{"id":17834,"date":"2020-04-11T22:46:18","date_gmt":"2020-04-11T22:46:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/?post_type=chapter&#038;p=17834"},"modified":"2020-10-22T09:13:58","modified_gmt":"2020-10-22T09:13:58","slug":"summary-solving-compound-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/summary-solving-compound-inequalities\/","title":{"raw":"8.2.d - Summary: Solving Compound Inequalities","rendered":"8.2.d &#8211; Summary: Solving Compound Inequalities"},"content":{"raw":"Key Concepts\r\n<h3>Writing Solutions to Absolute Value Inequalities<\/h3>\r\nFor any positive value of <i>a\u00a0<\/i>and\u00a0<em>x,<\/em>\u00a0a single variable, or any algebraic expression:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Absolute Value Inequality<\/strong><\/td>\r\n<td><strong>Equivalent Inequality<\/strong><\/td>\r\n<td><strong>Interval Notation<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left|{ x }\\right|\\le{ a}[\/latex]<\/td>\r\n<td>[latex]{ -a}\\le{x}\\le{ a}[\/latex]<\/td>\r\n<td>[latex]\\left[-a, a\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left| x \\right|\\lt{a}[\/latex]<\/td>\r\n<td>[latex]{ -a}\\lt{x}\\lt{ a}[\/latex]<\/td>\r\n<td>[latex]\\left(-a, a\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left| x \\right|\\ge{ a}[\/latex]<\/td>\r\n<td>[latex]{x}\\le\\text{\u2212a}[\/latex] or [latex]{x}\\ge{ a}[\/latex]<\/td>\r\n<td>\u00a0[latex]\\left(-\\infty,-a\\right]\\cup\\left[a,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left| x \\right|\\gt\\text{a}[\/latex]<\/td>\r\n<td>[latex]\\displaystyle{x}\\lt\\text{\u2212a}[\/latex]\u00a0or [latex]{x}\\gt{ a}[\/latex]<\/td>\r\n<td>\u00a0[latex]\\left(-\\infty,-a\\right)\\cup\\left(a,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Glossary<\/h2>\r\n<strong>Union<\/strong>\u00a0 The solution of a compound inequality that consists of two inequalities joined with the word <em>or<\/em> is the union of the solutions of each inequality.\r\n\r\n<strong>Intersection<\/strong>\u00a0 The solution of a compound inequality that consists of two inequalities joined with the word<i> and <\/i>is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an <i>and<\/i> compound inequality are all the solutions that the two inequalities have in common","rendered":"<p>Key Concepts<\/p>\n<h3>Writing Solutions to Absolute Value Inequalities<\/h3>\n<p>For any positive value of <i>a\u00a0<\/i>and\u00a0<em>x,<\/em>\u00a0a single variable, or any algebraic expression:<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Absolute Value Inequality<\/strong><\/td>\n<td><strong>Equivalent Inequality<\/strong><\/td>\n<td><strong>Interval Notation<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left|{ x }\\right|\\le{ a}[\/latex]<\/td>\n<td>[latex]{ -a}\\le{x}\\le{ a}[\/latex]<\/td>\n<td>[latex]\\left[-a, a\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left| x \\right|\\lt{a}[\/latex]<\/td>\n<td>[latex]{ -a}\\lt{x}\\lt{ a}[\/latex]<\/td>\n<td>[latex]\\left(-a, a\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left| x \\right|\\ge{ a}[\/latex]<\/td>\n<td>[latex]{x}\\le\\text{\u2212a}[\/latex] or [latex]{x}\\ge{ a}[\/latex]<\/td>\n<td>\u00a0[latex]\\left(-\\infty,-a\\right]\\cup\\left[a,\\infty\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left| x \\right|\\gt\\text{a}[\/latex]<\/td>\n<td>[latex]\\displaystyle{x}\\lt\\text{\u2212a}[\/latex]\u00a0or [latex]{x}\\gt{ a}[\/latex]<\/td>\n<td>\u00a0[latex]\\left(-\\infty,-a\\right)\\cup\\left(a,\\infty\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Glossary<\/h2>\n<p><strong>Union<\/strong>\u00a0 The solution of a compound inequality that consists of two inequalities joined with the word <em>or<\/em> is the union of the solutions of each inequality.<\/p>\n<p><strong>Intersection<\/strong>\u00a0 The solution of a compound inequality that consists of two inequalities joined with the word<i> and <\/i>is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an <i>and<\/i> compound inequality are all the solutions that the two inequalities have in common<\/p>\n","protected":false},"author":253111,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"2d6bfb728dbf420fa9179d0a71ecfc13","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17834","chapter","type-chapter","status-publish","hentry"],"part":18856,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17834","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/users\/253111"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17834\/revisions"}],"predecessor-version":[{"id":20326,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17834\/revisions\/20326"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/18856"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17834\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/media?parent=17834"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=17834"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=17834"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/license?post=17834"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}