{"id":63,"date":"2019-08-20T17:01:52","date_gmt":"2019-08-20T21:01:52","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/absolute-value-functions\/"},"modified":"2022-06-01T10:39:24","modified_gmt":"2022-06-01T14:39:24","slug":"absolute-value-functions","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/absolute-value-functions\/","title":{"raw":"Absolute Value Functions","rendered":"Absolute Value Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section you will:\n<ul>\n \t<li>Graph an absolute value function.<\/li>\n \t<li>Solve an absolute value equation.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_01_06_001\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141725\/CNX_Precalc_Figure_01_06_001n.jpg\" alt=\"The Milky Way.\" width=\"488\" height=\"338\"> <strong>Figure 1. <\/strong>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]\n\n<div class=\"wp-caption-text\"><\/div>\n<\/div>\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 continue our investigation of <span class=\"no-emphasis\">absolute value functions<\/span>.<\/p>\n\n<div id=\"fs-id1165137426078\" class=\"bc-section section\">\n<h3>Understanding Absolute Value<\/h3>\n<p id=\"fs-id1165135449691\">Recall that in its basic form[latex]\\,f\\left(x\\right)=|x|,\\,[\/latex]the absolute value function is one of our toolkit functions. The <span class=\"no-emphasis\">absolute value<\/span> 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. Knowing this, we can use absolute value functions to solve some kinds of real-world problems.<\/p>\n\n<div id=\"fs-id1165135404116\" class=\"textbox key-takeaways\">\n<h3>Absolute Value Function<\/h3>\n<p id=\"fs-id1165137832269\">The absolute value function can be defined as a piecewise function<\/p>\n\n<div id=\"fs-id1165137665354\" class=\"unnumbered aligncenter\">[latex]\\,f\\left(x\\right)=|x|=\\bigg\\{\\begin{array}{ccc}x&amp; \\text{if}&amp; x\\ge 0\\\\ -x&amp; \\text{if}&amp; x&lt;0\\end{array}\\,[\/latex]<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137657277\">\n<div id=\"fs-id1165137579723\">\n<h3>Using Absolute Value to Determine Resistance<\/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]\\,\\text{\u00b11%,}\\,\u00b1\\text{5%,}\\,[\/latex]or[latex]\\,\u00b1\\text{10%}\\text{.}[\/latex]<\/p>\n<p id=\"fs-id1165135175007\">Suppose we have a resistor rated at 680 ohms,[latex]\\,\u00b15%.\\,[\/latex]Use the absolute value function to express the range of possible values of the actual resistance.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137786481\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137786481\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137786481\"]\n<p id=\"fs-id1165137600783\">We can find that 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\n<div id=\"fs-id1165135176481\" class=\"unnumbered aligncenter\">[latex]|R-680|\\le 34[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137540249\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137828265\">\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\n<\/div>\n<div id=\"fs-id1165137715310\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137715310\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137715310\"]\n<p id=\"fs-id1165134090680\">using the variable[latex]\\,p\\,[\/latex]for passing,[latex]\\,|p-80|\\le 20[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135186288\" class=\"bc-section section\">\n<h3>Graphing an Absolute Value Function<\/h3>\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 <span class=\"no-emphasis\">origin<\/span> in <a class=\"autogenerated-content\" href=\"#Figure_01_06_003\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_06_003\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141728\/CNX_Precalc_Figure_01_06_003.jpg\" alt=\"Graph of an absolute function\" width=\"487\" height=\"251\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165135639350\"><a class=\"autogenerated-content\" href=\"#Figure_01_06_004\">(Figure)<\/a> shows the graph of[latex]\\,y=2|x\u20133|+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\n<div id=\"Figure_01_06_004\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141743\/CNX_Precalc_Figure_01_06_004.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"487\" height=\"486\"> <strong>Figure 3.<\/strong>[\/caption]\n\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135187768\">\n<div id=\"fs-id1165137741094\">\n<h3>Writing an Equation for an Absolute Value Function Given a Graph<\/h3>\n<p id=\"fs-id1165135414332\">Write an equation for the function graphed in <a class=\"autogenerated-content\" href=\"#Figure_01_06_005\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_06_005\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141801\/CNX_Precalc_Figure_01_06_005.jpg\" alt=\"Graph of an absolute function.\" width=\"487\" height=\"363\"> <strong>Figure 4.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736321\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137736321\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137736321\"]\n<p id=\"fs-id1165137680556\">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. See <a class=\"autogenerated-content\" href=\"#Figure_01_06_006\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_06_006\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141810\/CNX_Precalc_Figure_01_06_006.jpg\" alt=\"Graph of two transformations for an absolute function at (3, -2).\" width=\"487\" height=\"363\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137805107\">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 as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_06_007\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_06_007\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141818\/CNX_Precalc_Figure_01_06_007.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\"> <strong>Figure 6.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137732766\">From this information we can write the equation<\/p>\n\n<div id=\"fs-id1165137425569\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill f\\left(x\\right)&amp; =&amp; 2|x-3|-2,\\hfill &amp; \\phantom{\\rule{1em}{0ex}}\\text{treating the stretch as }a\\text{ vertical stretch,or}\\hfill \\\\ \\hfill f\\left(x\\right)&amp; =&amp; |2\\left(x-3\\right)|-2,\\hfill &amp; \\phantom{\\rule{1em}{0ex}}\\text{treating the stretch as }a\\text{ horizontal compression}.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137706602\">\n<h4>Analysis<\/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\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134377948\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165135245777\"><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]\n<\/em><\/p>\n\n<div id=\"fs-id1165135514699\" class=\"unnumbered aligncenter\">[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\n<div id=\"fs-id1165135173265\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 2&amp; =&amp; a|1-3|-2\\hfill \\\\ \\hfill 4&amp; =&amp; 2a\\hfill \\\\ \\hfill a&amp; =&amp; 2\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137611709\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165134312222\">\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\n<\/div>\n<div id=\"fs-id1165137405204\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137405204\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137405204\"]\n<p id=\"fs-id1165137758260\">[latex]f\\left(x\\right)=-|x+2|+3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203778\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137527840\"><strong>Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?