{"id":45,"date":"2025-02-13T22:43:07","date_gmt":"2025-02-13T22:43:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/absolute-value-functions\/"},"modified":"2025-12-30T17:42:51","modified_gmt":"2025-12-30T17:42:51","slug":"absolute-value-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/absolute-value-functions\/","title":{"raw":"Absolute Value Functions: Learn It 1","rendered":"Absolute Value Functions: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph an absolute value function.<\/li>\r\n \t<li>Solve an absolute value equation.<\/li>\r\n \t<li>Solve an absolute value inequality.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<figure id=\"Figure_01_06_001\" class=\"medium\"><\/figure>\r\n<h2>Understanding Absolute Value<\/h2>\r\n<p id=\"fs-id1165135449691\">Recall that in its basic form [latex]\\displaystyle{f}\\left({x}\\right)={|x|}[\/latex], the absolute value function, is one of our toolkit functions. The <strong>absolute value<\/strong> function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>absolute value<\/h3>\r\nThe <strong>absolute value<\/strong> of a number is a measure of its distance from zero on the number line, regardless of direction. It is always a non-negative value.\r\n\r\n&nbsp;\r\n\r\nFor a real number [latex]x[\/latex], the absolute value is denoted by [latex]|x|[\/latex] and is defined as:\r\n<p style=\"text-align: center;\">[latex]|x| = \\begin{cases} x &amp; \\text{if } x \\geq 0 \\\\ -x &amp; \\text{if } x &lt; 0 \\end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1165137761508\">Describe all values [latex]x[\/latex] within or including a distance of 4 from the number 5.<\/p>\r\n[reveal-answer q=\"617814\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617814\"]\r\n\r\n<img class=\"wp-image-4949 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02221742\/3.4.L1.Diagram-300x89.png\" alt=\"Number line describing the difference of the distance of 4 away from 5.\" width=\"384\" height=\"114\" \/>\r\n<p id=\"fs-id1165135424674\">We want the distance between [latex]x[\/latex] and 5 to be less than or equal to 4. We can draw a number line\u00a0to represent the condition to be satisfied.<span id=\"fs-id1165137761581\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165137772130\">The distance from [latex]x[\/latex] to 5 can be represented using the absolute value as [latex]|x - 5|[\/latex]. We want the values of [latex]x[\/latex] that satisfy the condition [latex]|x - 5|\\le 4[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165135161478\">Note that<\/p>\r\n\r\n<div id=\"fs-id1165134394601\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{-4}\\le{x - 5}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{1}\\le{x}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\">And:<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{x-5}\\le{4}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{x}\\le{9}[\/latex]<\/div>\r\n<p id=\"fs-id1165137569650\">So [latex]|x - 5|\\le 4[\/latex] is equivalent to [latex]1\\le x\\le 9[\/latex].<\/p>\r\n<p id=\"fs-id1165137539782\">However, mathematicians generally prefer absolute value notation.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p id=\"fs-id1165135394310\">[ohm_question hide_question_numbers=1]317630[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1165135203760\">Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often [latex]\\displaystyle\\pm\\text{1%,}\\pm\\text{5%,}[\/latex] or [latex]\\displaystyle\\pm\\text{10%}[\/latex].<\/p>\r\n<p id=\"fs-id1165135175007\">Suppose we have a resistor rated at 680 ohms, [latex]\\pm 5%[\/latex]. Use the absolute value function to express the range of possible values of the actual resistance.<\/p>\r\n[reveal-answer q=\"430965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"430965\"]\r\n<p id=\"fs-id1165137600783\">5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance [latex]R[\/latex] in ohms,<\/p>\r\n\r\n<div id=\"fs-id1165135176481\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|R - 680|\\le 34[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317631[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\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<\/section>&nbsp;\r\n<dl id=\"fs-id1165137560214\" class=\"definition\">\r\n \t<dd id=\"fs-id1165135173524\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\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<\/section>\n<\/div>\n<figure id=\"Figure_01_06_001\" class=\"medium\"><\/figure>\n<h2>Understanding Absolute Value<\/h2>\n<p id=\"fs-id1165135449691\">Recall that in its basic form [latex]\\displaystyle{f}\\left({x}\\right)={|x|}[\/latex], the absolute value function, is one of our toolkit functions. The <strong>absolute value<\/strong> function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>absolute value<\/h3>\n<p>The <strong>absolute value<\/strong> of a number is a measure of its distance from zero on the number line, regardless of direction. It is always a non-negative value.