{"id":17824,"date":"2020-04-11T22:28:43","date_gmt":"2020-04-11T22:28:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/?post_type=chapter&#038;p=17824"},"modified":"2020-10-22T09:13:05","modified_gmt":"2020-10-22T09:13:05","slug":"summary-solving-single-step-and-multi-step-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/summary-solving-single-step-and-multi-step-inequalities\/","title":{"raw":"8.1.c - Summary: Solving Single- and Multi-Step Inequalities","rendered":"8.1.c &#8211; Summary: Solving Single- and Multi-Step Inequalities"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<h3>Inequality Signs<\/h3>\r\nThe box below shows the symbol, meaning, and an example for each inequality sign.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Symbol<\/th>\r\n<th>Words<\/th>\r\n<th>Example<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\neq [\/latex]<\/td>\r\n<td>not equal to<\/td>\r\n<td>[latex]{2}\\neq{8}[\/latex], <i>2<\/i>\u00a0<b>is not equal to<\/b> <i>8<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\gt[\/latex]<\/td>\r\n<td>greater than<\/td>\r\n<td>[latex]{5}\\gt{1}[\/latex], <i>5<\/i>\u00a0<b>is greater than<\/b>\u00a0<i>1<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\lt[\/latex]<\/td>\r\n<td>less than<\/td>\r\n<td>[latex]{2}\\lt{11}[\/latex], <i>2<\/i> <b>is less than<\/b>\u00a0<i>11<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\geq [\/latex]<\/td>\r\n<td>greater than or equal to<\/td>\r\n<td>[latex]{4}\\geq{ 4}[\/latex], <i>4<\/i> <b>is greater than or equal to<\/b>\u00a0<i>4<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\leq [\/latex]<\/td>\r\n<td>less than or equal to<\/td>\r\n<td>[latex]{7}\\leq{9}[\/latex], <i>7<\/i>\u00a0<b>is less than or equal to<\/b>\u00a0<i>9<\/i><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe table below describes all the possible inequalities that can occur and how to write them using interval notation, where <em>a<\/em> and <em>b<\/em> are real numbers.\r\n<table summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\r\n<thead>\r\n<tr>\r\n<th>Inequality<\/th>\r\n<th>Words<\/th>\r\n<th>Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\r\n<td>all real numbers between\u00a0<em>a<\/em> and <em>b<\/em>, not including a and b<\/td>\r\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\gt{a}[\/latex]<\/td>\r\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\r\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\lt{b}[\/latex]<\/td>\r\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\ge{a}[\/latex]<\/td>\r\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\le{b}[\/latex]<\/td>\r\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{a}\\le{x}\\lt{ b}[\/latex]<\/td>\r\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{a}\\lt{x}\\le{ b}[\/latex]<\/td>\r\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{a}\\le{x}\\le{ b}[\/latex]<\/td>\r\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\r\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\lt{a}\\text{ or }{x}\\gt{ b}[\/latex]<\/td>\r\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\r\n<td>[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers<\/td>\r\n<td>All real numbers<\/td>\r\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Start With<\/strong><\/td>\r\n<td><strong>Multiply By<\/strong><\/td>\r\n<td><strong>Final Inequality<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex]ac&gt;bc[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5&gt;3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]15&gt;9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]-c[\/latex]<\/td>\r\n<td>[latex]-ac&lt;-bc[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5&gt;3[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]-15&lt;-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe following table illustrates how the division\u00a0property is applied to inequalities, and how dividing by a negative reverses the inequality:\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Start With<\/strong><\/td>\r\n<td><strong>Divide By<\/strong><\/td>\r\n<td><strong>Final Inequality<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{a}{c}&gt;\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4&gt;2[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{4}{2}&gt;\\frac{2}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]-c[\/latex]<\/td>\r\n<td>[latex] \\displaystyle -\\frac{a}{c}&lt;-\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4&gt;2[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex] \\displaystyle -\\frac{4}{2}&lt;-\\frac{2}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"<h2>Key Concepts<\/h2>\n<h3>Inequality Signs<\/h3>\n<p>The box below shows the symbol, meaning, and an example for each inequality sign.<\/p>\n<table>\n<thead>\n<tr>\n<th>Symbol<\/th>\n<th>Words<\/th>\n<th>Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\neq[\/latex]<\/td>\n<td>not equal to<\/td>\n<td>[latex]{2}\\neq{8}[\/latex], <i>2<\/i>\u00a0<b>is not equal to<\/b> <i>8<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\gt[\/latex]<\/td>\n<td>greater than<\/td>\n<td>[latex]{5}\\gt{1}[\/latex], <i>5<\/i>\u00a0<b>is greater than<\/b>\u00a0<i>1<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\lt[\/latex]<\/td>\n<td>less than<\/td>\n<td>[latex]{2}\\lt{11}[\/latex], <i>2<\/i> <b>is less than<\/b>\u00a0<i>11<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\geq[\/latex]<\/td>\n<td>greater than or equal to<\/td>\n<td>[latex]{4}\\geq{ 4}[\/latex], <i>4<\/i> <b>is greater than or equal to<\/b>\u00a0<i>4<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\leq[\/latex]<\/td>\n<td>less than or equal to<\/td>\n<td>[latex]{7}\\leq{9}[\/latex], <i>7<\/i>\u00a0<b>is less than or equal to<\/b>\u00a0<i>9<\/i><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The table below describes all the possible inequalities that can occur and how to write them using interval notation, where <em>a<\/em> and <em>b<\/em> are real numbers.<\/p>\n<table summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\n<thead>\n<tr>\n<th>Inequality<\/th>\n<th>Words<\/th>\n<th>Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\n<td>all real numbers between\u00a0<em>a<\/em> and <em>b<\/em>, not including a and b<\/td>\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\gt{a}[\/latex]<\/td>\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\lt{b}[\/latex]<\/td>\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\ge{a}[\/latex]<\/td>\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\le{b}[\/latex]<\/td>\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{a}\\le{x}\\lt{ b}[\/latex]<\/td>\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{a}\\lt{x}\\le{ b}[\/latex]<\/td>\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{a}\\le{x}\\le{ b}[\/latex]<\/td>\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\lt{a}\\text{ or }{x}\\gt{ b}[\/latex]<\/td>\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\n<td>[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers<\/td>\n<td>All real numbers<\/td>\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td><strong>Start With<\/strong><\/td>\n<td><strong>Multiply By<\/strong><\/td>\n<td><strong>Final Inequality<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]ac>bc[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5>3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]15>9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]-c[\/latex]<\/td>\n<td>[latex]-ac<-bc[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5>3[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-15<-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The following table illustrates how the division\u00a0property is applied to inequalities, and how dividing by a negative reverses the inequality:<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td><strong>Start With<\/strong><\/td>\n<td><strong>Divide By<\/strong><\/td>\n<td><strong>Final Inequality<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{a}{c}>\\frac{b}{c}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4>2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{4}{2}>\\frac{2}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]-c[\/latex]<\/td>\n<td>[latex]\\displaystyle -\\frac{a}{c}<-\\frac{b}{c}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4>2[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]\\displaystyle -\\frac{4}{2}<-\\frac{2}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":253111,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"2effe3dbd32040468a7ce546b498aa70","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17824","chapter","type-chapter","status-publish","hentry"],"part":18856,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17824","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":6,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17824\/revisions"}],"predecessor-version":[{"id":20321,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17824\/revisions\/20321"}],"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\/17824\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/media?parent=17824"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=17824"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=17824"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/license?post=17824"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}