\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.\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 (see <a class=\"autogenerated-content\" href=\"#Figure_01_06_008\">(Figure)<\/a>).\n<\/em><\/p>\n\n<\/div>\n<div id=\"Figure_01_06_008\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141824\/CNX_Precalc_Figure_01_06_008abc.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"975\" height=\"415\"> <strong>Figure 7. <\/strong>(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]\n\n<\/div>\n<div id=\"fs-id1165133257286\" class=\"bc-section section\">\n<h3>Solving an Absolute Value Equation<\/h3>\n<p id=\"fs-id1165137401775\">In <a class=\"target-chapter\" href=\"\/contents\/b88d40b3-a0d5-4a8f-8b2e-85993b911cc0\">Other Type of Equations<\/a>, we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. 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=|2x-6|,\\,[\/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\n<div id=\"fs-id1165137583696\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccccccc}\\hfill 2x-6&amp; =&amp; 8\\hfill &amp; \\phantom{\\rule{1em}{0ex}}\\text{or}\\phantom{\\rule{1em}{0ex}}&amp; \\hfill 2x-6&amp; =&amp; -8\\hfill \\\\ \\hfill 2x&amp; =&amp; 14\\hfill &amp; &amp; \\hfill 2x&amp; =&amp; -2\\hfill \\\\ \\hfill x&amp; =&amp; 7\\hfill &amp; &amp; \\hfill x&amp; =&amp; -1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137641126\">Knowing how to solve problems involving <span class=\"no-emphasis\">absolute value functions<\/span> 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>\nAn absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,\n<div id=\"fs-id1165137646929\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|x|=4,\\hfill \\\\ |2x-1|=3,\\text{or}\\hfill \\\\ |5x+2|-4=9\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1165137692078\" class=\"textbox key-takeaways\">\n<h3>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&lt;0,[\/latex] the equation [latex]|A|=B[\/latex] has no solution.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135160087\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135593248\"><strong>Given the formula for an absolute value function, find the horizontal intercepts of its graph<\/strong>.<\/p>\n\n<ol id=\"fs-id1165131968095\" type=\"1\">\n \t<li>Isolate the absolute value term.<\/li>\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>\n \t<li>Solve for[latex]\\,x.\\,[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137619575\">\n<div id=\"fs-id1165135309797\">\n<h3>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[latex]\\,x\\,[\/latex]such that[latex]\\,f\\left(x\\right)=0.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137405662\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137405662\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137405662\"]\n<div id=\"fs-id1165137618972\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccccccc}\\hfill 0&amp; =&amp; |4x+1|-7\\hfill &amp; &amp; &amp; &amp; &amp; \\text{Substitute 0 for }f\\left(x\\right).\\hfill \\\\ \\hfill 7&amp; =&amp; |4x+1|\\hfill &amp; &amp; &amp; &amp; &amp; \\text{Isolate the absolute value on one side of the equation}.\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\\\ &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\\\ &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\\\ \\hfill 7&amp; =&amp; 4x+1\\hfill &amp; \\text{or}&amp; \\hfill \\phantom{\\rule{2em}{0ex}}-7&amp; =&amp; 4x+1\\hfill &amp; \\text{Break into two separate equations and solve}.\\hfill \\\\ \\hfill 6&amp; =&amp; 4x\\hfill &amp; &amp; \\hfill -8&amp; =&amp; 4x\\hfill &amp; \\\\ &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\\\ \\hfill x&amp; =&amp; \\frac{6}{4}=1.5\\hfill &amp; &amp; \\hfill x&amp; =&amp; \\frac{-8}{4}=-2\\hfill &amp; \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137870931\">The function outputs 0 when[latex]\\,x=\\frac{3}{2}\\,[\/latex]or[latex]\\,x=-2.[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_06_011\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_01_06_011\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141827\/CNX_Precalc_Figure_01_06_011F.jpg\" alt=\"Graph an absolute function with x-intercepts at -2 and 1.5.\" width=\"731\" height=\"476\"> <strong>Figure 8.<\/strong>[\/caption]\n\n<span id=\"fs-id1165137662351\">[\/hidden-answer]<\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137599670\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165135191411\">\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\n<\/div>\n<div id=\"fs-id1165137939483\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137939483\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137939483\"]\n<p id=\"fs-id1165137481390\">[latex]x=-1\\,[\/latex]or[latex]\\,\\,x=2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135175321\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165135606935\"><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<\/em>[latex]\\,2+|3x-5|=1.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571678\" class=\"bc-section section\">\n<div id=\"fs-id1165134257615\" class=\"precalculus media\">\n<p id=\"fs-id1165134332731\">Access these online resources for additional instruction and practice with absolute value.<\/p>\n\n<ul id=\"fs-id1165137508064\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphabsvalue\">Graphing Absolute Value Functions<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphabsvalue2\">Graphing Absolute Value Functions 2<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133291312\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135332513\">\n \t<li>Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See <a class=\"autogenerated-content\" href=\"#Example_01_06_02\">(Figure)<\/a>.<\/li>\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. See <a class=\"autogenerated-content\" href=\"#Example_01_06_03\">(Figure)<\/a>.<\/li>\n \t<li>In an absolute value equation, an unknown variable is the input of an absolute value function.<\/li>\n \t<li>If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See <a class=\"autogenerated-content\" href=\"#Example_01_06_04\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135255406\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137406985\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137734873\">\n<div id=\"fs-id1165135510060\">\n<p id=\"fs-id1165134347447\">How do you solve an absolute value equation?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135206149\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135206149\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135206149\"]\n<p id=\"fs-id1165135189754\">Isolate the absolute value term so that the equation is of the form[latex]\\,|A|=B.