{"id":113,"date":"2023-06-21T13:22:34","date_gmt":"2023-06-21T13:22:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-compound-and-absolute-value-inequalities\/"},"modified":"2023-06-21T13:22:34","modified_gmt":"2023-06-21T13:22:34","slug":"summary-compound-and-absolute-value-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-compound-and-absolute-value-inequalities\/","title":{"raw":"Summary: Compound and Absolute Value Inequalities","rendered":"Summary: Compound and Absolute Value Inequalities"},"content":{"raw":"\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>Interval notation is a method to give the solution set of an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well.<\/li>\n \t<li>Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality.<\/li>\n \t<li>Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities.<\/li>\n \t<li>Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value with the inequality symbol flipped.\n<ul>\n \t<li>[latex]|X|&gt; k[\/latex] which is equivalent to: [latex]X&lt; -k\\text{, or }X&gt; k[\/latex]<\/li>\n<\/ul>\n<\/li>\n \t<li>Absolute value inequality solutions can be verified by graphing. We can check the algebraic solutions by graphing as we cannot depend on a visual for a precise solution.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165131990658\" class=\"definition\">\n \t<dt><strong>compound inequality<\/strong><\/dt>\n \t<dd id=\"fs-id1165131990661\">a problem or a statement that includes two inequalities<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n \t<dt><strong>interval<\/strong><\/dt>\n \t<dd id=\"fs-id1165132943525\">an interval describes a set of numbers where a solution falls<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n \t<dt><strong>interval notation<\/strong><\/dt>\n \t<dd id=\"fs-id1165134297639\">a mathematical statement that describes a solution set and uses parentheses or brackets to indicate where an interval begins and ends<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n \t<dt><strong>linear inequality<\/strong><\/dt>\n \t<dd id=\"fs-id1165135486042\">similar to a linear equation except that the solutions will include an interval of numbers<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>Interval notation is a method to give the solution set of an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well.<\/li>\n<li>Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality.<\/li>\n<li>Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities.<\/li>\n<li>Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value with the inequality symbol flipped.\n<ul>\n<li>[latex]|X|> k[\/latex] which is equivalent to: [latex]X< -k\\text{, or }X> k[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Absolute value inequality solutions can be verified by graphing. We can check the algebraic solutions by graphing as we cannot depend on a visual for a precise solution.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt>\n<\/dt>\n<dt><strong>compound inequality<\/strong><\/dt>\n<dd id=\"fs-id1165131990661\">a problem or a statement that includes two inequalities<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>interval<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">an interval describes a set of numbers where a solution falls<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt><strong>interval notation<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">a mathematical statement that describes a solution set and uses parentheses or brackets to indicate where an interval begins and ends<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>linear inequality<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">similar to a linear equation except that the solutions will include an interval of numbers<\/dd>\n<\/dl>\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-113\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":395986,"menu_order":28,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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