{"id":1500,"date":"2024-04-17T17:31:34","date_gmt":"2024-04-17T17:31:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1500"},"modified":"2025-02-12T17:32:43","modified_gmt":"2025-02-12T17:32:43","slug":"limits-and-continuity-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/limits-and-continuity-background-youll-need-2\/","title":{"raw":"Limits and Continuity: Background You'll Need 2","rendered":"Limits and Continuity: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Solve inequalities that include absolute values&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:513,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0}\">Solve inequalities that include absolute values<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Absolute Value Inequalities<\/h2>\r\n<p>Absolute value inequalities are pivotal in calculus for understanding limits, a foundational concept that delves into function behavior near specific points.<\/p>\r\n<p>An absolute value inequality, such as [latex]|A| &lt; B,|A|\\le B,|A| &gt; B,\\text{or }|A|\\ge B[\/latex], represents conditions where [latex]A[\/latex], an expression of the variable [latex]x[\/latex], falls within a specific range from zero. The inequality [latex]|A| &lt; B[\/latex] is mathematically equivalent to [latex]-B&lt; A&lt; B[\/latex].<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3><strong>absolute value inequality<\/strong><\/h3>\r\n<p>An <strong>absolute value inequality<\/strong> is an equation of the form<\/p>\r\n<div style=\"text-align: center;\">[latex]|A| &lt; B,|A|\\le B,|A| &gt; B,\\text{or }|A|\\ge B[\/latex],<\/div>\r\n<p>where [latex]A[\/latex], and sometimes [latex]B[\/latex], represents an algebraic expression dependent on a variable [latex]x[\/latex]<em>.<\/em><\/p>\r\n<\/section>\r\n<section class=\"textbox recall\">Remember that absolute value is like measuring how far a number is from zero on a number line. It doesn\u2019t matter which direction you go\u2014left or right\u2014the absolute value is always the distance without signs.<\/section>\r\n<p>Solving these inequalities is about determining all the possible values for [latex]x[\/latex] that meet the specified conditions, often leading to a specific interval or set of intervals.<\/p>\r\n<p>There are two basic approaches to solving absolute value inequalities: the graphical and the algebraic approach.<\/p>\r\n<p>The graphical method involves visually interpreting the solutions on a graph, which can give a good approximate understanding of where the solutions lie. However, the algebraic approach, though potentially more abstract, provides precise solutions that are sometimes challenging to discern graphically.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Solving Absolute Value Inequalities<\/strong><\/p>\r\n<p><strong>Algebraic Method<\/strong>:<\/p>\r\n<ol>\r\n\t<li>Isolate the absolute value expression on one side of the inequality.<\/li>\r\n\t<li>Set up two separate inequalities: one for the positive and one for the negative scenario.<\/li>\r\n\t<li>Solve both inequalities for [latex]x[\/latex] and combine the solution sets.<\/li>\r\n<\/ol>\r\n<strong>Graphical Method<\/strong>:\r\n\r\n<ol>\r\n\t<li>Graph the functions inside the absolute value and their opposites.<\/li>\r\n\t<li>Find the points of intersection with the reference value.<\/li>\r\n\t<li>The solution interval is between these intersection points.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<p><strong> Solving Inequalities Involving [latex]x[\/latex]<br \/>\r\n<br \/>\r\n<\/strong>To solve an inequality for [latex]x[\/latex], follow these steps:<\/p>\r\n<ol>\r\n\t<li><strong>Isolate [latex]x[\/latex]:<\/strong> Ensure [latex] x[\/latex] is by itself on one side of the inequality. If a number is added or subtracted from [latex]x[\/latex], counteract this by doing the opposite operation on both sides of the inequality.<\/li>\r\n\t<li><strong>Simplify<\/strong>: Combine like terms and simplify each side of the inequality.