\\,[\/latex]Form one equation by setting the expression inside the absolute value symbol,[latex]\\,A,\\,[\/latex]equal to the expression on the other side of the equation,[latex]\\,B.\\,[\/latex]Form a second equation by setting[latex]\\,A\\,[\/latex]equal to the opposite of the expression on the other side of the equation,[latex]\\,-B.\\,[\/latex]Solve each equation for the variable.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137593210\">\n<div id=\"fs-id1165131968049\">\n<p id=\"fs-id1165135440055\">How can you tell whether an absolute value function has two <em>x<\/em>-intercepts without graphing the function?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133103957\">\n<div id=\"fs-id1165133103959\">\n\nWhen solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?\n\n<\/div>\n<div id=\"fs-id1165134475281\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134475281\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134475281\"]\n<p id=\"fs-id1165137662762\">The graph of the absolute value function does not cross the[latex]\\,x[\/latex]-axis, so the graph is either completely above or completely below the[latex]\\,x[\/latex]-axis.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135264708\">\n<div id=\"fs-id1165135149122\">\n<p id=\"fs-id1165135149124\">How can you use the graph of an absolute value function to determine the <em>x<\/em>-values for which the function values are negative?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134273549\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<div id=\"fs-id1165137841613\">\n<div id=\"fs-id1165137841615\">\n<p id=\"fs-id1165137579699\">Describe all numbers[latex]\\,x\\,[\/latex]that are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135445894\">\n<div>\n<p id=\"fs-id1165135169187\">Describe all numbers[latex]\\,x\\,[\/latex]that are at a distance of[latex]\\,\\frac{1}{2}\\,[\/latex]from the number \u22124. Express this set of numbers using absolute value notation.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135356596\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135356596\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135356596\"]\n<p id=\"fs-id1165134278683\">[latex]\\,|x+4|=\\frac{1}{2}\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137542576\">\n<div id=\"fs-id1165137648320\">\n<p id=\"fs-id1165137648322\">Describe the situation in which the distance that point[latex]\\,x\\,[\/latex]is from 10 is at least 15 units. Express this set of numbers using absolute value notation.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134057540\">\n<div id=\"fs-id1165137464076\">\n<p id=\"fs-id1165137464078\">Find all function values[latex]\\,f\\left(x\\right)\\,[\/latex]such that the distance from[latex]\\,f\\left(x\\right)\\,[\/latex]to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137541376\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137541376\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137541376\"]\n<p id=\"fs-id1165137572565\">[latex]|f\\left(x\\right)-8|&lt;0.03[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137589731\">For the following exercises, find the <em>x<\/em>- and <em>y<\/em>-intercepts of the graphs of each function.<\/p>\n\n<div id=\"fs-id1165134401702\">\n<div id=\"fs-id1165135362510\">\n<p id=\"fs-id1165135362512\">[latex]f\\left(x\\right)=4|x-3|+4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137824535\">\n<div id=\"fs-id1165137824537\">\n<p id=\"fs-id1165134220856\">[latex]f\\left(x\\right)=-3|x-2|-1[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137433352\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137433352\"]\n<p id=\"fs-id1165137433352\">[latex]\\left(0,-7\\right);\\,[\/latex]no[latex]\\,x[\/latex]-intercepts<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134371172\">\n<div id=\"fs-id1165134371174\">\n<p id=\"fs-id1165137590699\">[latex]f\\left(x\\right)=-2|x+1|+6[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137656160\">\n<div id=\"fs-id1165137656163\">\n<p id=\"fs-id1165137938872\">[latex]f\\left(x\\right)=-5|x+2|+15[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137427184\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137427184\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137427184\"]\n<p id=\"fs-id1165137427186\">[latex]\\left(0,\\,5\\right),\\left(1,0\\right),\\left(-5,0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"eip-613\">\n<div id=\"eip-946\">\n<p id=\"eip-55\">[latex]f\\left(x\\right)=2|x-1|-6[\/latex]<\/p>\n\n<div class=\"textbox shaded\">[reveal-answer q=\"59208\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"59208\"][latex]\\left(0,-4\\right),\\left(4,0\\right),\\left(-2,0\\right)[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-240\">\n<div id=\"eip-147\">\n<p id=\"eip-934\">[latex]f\\left(x\\right)=|-2x+1|-13[\/latex]<\/p>\n\n<div class=\"textbox shaded\">[reveal-answer q=\"301101\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"301101\"]\n<div id=\"eip-240\">\n<div id=\"eip-302\">\n<p id=\"eip-20\">[latex]\\left(0,-12\\right),\\left(-6,0\\right),\\left(7,0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-452\">\n<div id=\"eip-163\">\n<p id=\"eip-223\">[latex]f\\left(x\\right)=-|x-9|+16[\/latex]<\/p>\n\n<div class=\"textbox shaded\">[reveal-answer q=\"155957\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"155957\"]\n<div id=\"fs-id1165134273549\" class=\"bc-section section\">\n<div id=\"eip-452\">\n<div>[latex]\\left(0,7\\right),\\left(25,0\\right),\\left(-7,0\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165133047532\">For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.<\/p>\n\n<div id=\"fs-id1165137891404\">\n<div id=\"fs-id1165137817696\">\n<p id=\"fs-id1165137817699\">[latex]y=|x-1|[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"637472\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"637472\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141838\/CNX_Precalc_Figure_01_06_201.jpg\" alt=\"Graph of an absolute function with points at (-1, 2), (0, 1), (1, 0), (2, 1), and (3, 2).\">[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137679099\">\n<div id=\"fs-id1165137679101\">\n<p id=\"fs-id1165137418760\">[latex]y=|x+1|[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135422938\">\n<div id=\"fs-id1165135422940\">\n<p id=\"fs-id1165137652979\">[latex]y=|x|+1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137414774\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137414774\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137414774\"]<span id=\"fs-id1165137658298\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141840\/CNX_Precalc_Figure_01_06_203.jpg\" alt=\"Graph of an absolute function with points at (-2, 3), (-1, 2), (0, 1), (1, 2), and (2, 3).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1165137406944\">For the following exercises, graph the given functions by hand.