{"id":372,"date":"2019-07-15T22:44:32","date_gmt":"2019-07-15T22:44:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/solution-sets-of-inequalities\/"},"modified":"2019-07-15T22:44:32","modified_gmt":"2019-07-15T22:44:32","slug":"solution-sets-of-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/solution-sets-of-inequalities\/","title":{"raw":"Solution Sets of Inequalities","rendered":"Solution Sets of Inequalities"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n \t<li>Identify whether an ordered pair is in the solution set of a linear inequality<\/li>\n<\/ul>\n<\/div>\nThe graph below shows the region of values that makes the inequality [latex]3x+2y\\leq6[\/latex] true (shaded red), the boundary line [latex]3x+2y=6[\/latex], as well as a handful of ordered pairs. The boundary line is solid because points on the boundary line [latex]3x+2y=6[\/latex]&nbsp;will make the inequality [latex]3x+2y\\leq6[\/latex]&nbsp;true.\n\n<img class=\"aligncenter wp-image-2873 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19172056\/Screen-Shot-2016-04-19-at-10.20.21-AM.png\" alt=\"A solid downward-sloping line running. The region below the line is shaded and is labeled 3x+2y is less than or equal to 6. The region above the line is unshaded and is labeled 3x+2y=6. The points (-5,5) and (-2,-2) are in the shaded region. The points (2,3) and (4,-1) are in the unshaded region. The point (2,0) is on the line.\" width=\"464\" height=\"472\">\n\nYou can substitute the <i>x<\/i>&nbsp;and <i>y-<\/i>values of each of the [latex](x,y)[\/latex] ordered pairs into the inequality to find solutions. Sometimes making a table of values makes sense for more complicated inequalities.\n<table>\n<thead>\n<tr>\n<th style=\"text-align: left\"><strong>Ordered Pair<\/strong><\/th>\n<th style=\"text-align: left\"><strong>Makes the inequality&nbsp;<\/strong><strong>[latex]3x+2y\\leq6[\/latex]&nbsp;<\/strong><strong>a true statement<\/strong><\/th>\n<th style=\"text-align: left\"><strong>Makes the inequality&nbsp;<\/strong><strong>[latex]3x+2y\\leq6[\/latex]&nbsp;<\/strong><strong>a false statement<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: left\">[latex](\u22125, 5)[\/latex]<\/td>\n<td style=\"text-align: left\">[latex]\\begin{array}{r}3\\left(\u22125\\right)+2\\left(5\\right)\\leq6\\\\\u221215+10\\leq6\\\\\u22125\\leq6\\end{array}[\/latex]<\/td>\n<td style=\"text-align: left\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left\">[latex](\u22122,\u22122)[\/latex]<\/td>\n<td style=\"text-align: left\">[latex]\\begin{array}{r}3\\left(\u22122\\right)+2\\left(\u20132\\right)\\leq6\\\\\u22126+\\left(\u22124\\right)\\leq6\\\\\u201310\\leq6\\end{array}[\/latex]<\/td>\n<td style=\"text-align: left\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left\">[latex](2,3)[\/latex]<\/td>\n<td style=\"text-align: left\"><\/td>\n<td style=\"text-align: left\">[latex]\\begin{array}{r}3\\left(2\\right)+2\\left(3\\right)\\leq6\\\\6+6\\leq6\\\\12\\leq6\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left\">[latex](2,0)[\/latex]<\/td>\n<td style=\"text-align: left\">[latex]\\begin{array}{r}3\\left(2\\right)+2\\left(0\\right)\\leq6\\\\6+0\\leq6\\\\6\\leq6\\end{array}[\/latex]<\/td>\n<td style=\"text-align: left\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left\">[latex](4,\u22121)[\/latex]<\/td>\n<td style=\"text-align: left\"><\/td>\n<td style=\"text-align: left\">[latex]\\begin{array}{r}3\\left(4\\right)+2\\left(\u22121\\right)\\leq6\\\\12+\\left(\u22122\\right)\\leq6\\\\10\\leq6\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nIf substituting [latex](x,y)[\/latex] into the inequality yields a true statement, then the ordered pair is a solution to the inequality, and the point will be plotted within the shaded region or the point will be part of a solid boundary line. A false statement means that the ordered pair is not a solution, and the point will graph outside the shaded region, or the point will be part of a dotted boundary line.\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nUse the graph to determine which ordered pairs plotted below are solutions of the inequality&nbsp;[latex]x\u2013y&lt;3[\/latex].\n\n<img class=\"aligncenter wp-image-2876 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19172536\/Screen-Shot-2016-04-19-at-10.25.12-AM.png\" alt=\"Upward-sloping dotted line. The region above the line is shaded and labeled x-y<3. The points (4,0) and (3,-2) are in the unshaded region. The point (1,-2) is on the dotted line. The points (-1,1) and (-2,-2) are in the shaded region.