<\/li>\r\n\t<li><strong>Divide or Multiply:<\/strong> If [latex]x[\/latex] is multiplied by a coefficient, divide both sides of the inequality by that number to solve for [latex]x[\/latex]. Remember, if you multiply or divide by a negative number, you must flip the direction of the inequality sign!<\/li>\r\n\t<li><strong>Check Your Solution<\/strong>: Substitute your solution back into the original inequality to verify it.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Suppose we want to determine the range of possible returns on an investment where the amount earned is no more than [latex]$200[\/latex] above or below [latex]$600[\/latex].<\/p>\r\n<p>Solving algebraically:<\/p>\r\n<ol>\r\n\t<li>Write down the absolute value inequality: [latex]|x - 600|\\le 200[\/latex]<\/li>\r\n\t<li>Create two separate inequalities:\r\n\r\n<ul>\r\n\t<li>[latex]x - 600\\le 200[\/latex] (For the positive scenario)<\/li>\r\n\t<li>[latex]x - 600\\ge -200[\/latex] (For the negative scenario)<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Solve for [latex]x[\/latex] in both cases:\r\n\r\n<ul>\r\n\t<li>[latex]x \\le 800[\/latex]<\/li>\r\n\t<li>[latex]x \\ge 400[\/latex]\u00a0<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Combine the solutions to state the final range: [latex]400\\le x\\le 800[\/latex]<\/li>\r\n<\/ol>\r\n<p>This means our returns would be between [latex]$400[\/latex] and [latex]$800[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Solve [latex]|x - 1|\\le 3[\/latex].<\/p>\r\n<p>[reveal-answer q=\"4865\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4865\"]<\/p>\r\n<p>[latex]|x - 1|\\le 3[\/latex] is equivalent to [latex]-3\\le x - 1\\le 3[\/latex].<br \/>\r\n<br \/>\r\nSeparating the two inequalities we get:<\/p>\r\n<center>[latex]-3\\le x - 1 \\text{ and } x - 1\\le 3[\/latex]<\/center>\r\n<p>Solving these for [latex]x[\/latex] gives:<\/p>\r\n<center>[latex]x=-2 \\text{ and } x=4[\/latex]<\/center>\r\n<p>This can be combined to:<\/p>\r\n<p>&nbsp;<\/p>\r\n<center>[latex]-2\\le x\\le 4[\/latex]<\/center>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]288274[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Solve inequalities that include absolute values&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:513,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0}\">Solve inequalities that include absolute values<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Solving Absolute Value Inequalities<\/h2>\n<p>Absolute value inequalities are pivotal in calculus for understanding limits, a foundational concept that delves into function behavior near specific points.<\/p>\n<p>An absolute value inequality, such as [latex]|A| < B,|A|\\le B,|A| > B,\\text{or }|A|\\ge B[\/latex], represents conditions where [latex]A[\/latex], an expression of the variable [latex]x[\/latex], falls within a specific range from zero. The inequality [latex]|A| < B[\/latex] is mathematically equivalent to [latex]-B< A< B[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>absolute value inequality<\/strong><\/h3>\n<p>An <strong>absolute value inequality<\/strong> is an equation of the form<\/p>\n<div style=\"text-align: center;\">[latex]|A| < B,|A|\\le B,|A| > B,\\text{or }|A|\\ge B[\/latex],<\/div>\n<p>where [latex]A[\/latex], and sometimes [latex]B[\/latex], represents an algebraic expression dependent on a variable [latex]x[\/latex]<em>.<\/em><\/p>\n<\/section>\n<section class=\"textbox recall\">Remember that absolute value is like measuring how far a number is from zero on a number line. It doesn\u2019t matter which direction you go\u2014left or right\u2014the absolute value is always the distance without signs.<\/section>\n<p>Solving these inequalities is about determining all the possible values for [latex]x[\/latex] that meet the specified conditions, often leading to a specific interval or set of intervals.<\/p>\n<p>There are two basic approaches to solving absolute value inequalities: the graphical and the algebraic approach.<\/p>\n<p>The graphical method involves visually interpreting the solutions on a graph, which can give a good approximate understanding of where the solutions lie. However, the algebraic approach, though potentially more abstract, provides precise solutions that are sometimes challenging to discern graphically.