<\/p>\n\n<div id=\"fs-id1165135332726\">\n<div id=\"fs-id1165135332729\">\n<p id=\"fs-id1165135251459\">[latex]y=|x|-2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137601710\">\n<div id=\"fs-id1165137601713\">\n<p id=\"fs-id1165137419974\">[latex]y=-|x|[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137400044\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137400044\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137400044\"]<span id=\"fs-id1165135160170\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141846\/CNX_Precalc_Figure_01_06_205.jpg\" alt=\"Graph of an absolute function.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137735265\">\n<div id=\"fs-id1165137431347\">\n<p id=\"fs-id1165137431349\">[latex]y=-|x|-2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731590\">\n<div id=\"fs-id1165137603675\">\n<p id=\"fs-id1165137603678\">[latex]y=-|x-3|-2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137645253\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137645253\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137645253\"]<span id=\"fs-id1165135653964\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141850\/CNX_Precalc_Figure_01_06_207.jpg\" alt=\"Graph of an absolute function.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137394585\">\n<div id=\"fs-id1165137394587\">\n<p id=\"fs-id1165135572124\">[latex]f\\left(x\\right)=-|x-1|-2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135335986\">\n<div id=\"fs-id1165137651575\">\n<p id=\"fs-id1165137651578\">[latex]f\\left(x\\right)=-|x+3|+4[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137434149\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137434149\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137434149\"]<span id=\"fs-id1165137553071\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141852\/CNX_Precalc_Figure_01_06_209.jpg\" alt=\"Graph of an absolute function.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137705796\">\n<div id=\"fs-id1165137705798\">\n<p id=\"fs-id1165137469167\">[latex]f\\left(x\\right)=2|x+3|+1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137427199\">\n<div id=\"fs-id1165137619904\">\n<p id=\"fs-id1165137619906\">[latex]f\\left(x\\right)=3|x-2|+3[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137436399\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137436399\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137436399\"]<span id=\"fs-id1165137410325\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141855\/CNX_Precalc_Figure_01_06_211.jpg\" alt=\"Graph of an absolute function.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137715460\">\n<div id=\"fs-id1165137715462\">\n<p id=\"fs-id1165137469722\">[latex]f\\left(x\\right)=|2x-4|-3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135258293\">\n<div id=\"fs-id1165135258295\">\n<p id=\"fs-id1165137452029\">[latex]f\\left(x\\right)=|3x+9|+2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137433126\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137433126\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137433126\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141857\/CNX_Precalc_Figure_01_06_213.jpg\" alt=\"Graph of an absolute function.\">[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137464095\">\n<div id=\"fs-id1165137464097\">\n<p id=\"fs-id1165137470140\">[latex]f\\left(x\\right)=-|x-1|-3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137443657\">\n<div id=\"fs-id1165137911316\">\n<p id=\"fs-id1165137911318\">[latex]f\\left(x\\right)=-|x+4|-3[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137399944\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137399944\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137399944\"]<span id=\"fs-id1165137451010\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141859\/CNX_Precalc_Figure_01_06_215.jpg\" alt=\"Graph of an absolute function.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137803326\">\n<div id=\"fs-id1165137803328\">\n<p id=\"fs-id1165137824374\">[latex]f\\left(x\\right)=\\frac{1}{2}|x+4|-3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137897208\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<div id=\"fs-id1165137749758\">\n<div id=\"fs-id1165137749760\">\n<p id=\"fs-id1165137460158\">Use a graphing utility to graph [latex]f\\left(x\\right)=10|x-2|[\/latex] on the viewing window [latex]\\left[0,4\\right].[\/latex] Identify the corresponding range. Show the graph.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134042934\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134042934\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134042934\"]\n<p id=\"fs-id1165134042935\">range:[latex]\\,\\left[0,20\\right][\/latex]<\/p>\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141901\/CNX_Precalc_Figure_01_06_217.jpg\" alt=\"Graph of an absolute function.\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137413783\">\n<div id=\"fs-id1165137434783\">\n<p id=\"fs-id1165137434785\">Use a graphing utility to graph[latex]\\,f\\left(x\\right)=-100|x|+100\\,[\/latex]on the viewing window[latex]\\,\\left[-5,5\\right].\\,[\/latex]Identify the corresponding range. Show the graph.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137831208\">For the following exercises, graph each function using a graphing utility. Specify the viewing window.<\/p>\n\n<div id=\"fs-id1165137762283\">\n<div id=\"fs-id1165135464843\">\n<p id=\"fs-id1165137724085\">[latex]f\\left(x\\right)=-0.1|0.1\\left(0.2-x\\right)|+0.3[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137812573\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137812573\"]\n<p id=\"fs-id1165137812573\">[latex]x\\text{-}[\/latex]intercepts:<\/p>\n<span id=\"fs-id1165137784866\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141903\/CNX_Precalc_Figure_01_06_219.jpg\" alt=\"Graph of an absolute function.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165134039354\">\n<div id=\"eip-id1165134039356\">\n<p id=\"fs-id1165137483195\">[latex]f\\left(x\\right)=4\u00d7{10}^{9}|x-\\left(5\u00d7{10}^{9}\\right)|+2\u00d7{10}^{9}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137419467\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165137901338\">For the following exercises, solve the inequality.<\/p>\n\n<div id=\"fs-id1165137434569\">\n<div id=\"fs-id1165137434570\">\n<p id=\"fs-id1165137434571\">If possible, find all values of [latex]a[\/latex] such that there are no [latex]x\\text{-}[\/latex]intercepts for [latex]f\\left(x\\right)=2|x+1|+a.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137639316\">\n<div id=\"fs-id1165137652958\">\n<p id=\"fs-id1165137652960\">If possible, find all values of[latex]\\,a\\,[\/latex]such that there are no [latex]\\,y[\/latex]-intercepts for[latex]\\,f\\left(x\\right)=2|x+1|+a.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137648025\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137648025\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137648025\"]\n<p id=\"fs-id1165137454792\">There is no solution for[latex]\\,a\\,[\/latex]that will keep the function from having a[latex]\\,y[\/latex]-intercept. The absolute value function always crosses the [latex]\\,y[\/latex]-intercept when[latex]\\,x=0.