{"id":995,"date":"2015-11-12T18:37:57","date_gmt":"2015-11-12T18:37:57","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=995"},"modified":"2017-03-31T21:16:15","modified_gmt":"2017-03-31T21:16:15","slug":"introduction-to-absolute-value-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/introduction-to-absolute-value-functions\/","title":{"raw":"Introduction to Absolute Value Functions","rendered":"Introduction to Absolute Value Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>LEARNING OBJECTIVES<\/h3>\r\nBy the end of this lesson, you will be able to:\r\n<ul><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><\/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\/924\/2015\/11\/25200920\/CNX_Precalc_Figure_01_06_001n2.jpg\" alt=\"The Milky Way.\" width=\"488\" height=\"338\" data-media-type=\"image\/jpg\"\/><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]\r\n\r\n<\/figure><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 data-type=\"title\">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\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"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\" data-type=\"example\">\r\n<div id=\"fs-id1165137437173\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137618976\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">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\r\n<\/div>\r\n<div id=\"fs-id1165135193649\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\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\/924\/2015\/11\/25200921\/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\" data-media-type=\"image\/jpg\"\/><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\" data-type=\"media\" data-alt=\"Number line describing the difference of the distance of 4 away from 5.\">\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<\/div>\r\n<div id=\"fs-id1165137656116\" class=\"commentary\" data-type=\"commentary\">\r\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\r\n<p id=\"fs-id1165135161478\">Note that<\/p>\r\n\r\n<div id=\"fs-id1165134394601\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle{-4}\\le{x - 5}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle{1}\\le{x}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" data-type=\"equation\" data-label=\"\">And:<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle{x-5}\\le{4}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[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\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 1<\/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<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-7\/\" target=\"_blank\">Solution<\/a>\r\n\r\n<\/div>\r\n<div id=\"Example_01_06_02\" class=\"example\" data-type=\"example\">\r\n<div id=\"fs-id1165137657277\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137579723\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">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\\text{\\pm 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\r\n<\/div>\r\n<div id=\"fs-id1165137786481\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\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;\" data-type=\"equation\" data-label=\"\">[latex]|R - 680|\\le 34[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 2<\/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<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-7\/\" target=\"_blank\">Solution<\/a>\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>LEARNING OBJECTIVES<\/h3>\n<p>By the end of this lesson, you will be able to:<\/p>\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\/924\/2015\/11\/25200920\/CNX_Precalc_Figure_01_06_001n2.jpg\" alt=\"The Milky Way.\" width=\"488\" height=\"338\" data-media-type=\"image\/jpg\" \/><\/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 data-type=\"title\">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\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"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\" data-type=\"example\">\n<div id=\"fs-id1165137437173\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137618976\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">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>\n<div id=\"fs-id1165135193649\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\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\/924\/2015\/11\/25200921\/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\" data-media-type=\"image\/jpg\" \/><\/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\" data-type=\"media\" data-alt=\"Number line describing the difference of the distance of 4 away from 5.\"><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<\/div>\n<div id=\"fs-id1165137656116\" class=\"commentary\" data-type=\"commentary\">\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\n<p id=\"fs-id1165135161478\">Note that<\/p>\n<div id=\"fs-id1165134394601\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle{-4}\\le{x - 5}[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle{1}\\le{x}[\/latex]<\/div>\n<div class=\"equation unnumbered\" data-type=\"equation\" data-label=\"\">And:<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle{x-5}\\le{4}[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[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 class=\"bcc-box bcc-success\">\n<h3>Try It 1<\/h3>\n<p id=\"fs-id1165135394310\">Describe all values [latex]x[\/latex] within a distance of 3 from the number 2.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-7\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<div id=\"Example_01_06_02\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137657277\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137579723\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">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\\text{\\pm 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>\n<div id=\"fs-id1165137786481\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\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;\" data-type=\"equation\" data-label=\"\">[latex]|R - 680|\\le 34[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 2<\/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<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-7\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-995\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":276,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-995","chapter","type-chapter","status-publish","hentry"],"part":992,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/995","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/995\/revisions"}],"predecessor-version":[{"id":2825,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/995\/revisions\/2825"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/992"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/995\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=995"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=995"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=995"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=995"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}