{"id":2514,"date":"2025-08-13T17:21:45","date_gmt":"2025-08-13T17:21:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2514"},"modified":"2025-09-23T20:56:08","modified_gmt":"2025-09-23T20:56:08","slug":"linear-functions-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/linear-functions-background-youll-need-3\/","title":{"raw":"Linear Functions: Background You'll Need 3","rendered":"Linear Functions: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Simplify expressions with absolute value.<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>absolute value<\/h3>\r\n<p class=\"whitespace-normal break-words\">Absolute value represents the distance of a number from zero on the number line, always resulting in a non-negative value. The absolute value of a number [latex]x[\/latex] is written as [latex]|x|[\/latex].<\/p>\r\n\r\n<\/section>In simple terms: [latex]|x|[\/latex] equals [latex]x[\/latex] if [latex]x[\/latex] is positive or zero, and equals [latex]-x[\/latex] if [latex]x[\/latex] is negative.\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\">To simplify expressions with absolute value:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Treat absolute value bars as parenthesis and simplify what's inside the absolute value bars first<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine if the expression inside is positive, negative, or zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the absolute value definition:\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">If positive or zero: remove the bars, keep the number<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If negative: remove the bars, change the sign of the number<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify any remaining arithmetic following the order of operations<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Simplify each expression:<\/p>\r\n<p class=\"whitespace-normal break-words\">a) [latex]|7|[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"padding-left: 40px;\">Since [latex]7 &gt; 0[\/latex], we have [latex]|7| = 7[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">b) [latex]|-12|[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"padding-left: 40px;\">Since [latex]-12 &lt; 0[\/latex], we have [latex]|-12| = 12[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">c) [latex]|3 - 8|[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"padding-left: 40px;\">[latex]\\begin{align} |3 - 8| &amp;= |-5| \\quad \\text{(simplify inside first)} \\\\ &amp;= 5 \\quad \\text{(since -5 &lt; 0)}\u00a0 \\end{align}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">d) [latex]2|4 - 9| + 3[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"padding-left: 40px;\">[latex]\\begin{align} 2|4 - 9| + 3 &amp;= 2|-5| + 3 \\quad \\text{(simplify inside first)} \\\\ &amp;= 2(5) + 3 \\quad \\text{(since -5 &lt; 0)} \\\\ &amp;= 10 + 3 \\\\ &amp;= 13 \\end{align}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">Remember that absolute value measures distance, which is always positive or zero. Think of [latex]|-3|[\/latex] as \"how far is -3 from zero?\" The answer is 3 units, so [latex]|-3| = 3[\/latex].<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\"><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"whitespace-normal break-words\">When you see nested operations like [latex]|a - b|[\/latex], always compute what's inside the bars first, then apply the absolute value. Don't try to \"distribute\" the absolute value bars to individual terms.<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]312646[\/ohm_question]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Simplify expressions with absolute value.<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>absolute value<\/h3>\n<p class=\"whitespace-normal break-words\">Absolute value represents the distance of a number from zero on the number line, always resulting in a non-negative value. The absolute value of a number [latex]x[\/latex] is written as [latex]|x|[\/latex].<\/p>\n<\/section>\n<p>In simple terms: [latex]|x|[\/latex] equals [latex]x[\/latex] if [latex]x[\/latex] is positive or zero, and equals [latex]-x[\/latex] if [latex]x[\/latex] is negative.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\">To simplify expressions with absolute value:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Treat absolute value bars as parenthesis and simplify what&#8217;s inside the absolute value bars first<\/li>\n<li class=\"whitespace-normal break-words\">Determine if the expression inside is positive, negative, or zero<\/li>\n<li class=\"whitespace-normal break-words\">Apply the absolute value definition:\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">If positive or zero: remove the bars, keep the number<\/li>\n<li class=\"whitespace-normal break-words\">If negative: remove the bars, change the sign of the number<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Simplify any remaining arithmetic following the order of operations<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Simplify each expression:<\/p>\n<p class=\"whitespace-normal break-words\">a) [latex]|7|[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\" style=\"padding-left: 40px;\">Since [latex]7 > 0[\/latex], we have [latex]|7| = 7[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">b) [latex]|-12|[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\" style=\"padding-left: 40px;\">Since [latex]-12 < 0[\/latex], we have [latex]|-12| = 12[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">c) [latex]|3 - 8|[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\" style=\"padding-left: 40px;\">[latex]\\begin{align} |3 - 8| &= |-5| \\quad \\text{(simplify inside first)} \\\\ &= 5 \\quad \\text{(since -5 < 0)}\u00a0 \\end{align}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">d) [latex]2|4 - 9| + 3[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\" style=\"padding-left: 40px;\">[latex]\\begin{align} 2|4 - 9| + 3 &= 2|-5| + 3 \\quad \\text{(simplify inside first)} \\\\ &= 2(5) + 3 \\quad \\text{(since -5 < 0)} \\\\ &= 10 + 3 \\\\ &= 13 \\end{align}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">Remember that absolute value measures distance, which is always positive or zero. Think of [latex]|-3|[\/latex] as &#8220;how far is -3 from zero?&#8221; The answer is 3 units, so [latex]|-3| = 3[\/latex].<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"whitespace-normal break-words\">When you see nested operations like [latex]|a - b|[\/latex], always compute what&#8217;s inside the bars first, then apply the absolute value. Don&#8217;t try to &#8220;distribute&#8221; the absolute value bars to individual terms.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm312646\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312646&theme=lumen&iframe_resize_id=ohm312646&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":61,"module-header":"background_you_need","content_attributions":[],"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\/2514"}],"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\/67"}],"version-history":[{"count":17,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2514\/revisions"}],"predecessor-version":[{"id":4312,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2514\/revisions\/4312"}],"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\/2514\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2514"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2514"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2514"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}