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135172151\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165137641899\">\n<div id=\"fs-id1165137641901\">\n<p id=\"fs-id1165137459748\">Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and[latex]\\,x\\,[\/latex]represents the distance from city B to city A, express this using absolute value notation.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137812302\">\n<div id=\"fs-id1165137812304\">\n<p id=\"fs-id1165137431941\">The true proportion[latex]\\,p\\,[\/latex]of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135431083\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135431083\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135431083\"]\n<p id=\"fs-id1165134042456\">[latex]|p-0.08|\\le 0.015[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137598001\">\n<div id=\"fs-id1165137562568\">\n<p id=\"fs-id1165137562570\">Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable[latex]\\,x\\,[\/latex]for the score.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137758810\">\n<div id=\"fs-id1165137758812\">\n<p id=\"fs-id1165135332394\">A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using[latex]\\,x\\,[\/latex]as the diameter of the bearing, write this statement using absolute value notation.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135192953\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135192953\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135192953\"]\n[latex]|x-5.0|\\le 0.01[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137732323\">\n<div id=\"fs-id1165137732325\">\n<p id=\"fs-id1165137673610\">The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is[latex]\\,x\\,[\/latex]inches, express the tolerance using absolute value notation.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section you will:<\/p>\n<ul>\n<li>Graph an absolute value function.<\/li>\n<li>Solve an absolute value equation.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_01_06_001\" class=\"medium\">\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141725\/CNX_Precalc_Figure_01_06_001n.jpg\" alt=\"The Milky Way.\" width=\"488\" height=\"338\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>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)<\/figcaption><\/figure>\n<div class=\"wp-caption-text\"><\/div>\n<\/div>\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 continue our investigation of <span class=\"no-emphasis\">absolute value functions<\/span>.<\/p>\n<div id=\"fs-id1165137426078\" class=\"bc-section section\">\n<h3>Understanding Absolute Value<\/h3>\n<p id=\"fs-id1165135449691\">Recall that in its basic form[latex]\\,f\\left(x\\right)=|x|,\\,[\/latex]the absolute value function is one of our toolkit functions. The <span class=\"no-emphasis\">absolute value<\/span> 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. Knowing this, we can use absolute value functions to solve some kinds of real-world problems.<\/p>\n<div id=\"fs-id1165135404116\" class=\"textbox key-takeaways\">\n<h3>Absolute Value Function<\/h3>\n<p id=\"fs-id1165137832269\">The absolute value function can be defined as a piecewise function<\/p>\n<div id=\"fs-id1165137665354\" class=\"unnumbered aligncenter\">[latex]\\,f\\left(x\\right)=|x|=\\bigg\\{\\begin{array}{ccc}x& \\text{if}& x\\ge 0\\\\ -x& \\text{if}& x<0\\end{array}\\,[\/latex]<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137657277\">\n<div id=\"fs-id1165137579723\">\n<h3>Using Absolute Value to Determine Resistance<\/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]\\,\\text{\u00b11%,}\\,\u00b1\\text{5%,}\\,[\/latex]or[latex]\\,\u00b1\\text{10%}\\text{.}[\/latex]<\/p>\n<p id=\"fs-id1165135175007\">Suppose we have a resistor rated at 680 ohms,[latex]\\,\u00b15%.\\,[\/latex]Use the absolute value function to express the range of possible values of the actual resistance.<\/p>\n<\/div>\n<div id=\"fs-id1165137786481\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137600783\">We can find that 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=\"unnumbered aligncenter\">[latex]|R-680|\\le 34[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137540249\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137828265\">\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>\n<div id=\"fs-id1165137715310\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134090680\">using the variable[latex]\\,p\\,[\/latex]for passing,[latex]\\,|p-80|\\le 20[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135186288\" class=\"bc-section section\">\n<h3>Graphing an Absolute Value Function<\/h3>\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 <span class=\"no-emphasis\">origin<\/span> in <a class=\"autogenerated-content\" href=\"#Figure_01_06_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_06_003\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141728\/CNX_Precalc_Figure_01_06_003.jpg\" alt=\"Graph of an absolute function\" width=\"487\" height=\"251\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135639350\"><a class=\"autogenerated-content\" href=\"#Figure_01_06_004\">(Figure)<\/a> shows the graph of[latex]\\,y=2|x\u20133|+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 id=\"Figure_01_06_004\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141743\/CNX_Precalc_Figure_01_06_004.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"487\" height=\"486\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135187768\">\n<div id=\"fs-id1165137741094\">\n<h3>Writing an Equation for an Absolute Value Function Given a Graph<\/h3>\n<p id=\"fs-id1165135414332\">Write an equation for the function graphed in <a class=\"autogenerated-content\" href=\"#Figure_01_06_005\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_06_005\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141801\/CNX_Precalc_Figure_01_06_005.jpg\" alt=\"Graph of an absolute function.\" width=\"487\" height=\"363\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736321\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137680556\">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. See <a class=\"autogenerated-content\" href=\"#Figure_01_06_006\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_06_006\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141810\/CNX_Precalc_Figure_01_06_006.jpg\" alt=\"Graph of two transformations for an absolute function at (3, -2).\" width=\"487\" height=\"363\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137805107\">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 as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_06_007\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_06_007\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141818\/CNX_Precalc_Figure_01_06_007.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\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137732766\">From this information we can write the equation<\/p>\n<div id=\"fs-id1165137425569\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill f\\left(x\\right)& =& 2|x-3|-2,\\hfill & \\phantom{\\rule{1em}{0ex}}\\text{treating the stretch as }a\\text{ vertical stretch,or}\\hfill \\\\ \\hfill f\\left(x\\right)& =& |2\\left(x-3\\right)|-2,\\hfill & \\phantom{\\rule{1em}{0ex}}\\text{treating the stretch as }a\\text{ horizontal compression}.