9[\/latex], we know the graph can change only from being less than 4 to greater than 4 at these values. This divides the number line up into three intervals:<\/p>\n<p style=\"text-align: center;\">[latex]{ x }<{ 1 },\\text{ }{ 1 }<{ x }<{ 9 },\\text{ and }{ x }>{ 9 }[\/latex].<\/p>\n<p id=\"fs-id1165137422669\">To determine when the function is less than 4, we could choose a value in each interval and see if the output is less than or greater than 4, as shown in the table below.<\/p>\n<table id=\"Table_01_06_01\" style=\"border: 1px dashed #bbbbbb;\" summary=\"Table describing the interval test for certain inequalities for x. So if x&lt;1 and f(x)=0, then |0-5|&gt;4. If1&lt; x&lt;9 and f(x)=6, then |6-5|&lt;4. If x&lt;9 and f(x)=11, then |11-5|&gt;4.\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th>Interval test [latex]x[\/latex]<\/th>\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\n<th colspan=\"2\">[latex]<4[\/latex] or [latex]>4?[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{ x }<{ 1 }[\/latex]<\/td>\n<td>0<\/td>\n<td>[latex]|0 - 5|=5[\/latex]<\/td>\n<td>Greater than<\/td>\n<\/tr>\n<tr>\n<td>[latex]{ 1 }<{ x }<{ 9 }[\/latex]<\/td>\n<td>6<\/td>\n<td>[latex]|6 - 5|=1[\/latex]<\/td>\n<td>Less than<\/td>\n<\/tr>\n<tr>\n<td>[latex]{ x }>{ 9 }[\/latex]<\/td>\n<td>11<\/td>\n<td>[latex]|11 - 5|=6[\/latex]<\/td>\n<td>Greater than<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137558949\">Because [latex]1\\le x\\le 9[\/latex] is the only interval in which the output at the test value is less than 4, we can conclude that the solution to [latex]|x - 5|\\le 4[\/latex] is [latex]1\\le x\\le 9[\/latex], or [latex]\\left[1,9\\right][\/latex].<\/p>\n<p>To use a graph, we can sketch the function [latex]f\\left(x\\right)=|x - 5|[\/latex]. To help us see where the outputs are 4, the line [latex]g\\left(x\\right)=4[\/latex] could also be sketched.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010620\/CNX_Precalc_Figure_01_06_0132.jpg\" alt=\"Graph of an absolute function and a vertical line, demonstrating how to see what outputs are less than the vertical line.\" width=\"487\" height=\"288\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11.<\/b> Graph to find the points satisfying an absolute value inequality.<\/p>\n<\/div>\n<p id=\"fs-id1165137874583\">We can see the following:<\/p>\n<ul id=\"fs-id1165134148370\">\n<li>The output values of the absolute value are equal to 4 at [latex]x=1[\/latex] and [latex]x=9[\/latex].<\/li>\n<li>The graph of [latex]f[\/latex] is below the graph of [latex]g[\/latex] on [latex]1<x<9[\/latex]. This means the output values of [latex]f\\left(x\\right)[\/latex] are less than the output values of [latex]g\\left(x\\right)[\/latex].<\/li>\n<li>The absolute value is less than or equal to 4 between these two points, when [latex]1\\le x\\le 9[\/latex]. In interval notation, this would be the interval [latex]\\left[1,9\\right][\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135381301\" class=\"commentary\">\n<h3>Analysis of the Solution<\/h3>\n<p id=\"fs-id1165135689465\">For absolute value inequalities,<\/p>\n<div id=\"fs-id1165135650752\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]|x-A|<C[latex] can be rewritten [latex]-C<x-A<C[\/latex] and [latex]|x-A| > C[\/latex] can be rewritten [latex]x-A < -C \\text{ or } x-A > C[\/latex].<\/div>\n<p id=\"fs-id1165135195336\">The [latex]<[\/latex] or [latex]>[\/latex] symbol may be replaced by [latex]\\le \\text{ or }\\ge[\/latex].<\/p>\n<p id=\"fs-id1165135524557\">So, for this example, we could use this alternative approach.<\/p>\n<div id=\"fs-id1165134226778\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{gathered}|x - 5|\\le 4 \\\\ -4\\le x - 5\\le 4 \\\\ -4+5\\le x - 5+5\\le 4+5 \\\\ 1\\le x\\le 9 \\end{gathered}[\/latex] [latex]\\begin{align} &\\\\&&& \\text{Rewrite by removing the absolute value bars}. \\\\ &&& \\text{Isolate the }x. \\\\& \\end{align}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137450875\">Solve [latex]|x+2|\\le 6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q606238\">Show Solution<\/span><\/p>\n<div id=\"q606238\" class=\"hidden-answer\" style=\"display: none\">\n<p><span class=\"s1\">[latex]4\\le x\\le 8[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137530158\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135445865\">How To: Given an absolute value function, solve for the set of inputs where the output is positive (or negative).<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1165137572514\">\n<li>Set the function equal to zero and solve for the boundary points of the solution set.<\/li>\n<li>Use test points or a graph to determine where the function\u2019s output is positive or negative.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_07\" class=\"example\">\n<div id=\"fs-id1165137409791\" class=\"exercise\">\n<div id=\"fs-id1165137838822\" class=\"problem textbox shaded\">\n<h3>Example 7: Using a Graphical Approach to Solve Absolute Value Inequalities<\/h3>\n<p id=\"fs-id1165137933778\">Given the function [latex]f\\left(x\\right)=-\\frac{1}{2}|4x - 5|+3[\/latex], determine the [latex]x\\text{-}[\/latex] values for which the function values are negative.