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Solving Absolute Value Inequalities<\/strong><\/p>\n<p><strong>Algebraic Method<\/strong>:<\/p>\n<ol>\n<li>Isolate the absolute value expression on one side of the inequality.<\/li>\n<li>Set up two separate inequalities: one for the positive and one for the negative scenario.<\/li>\n<li>Solve both inequalities for [latex]x[\/latex] and combine the solution sets.<\/li>\n<\/ol>\n<p><strong>Graphical Method<\/strong>:<\/p>\n<ol>\n<li>Graph the functions inside the absolute value and their opposites.<\/li>\n<li>Find the points of intersection with the reference value.<\/li>\n<li>The solution interval is between these intersection points.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox recall\">\n<p><strong> Solving Inequalities Involving [latex]x[\/latex]<\/p>\n<p><\/strong>To solve an inequality for [latex]x[\/latex], follow these steps:<\/p>\n<ol>\n<li><strong>Isolate [latex]x[\/latex]:<\/strong> Ensure [latex]x[\/latex] is by itself on one side of the inequality. If a number is added or subtracted from [latex]x[\/latex], counteract this by doing the opposite operation on both sides of the inequality.<\/li>\n<li><strong>Simplify<\/strong>: Combine like terms and simplify each side of the inequality.<\/li>\n<li><strong>Divide or Multiply:<\/strong> If [latex]x[\/latex] is multiplied by a coefficient, divide both sides of the inequality by that number to solve for [latex]x[\/latex]. Remember, if you multiply or divide by a negative number, you must flip the direction of the inequality sign!<\/li>\n<li><strong>Check Your Solution<\/strong>: Substitute your solution back into the original inequality to verify it.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>Suppose we want to determine the range of possible returns on an investment where the amount earned is no more than [latex]$200[\/latex] above or below [latex]$600[\/latex].<\/p>\n<p>Solving algebraically:<\/p>\n<ol>\n<li>Write down the absolute value inequality: [latex]|x - 600|\\le 200[\/latex]<\/li>\n<li>Create two separate inequalities:\n<ul>\n<li>[latex]x - 600\\le 200[\/latex] (For the positive scenario)<\/li>\n<li>[latex]x - 600\\ge -200[\/latex] (For the negative scenario)<\/li>\n<\/ul>\n<\/li>\n<li>Solve for [latex]x[\/latex] in both cases:\n<ul>\n<li>[latex]x \\le 800[\/latex]<\/li>\n<li>[latex]x \\ge 400[\/latex]\u00a0<\/li>\n<\/ul>\n<\/li>\n<li>Combine the solutions to state the final range: [latex]400\\le x\\le 800[\/latex]<\/li>\n<\/ol>\n<p>This means our returns would be between [latex]$400[\/latex] and [latex]$800[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Solve [latex]|x - 1|\\le 3[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4865\">Show Solution<\/button><\/p>\n<div id=\"q4865\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]|x - 1|\\le 3[\/latex] is equivalent to [latex]-3\\le x - 1\\le 3[\/latex].<\/p>\n<p>Separating the two inequalities we get:<\/p>\n<div style=\"text-align: center;\">[latex]-3\\le x - 1 \\text{ and } x - 1\\le 3[\/latex]<\/div>\n<p>Solving these for [latex]x[\/latex] gives:<\/p>\n<div style=\"text-align: center;\">[latex]x=-2 \\text{ and } x=4[\/latex]<\/div>\n<p>This can be combined to:<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]-2\\le x\\le 4[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288274\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288274&theme=lumen&iframe_resize_id=ohm288274&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1500"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":27,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1500\/revisions"}],"predecessor-version":[{"id":4672,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1500\/revisions\/4672"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1500\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1500"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1500"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1500"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1500"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}