\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137706602\">\n<h4>Analysis<\/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 id=\"fs-id1165134377948\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165135245777\"><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]<br \/>\n<\/em><\/p>\n<div id=\"fs-id1165135514699\" class=\"unnumbered aligncenter\">[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=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 2& =& a|1-3|-2\\hfill \\\\ \\hfill 4& =& 2a\\hfill \\\\ \\hfill a& =& 2\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137611709\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165134312222\">\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>\n<div id=\"fs-id1165137405204\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137758260\">[latex]f\\left(x\\right)=-|x+2|+3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135203778\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137527840\"><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 (see <a class=\"autogenerated-content\" href=\"#Figure_01_06_008\">(Figure)<\/a>).<br \/>\n<\/em><\/p>\n<\/div>\n<div id=\"Figure_01_06_008\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141824\/CNX_Precalc_Figure_01_06_008abc.jpg\" alt=\"Graph of the different types of transformations for an absolute function.\" width=\"975\" height=\"415\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7. <\/strong>(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.<\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165133257286\" class=\"bc-section section\">\n<h3>Solving an Absolute Value Equation<\/h3>\n<p id=\"fs-id1165137401775\">In <a class=\"target-chapter\" href=\"\/contents\/b88d40b3-a0d5-4a8f-8b2e-85993b911cc0\">Other Type of Equations<\/a>, we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. 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=|2x-6|,\\,[\/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=\"unnumbered aligncenter\">[latex]\\begin{array}{ccccccc}\\hfill 2x-6& =& 8\\hfill & \\phantom{\\rule{1em}{0ex}}\\text{or}\\phantom{\\rule{1em}{0ex}}& \\hfill 2x-6& =& -8\\hfill \\\\ \\hfill 2x& =& 14\\hfill & & \\hfill 2x& =& -2\\hfill \\\\ \\hfill x& =& 7\\hfill & & \\hfill x& =& -1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137641126\">Knowing how to solve problems involving <span class=\"no-emphasis\">absolute value functions<\/span> 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>An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,<\/p>\n<div id=\"fs-id1165137646929\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|x|=4,\\hfill \\\\ |2x-1|=3,\\text{or}\\hfill \\\\ |5x+2|-4=9\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1165137692078\" class=\"textbox key-takeaways\">\n<h3>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=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135593248\"><strong>Given the formula for an absolute value function, find the horizontal intercepts of its graph<\/strong>.<\/p>\n<ol id=\"fs-id1165131968095\" type=\"1\">\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 class=\"textbox examples\">\n<div id=\"fs-id1165137619575\">\n<div id=\"fs-id1165135309797\">\n<h3>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[latex]\\,x\\,[\/latex]such that[latex]\\,f\\left(x\\right)=0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137405662\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1165137618972\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccccccc}\\hfill 0& =& |4x+1|-7\\hfill & & & & & \\text{Substitute 0 for }f\\left(x\\right).\\hfill \\\\ \\hfill 7& =& |4x+1|\\hfill & & & & & \\text{Isolate the absolute value on one side of the equation}.\\hfill \\\\ & & & & & & & \\\\ & & & & & & & \\\\ & & & & & & & \\\\ \\hfill 7& =& 4x+1\\hfill & \\text{or}& \\hfill \\phantom{\\rule{2em}{0ex}}-7& =& 4x+1\\hfill & \\text{Break into two separate equations and solve}.\\hfill \\\\ \\hfill 6& =& 4x\\hfill & & \\hfill -8& =& 4x\\hfill & \\\\ & & & & & & & \\\\ \\hfill x& =& \\frac{6}{4}=1.5\\hfill & & \\hfill x& =& \\frac{-8}{4}=-2\\hfill & \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137870931\">The function outputs 0 when[latex]\\,x=\\frac{3}{2}\\,[\/latex]or[latex]\\,x=-2.[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_06_011\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_01_06_011\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141827\/CNX_Precalc_Figure_01_06_011F.jpg\" alt=\"Graph an absolute function with x-intercepts at -2 and 1.5.\" width=\"731\" height=\"476\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/figcaption><\/figure>\n<p><span id=\"fs-id1165137662351\"><\/details>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137599670\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165135191411\">\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>\n<div id=\"fs-id1165137939483\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137481390\">[latex]x=-1\\,[\/latex]or[latex]\\,\\,x=2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135175321\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165135606935\"><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<\/em>[latex]\\,2+|3x-5|=1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571678\" class=\"bc-section section\">\n<div id=\"fs-id1165134257615\" class=\"precalculus media\">\n<p id=\"fs-id1165134332731\">Access these online resources for additional instruction and practice with absolute value.<\/p>\n<ul id=\"fs-id1165137508064\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphabsvalue\">Graphing Absolute Value Functions<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphabsvalue2\">Graphing Absolute Value Functions 2<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133291312\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135332513\">\n<li>Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See <a class=\"autogenerated-content\" href=\"#Example_01_06_02\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_01_06_03\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_01_06_04\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135255406\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137406985\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137734873\">\n<div id=\"fs-id1165135510060\">\n<p id=\"fs-id1165134347447\">How do you solve an absolute value equation?<\/p>\n<\/div>\n<div id=\"fs-id1165135206149\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135189754\">Isolate the absolute value term so that the equation is of the form[latex]\\,|A|=B.\\,[\/latex]Form one equation by setting the expression inside the absolute value symbol,[latex]\\,A,\\,[\/latex]equal to the expression on the other side of the equation,[latex]\\,B.\\,[\/latex]Form a second equation by setting[latex]\\,A\\,[\/latex]equal to the opposite of the expression on the other side of the equation,[latex]\\,-B.\\,[\/latex]Solve each equation for the variable.