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q767825\">Show Solution<\/span><\/p>\n<div id=\"q767825\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137431799\">We are trying to determine where [latex]f\\left(x\\right)<0[\/latex], which is when [latex]-\\frac{1}{2}|4x - 5|+3<0[\/latex]. We begin by isolating the absolute value.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}-\\frac{1}{2}|4x - 5|&<-3 \\hfill && \\text{Multiply both sides by -2, and reverse the inequality}. \\\\ |4x - 5|&>6 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135317447\">Next we solve for the equality [latex]|4x - 5|=6[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}4x - 5&=6 & \\text{or} && 4x - 5&=-6 \\\\ 4x - 5&=6 &&& 4x&=-1 \\\\ x&=\\frac{11}{4} &&& x&=-\\frac{1}{4} \\end{align}[\/latex]<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010620\/CNX_Precalc_Figure_01_06_0142.jpg\" alt=\"Graph of an absolute function with x-intercepts at -0.25 and 2.75.\" width=\"487\" height=\"365\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 12<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135344887\">Now, we can examine the graph of [latex]f[\/latex] to observe where the output is negative. We will observe where the branches are below the <em>x<\/em>-axis. Notice that it is not even 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 has been reflected vertically.<span id=\"fs-id1165137728256\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137725461\">We observe that the graph of the function is below the <em>x<\/em>-axis left of [latex]x=-\\frac{1}{4}[\/latex] and right of [latex]x=\\frac{11}{4}[\/latex]. This means the function values are negative to the left of the first horizontal intercept at [latex]x=-\\frac{1}{4}[\/latex], and negative to the right of the second intercept at [latex]x=\\frac{11}{4}[\/latex]. This gives us the solution to the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]x<-\\frac{1}{4}\\text{ }\\text{or}\\text{ }x>\\frac{11}{4}[\/latex]<\/p>\n<p id=\"fs-id1165135502945\">In interval notation, this would be [latex]\\left(-\\infty ,-0.25\\right)\\cup \\left(2.75,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137784438\">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=\"q416287\">Show Solution<\/span><\/p>\n<div id=\"q416287\" class=\"hidden-answer\" style=\"display: none\">\n<p><span class=\"s1\">[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]<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm86181\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=86181&theme=oea&iframe_resize_id=ohm86181\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135332513\">\n<li>The absolute value function is commonly used to measure distances between points.<\/li>\n<li>Applied problems, such as ranges of possible values, can also be solved using the absolute value function.<\/li>\n<li>The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction.<\/li>\n<li>In an absolute value equation, an unknown variable is the input of an absolute value function.<\/li>\n<li>If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable.<\/li>\n<li>An absolute value equation may have one solution, two solutions, or no solutions.<\/li>\n<li>An absolute value inequality is similar to an absolute value equation but takes the form [latex]|A|<B,|A|\\le B,|A|>B,\\text{ or }|A|\\ge B[\/latex]. It can be solved by determining the boundaries of the solution set and then testing which segments are in the set.<\/li>\n<li>Absolute value inequalities can also be solved graphically.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135191341\" class=\"definition\">\n<dt>absolute value equation<\/dt>\n<dd id=\"fs-id1165137627032\">an equation of the form [latex]|A|=B[\/latex], with [latex]B\\ge 0[\/latex]; it will have solutions when [latex]A=B[\/latex] or [latex]A=-B[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137560214\" class=\"definition\">\n<dt>absolute value inequality<\/dt>\n<dd id=\"fs-id1165135173524\">a relationship in the form [latex]|{ A }|<{ B },|{ A }|\\le { B },|{ A }|>{ B },\\text{or }|{ A }|\\ge{ B }[\/latex]<\/dd>\n<\/dl>\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-13736\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":23588,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-13736","chapter","type-chapter","status-publish","hentry"],"part":10705,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13736","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/23588"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13736\/revisions"}],"predecessor-version":[{"id":15828,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13736\/revisions\/15828"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/10705"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13736\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=13736"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=13736"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=13736"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=13736"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}