<\/p>\n<p>&nbsp;<\/p>\n<p>For a real number [latex]x[\/latex], the absolute value is denoted by [latex]|x|[\/latex] and is defined as:<\/p>\n<p style=\"text-align: center;\">[latex]|x| = \\begin{cases} x & \\text{if } x \\geq 0 \\\\ -x & \\text{if } x < 0 \\end{cases}[\/latex]<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1165137761508\">Describe all values [latex]x[\/latex] within or including a distance of 4 from the number 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q617814\">Show Solution<\/button><\/p>\n<div id=\"q617814\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4949 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02221742\/3.4.L1.Diagram-300x89.png\" alt=\"Number line describing the difference of the distance of 4 away from 5.\" width=\"384\" height=\"114\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02221742\/3.4.L1.Diagram-300x89.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02221742\/3.4.L1.Diagram-65x19.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02221742\/3.4.L1.Diagram-225x67.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02221742\/3.4.L1.Diagram-350x104.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02221742\/3.4.L1.Diagram.png 437w\" sizes=\"(max-width: 384px) 100vw, 384px\" \/><\/p>\n<p id=\"fs-id1165135424674\">We want the distance between [latex]x[\/latex] and 5 to be less than or equal to 4. We can draw a number line\u00a0to represent the condition to be satisfied.<span id=\"fs-id1165137761581\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137772130\">The distance from [latex]x[\/latex] to 5 can be represented using the absolute value as [latex]|x - 5|[\/latex]. We want the values of [latex]x[\/latex] that satisfy the condition [latex]|x - 5|\\le 4[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165135161478\">Note that<\/p>\n<div id=\"fs-id1165134394601\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{-4}\\le{x - 5}[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{1}\\le{x}[\/latex]<\/div>\n<div class=\"equation unnumbered\">And:<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{x-5}\\le{4}[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle{x}\\le{9}[\/latex]<\/div>\n<p id=\"fs-id1165137569650\">So [latex]|x - 5|\\le 4[\/latex] is equivalent to [latex]1\\le x\\le 9[\/latex].<\/p>\n<p id=\"fs-id1165137539782\">However, mathematicians generally prefer absolute value notation.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p id=\"fs-id1165135394310\"><iframe loading=\"lazy\" id=\"ohm317630\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317630&theme=lumen&iframe_resize_id=ohm317630&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1165135203760\">Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often [latex]\\displaystyle\\pm\\text{1%,}\\pm\\text{5%,}[\/latex] or [latex]\\displaystyle\\pm\\text{10%}[\/latex].<\/p>\n<p id=\"fs-id1165135175007\">Suppose we have a resistor rated at 680 ohms, [latex]\\pm 5%[\/latex]. Use the absolute value function to express the range of possible values of the actual resistance.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q430965\">Show Solution<\/button><\/p>\n<div id=\"q430965\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137600783\">5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance [latex]R[\/latex] in ohms,<\/p>\n<div id=\"fs-id1165135176481\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|R - 680|\\le 34[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317631\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317631&theme=lumen&iframe_resize_id=ohm317631&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\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<\/section>\n<p>&nbsp;<\/p>\n<dl id=\"fs-id1165137560214\" class=\"definition\">\n<dd id=\"fs-id1165135173524\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":30,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":61,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/45"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":11,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/45\/revisions"}],"predecessor-version":[{"id":5157,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/45\/revisions\/5157"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/61"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/45\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=45"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=45"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=45"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=45"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}