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137593210\">\n<div id=\"fs-id1165131968049\">\n<p id=\"fs-id1165135440055\">How can you tell whether an absolute value function has two <em>x<\/em>-intercepts without graphing the function?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133103957\">\n<div id=\"fs-id1165133103959\">\n<p>When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?<\/p>\n<\/div>\n<div id=\"fs-id1165134475281\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137662762\">The graph of the absolute value function does not cross the[latex]\\,x[\/latex]-axis, so the graph is either completely above or completely below the[latex]\\,x[\/latex]-axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135264708\">\n<div id=\"fs-id1165135149122\">\n<p id=\"fs-id1165135149124\">How can you use the graph of an absolute value function to determine the <em>x<\/em>-values for which the function values are negative?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134273549\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<div id=\"fs-id1165137841613\">\n<div id=\"fs-id1165137841615\">\n<p id=\"fs-id1165137579699\">Describe all numbers[latex]\\,x\\,[\/latex]that are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135445894\">\n<div>\n<p id=\"fs-id1165135169187\">Describe all numbers[latex]\\,x\\,[\/latex]that are at a distance of[latex]\\,\\frac{1}{2}\\,[\/latex]from the number \u22124. Express this set of numbers using absolute value notation.<\/p>\n<\/div>\n<div id=\"fs-id1165135356596\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134278683\">[latex]\\,|x+4|=\\frac{1}{2}\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137542576\">\n<div id=\"fs-id1165137648320\">\n<p id=\"fs-id1165137648322\">Describe the situation in which the distance that point[latex]\\,x\\,[\/latex]is from 10 is at least 15 units. Express this set of numbers using absolute value notation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134057540\">\n<div id=\"fs-id1165137464076\">\n<p id=\"fs-id1165137464078\">Find all function values[latex]\\,f\\left(x\\right)\\,[\/latex]such that the distance from[latex]\\,f\\left(x\\right)\\,[\/latex]to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.<\/p>\n<\/div>\n<div id=\"fs-id1165137541376\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137572565\">[latex]|f\\left(x\\right)-8|<0.03[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137589731\">For the following exercises, find the <em>x<\/em>&#8211; and <em>y<\/em>-intercepts of the graphs of each function.<\/p>\n<div id=\"fs-id1165134401702\">\n<div id=\"fs-id1165135362510\">\n<p id=\"fs-id1165135362512\">[latex]f\\left(x\\right)=4|x-3|+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137824535\">\n<div id=\"fs-id1165137824537\">\n<p id=\"fs-id1165134220856\">[latex]f\\left(x\\right)=-3|x-2|-1[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137433352\">[latex]\\left(0,-7\\right);\\,[\/latex]no[latex]\\,x[\/latex]-intercepts<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134371172\">\n<div id=\"fs-id1165134371174\">\n<p id=\"fs-id1165137590699\">[latex]f\\left(x\\right)=-2|x+1|+6[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137656160\">\n<div id=\"fs-id1165137656163\">\n<p id=\"fs-id1165137938872\">[latex]f\\left(x\\right)=-5|x+2|+15[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137427184\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137427186\">[latex]\\left(0,\\,5\\right),\\left(1,0\\right),\\left(-5,0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"eip-613\">\n<div id=\"eip-946\">\n<p id=\"eip-55\">[latex]f\\left(x\\right)=2|x-1|-6[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]\\left(0,-4\\right),\\left(4,0\\right),\\left(-2,0\\right)[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-240\">\n<div id=\"eip-147\">\n<p id=\"eip-934\">[latex]f\\left(x\\right)=|-2x+1|-13[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"eip-240\">\n<div id=\"eip-302\">\n<p id=\"eip-20\">[latex]\\left(0,-12\\right),\\left(-6,0\\right),\\left(7,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-452\">\n<div id=\"eip-163\">\n<p id=\"eip-223\">[latex]f\\left(x\\right)=-|x-9|+16[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1165134273549\" class=\"bc-section section\">\n<div id=\"eip-452\">\n<div>[latex]\\left(0,7\\right),\\left(25,0\\right),\\left(-7,0\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165133047532\">For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.<\/p>\n<div id=\"fs-id1165137891404\">\n<div id=\"fs-id1165137817696\">\n<p id=\"fs-id1165137817699\">[latex]y=|x-1|[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141838\/CNX_Precalc_Figure_01_06_201.jpg\" alt=\"Graph of an absolute function with points at (-1, 2), (0, 1), (1, 0), (2, 1), and (3, 2).\" \/><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137679099\">\n<div id=\"fs-id1165137679101\">\n<p id=\"fs-id1165137418760\">[latex]y=|x+1|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135422938\">\n<div id=\"fs-id1165135422940\">\n<p id=\"fs-id1165137652979\">[latex]y=|x|+1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137414774\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137658298\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141840\/CNX_Precalc_Figure_01_06_203.jpg\" alt=\"Graph of an absolute function with points at (-2, 3), (-1, 2), (0, 1), (1, 2), and (2, 3).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137406944\">For the following exercises, graph the given functions by hand.<\/p>\n<div id=\"fs-id1165135332726\">\n<div id=\"fs-id1165135332729\">\n<p id=\"fs-id1165135251459\">[latex]y=|x|-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137601710\">\n<div id=\"fs-id1165137601713\">\n<p id=\"fs-id1165137419974\">[latex]y=-|x|[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137400044\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135160170\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141846\/CNX_Precalc_Figure_01_06_205.jpg\" alt=\"Graph of an absolute function.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735265\">\n<div id=\"fs-id1165137431347\">\n<p id=\"fs-id1165137431349\">[latex]y=-|x|-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731590\">\n<div id=\"fs-id1165137603675\">\n<p id=\"fs-id1165137603678\">[latex]y=-|x-3|-2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137645253\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135653964\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141850\/CNX_Precalc_Figure_01_06_207.jpg\" alt=\"Graph of an absolute function.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137394585\">\n<div id=\"fs-id1165137394587\">\n<p id=\"fs-id1165135572124\">[latex]f\\left(x\\right)=-|x-1|-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135335986\">\n<div id=\"fs-id1165137651575\">\n<p id=\"fs-id1165137651578\">[latex]f\\left(x\\right)=-|x+3|+4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137434149\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137553071\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141852\/CNX_Precalc_Figure_01_06_209.jpg\" alt=\"Graph of an absolute function.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705796\">\n<div id=\"fs-id1165137705798\">\n<p id=\"fs-id1165137469167\">[latex]f\\left(x\\right)=2|x+3|+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137427199\">\n<div id=\"fs-id1165137619904\">\n<p id=\"fs-id1165137619906\">[latex]f\\left(x\\right)=3|x-2|+3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137436399\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137410325\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141855\/CNX_Precalc_Figure_01_06_211.jpg\" alt=\"Graph of an absolute function.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137715460\">\n<div id=\"fs-id1165137715462\">\n<p id=\"fs-id1165137469722\">[latex]f\\left(x\\right)=|2x-4|-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135258293\">\n<div id=\"fs-id1165135258295\">\n<p id=\"fs-id1165137452029\">[latex]f\\left(x\\right)=|3x+9|+2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137433126\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141857\/CNX_Precalc_Figure_01_06_213.jpg\" alt=\"Graph of an absolute function.\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137464095\">\n<div id=\"fs-id1165137464097\">\n<p id=\"fs-id1165137470140\">[latex]f\\left(x\\right)=-|x-1|-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137443657\">\n<div id=\"fs-id1165137911316\">\n<p id=\"fs-id1165137911318\">[latex]f\\left(x\\right)=-|x+4|-3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137399944\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137451010\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141859\/CNX_Precalc_Figure_01_06_215.jpg\" alt=\"Graph of an absolute function.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137803326\">\n<div id=\"fs-id1165137803328\">\n<p id=\"fs-id1165137824374\">[latex]f\\left(x\\right)=\\frac{1}{2}|x+4|-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137897208\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<div id=\"fs-id1165137749758\">\n<div id=\"fs-id1165137749760\">\n<p id=\"fs-id1165137460158\">Use a graphing utility to graph [latex]f\\left(x\\right)=10|x-2|[\/latex] on the viewing window [latex]\\left[0,4\\right].[\/latex] Identify the corresponding range. Show the graph.<\/p>\n<\/div>\n<div id=\"fs-id1165134042934\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134042935\">range:[latex]\\,\\left[0,20\\right][\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141901\/CNX_Precalc_Figure_01_06_217.jpg\" alt=\"Graph of an absolute function.\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137413783\">\n<div id=\"fs-id1165137434783\">\n<p id=\"fs-id1165137434785\">Use a graphing utility to graph[latex]\\,f\\left(x\\right)=-100|x|+100\\,[\/latex]on the viewing window[latex]\\,\\left[-5,5\\right].\\,[\/latex]Identify the corresponding range. Show the graph.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137831208\">For the following exercises, graph each function using a graphing utility. Specify the viewing window.<\/p>\n<div id=\"fs-id1165137762283\">\n<div id=\"fs-id1165135464843\">\n<p id=\"fs-id1165137724085\">[latex]f\\left(x\\right)=-0.1|0.1\\left(0.2-x\\right)|+0.3[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137812573\">[latex]x\\text{-}[\/latex]intercepts:<\/p>\n<p><span id=\"fs-id1165137784866\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19141903\/CNX_Precalc_Figure_01_06_219.jpg\" alt=\"Graph of an absolute function.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-id1165134039354\">\n<div id=\"eip-id1165134039356\">\n<p id=\"fs-id1165137483195\">[latex]f\\left(x\\right)=4\u00d7{10}^{9}|x-\\left(5\u00d7{10}^{9}\\right)|+2\u00d7{10}^{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137419467\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165137901338\">For the following exercises, solve the inequality.<\/p>\n<div id=\"fs-id1165137434569\">\n<div id=\"fs-id1165137434570\">\n<p id=\"fs-id1165137434571\">If possible, find all values of [latex]a[\/latex] such that there are no [latex]x\\text{-}[\/latex]intercepts for [latex]f\\left(x\\right)=2|x+1|+a.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137639316\">\n<div id=\"fs-id1165137652958\">\n<p id=\"fs-id1165137652960\">If possible, find all values of[latex]\\,a\\,[\/latex]such that there are no [latex]\\,y[\/latex]-intercepts for[latex]\\,f\\left(x\\right)=2|x+1|+a.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137648025\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137454792\">There is no solution for[latex]\\,a\\,[\/latex]that will keep the function from having a[latex]\\,y[\/latex]-intercept. The absolute value function always crosses the [latex]\\,y[\/latex]-intercept when[latex]\\,x=0.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135172151\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165137641899\">\n<div id=\"fs-id1165137641901\">\n<p id=\"fs-id1165137459748\">Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and[latex]\\,x\\,[\/latex]represents the distance from city B to city A, express this using absolute value notation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137812302\">\n<div id=\"fs-id1165137812304\">\n<p id=\"fs-id1165137431941\">The true proportion[latex]\\,p\\,[\/latex]of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.<\/p>\n<\/div>\n<div id=\"fs-id1165135431083\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134042456\">[latex]|p-0.08|\\le 0.015[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137598001\">\n<div id=\"fs-id1165137562568\">\n<p id=\"fs-id1165137562570\">Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable[latex]\\,x\\,[\/latex]for the score.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137758810\">\n<div id=\"fs-id1165137758812\">\n<p id=\"fs-id1165135332394\">A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using[latex]\\,x\\,[\/latex]as the diameter of the bearing, write this statement using absolute value notation.<\/p>\n<\/div>\n<div id=\"fs-id1165135192953\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]|x-5.0|\\le 0.01[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137732323\">\n<div id=\"fs-id1165137732325\">\n<p id=\"fs-id1165137673610\">The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is[latex]\\,x\\,[\/latex]inches, express the tolerance using absolute value notation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":291,"menu_order":7,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-63","chapter","type-chapter","status-publish","hentry"],"part":50,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/63","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/63\/revisions"}],"predecessor-version":[{"id":64,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/63\/revisions\/64"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/50"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/63\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=63"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=63"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=63"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=63"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}