\" width=\"410\" height=\"415\">\n\n[reveal-answer q=\"840389\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"840389\"]\n\nSolutions will be located in the shaded region. Since this is a \u201cless than\u201d problem, ordered pairs on the boundary line are not included in the solution set.\n\nThe values below are located in the shaded region so they are solutions. When substituted into the inequality&nbsp;[latex]x\u2013y&lt;3[\/latex], they produce true statements.\n<p style=\"text-align: center\">[latex](\u22121,1)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](\u22122,\u22122)[\/latex]<\/p>\nThese values are not located in the shaded region, so are not solutions. When substituted into the inequality [latex]x-y&lt;3[\/latex], they produce false statements.\n<p style=\"text-align: center\">[latex](1,\u22122)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](3,\u22122)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](4,0)[\/latex]<\/p>\n[latex](\u22121,1)[\/latex] and [latex](\u22122,\u22122)[\/latex] are the ordered pairs indicated in the graph that are solutions to the inequality.[\/hidden-answer]\n\n<\/div>\nThe following video shows an example of determining whether an ordered pair is a solution to an inequality.\n\nhttps:\/\/youtu.be\/GQVdDRVq5_o\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nIs [latex](2,\u22123)[\/latex] a solution of the inequality [latex]y&lt;\u22123x+1[\/latex]?\n\n[reveal-answer q=\"746731\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"746731\"]\n\nIf [latex](2,\u22123)[\/latex] is a solution, then it will yield a true statement when substituted into the inequality&nbsp;[latex]y&lt;\u22123x+1[\/latex].\n<p style=\"text-align: center\">[latex]y&lt;\u22123x+1[\/latex]<\/p>\nSubstitute&nbsp;[latex]x=2[\/latex] and [latex]y=\u22123[\/latex]&nbsp;into inequality.\n<p style=\"text-align: center\">[latex]\u22123&lt;\u22123\\left(2\\right)+1[\/latex]<\/p>\nEvaluate.\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\u22123&lt;\u22126+1\\\\\u22123&lt;\u22125\\end{array}[\/latex]<\/p>\nThis statement is <b>not <\/b>true, so the ordered pair [latex](2,\u22123)[\/latex] is <b>not <\/b>a solution.\n\n[\/hidden-answer]\n\n<\/div>\nThe following video shows another example of determining whether an ordered pair is a solution to an inequality.\n\nhttps:\/\/youtu.be\/-x-zt_yM0RM\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Identify whether an ordered pair is in the solution set of a linear inequality<\/li>\n<\/ul>\n<\/div>\n<p>The graph below shows the region of values that makes the inequality [latex]3x+2y\\leq6[\/latex] true (shaded red), the boundary line [latex]3x+2y=6[\/latex], as well as a handful of ordered pairs. The boundary line is solid because points on the boundary line [latex]3x+2y=6[\/latex]&nbsp;will make the inequality [latex]3x+2y\\leq6[\/latex]&nbsp;true.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2873 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19172056\/Screen-Shot-2016-04-19-at-10.20.21-AM.png\" alt=\"A solid downward-sloping line running. The region below the line is shaded and is labeled 3x+2y is less than or equal to 6. The region above the line is unshaded and is labeled 3x+2y=6. The points (-5,5) and (-2,-2) are in the shaded region. The points (2,3) and (4,-1) are in the unshaded region. The point (2,0) is on the line.\" width=\"464\" height=\"472\" \/><\/p>\n<p>You can substitute the <i>x<\/i>&nbsp;and <i>y-<\/i>values of each of the [latex](x,y)[\/latex] ordered pairs into the inequality to find solutions. Sometimes making a table of values makes sense for more complicated inequalities.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: left\"><strong>Ordered Pair<\/strong><\/th>\n<th style=\"text-align: left\"><strong>Makes the inequality&nbsp;<\/strong><strong>[latex]3x+2y\\leq6[\/latex]&nbsp;<\/strong><strong>a true statement<\/strong><\/th>\n<th style=\"text-align: left\"><strong>Makes the inequality&nbsp;<\/strong><strong>[latex]3x+2y\\leq6[\/latex]&nbsp;<\/strong><strong>a false statement<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: left\">[latex](\u22125, 5)[\/latex]<\/td>\n<td style=\"text-align: left\">[latex]\\begin{array}{r}3\\left(\u22125\\right)+2\\left(5\\right)\\leq6\\\\\u221215+10\\leq6\\\\\u22125\\leq6\\end{array}[\/latex]<\/td>\n<td style=\"text-align: left\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left\">[latex](\u22122,\u22122)[\/latex]<\/td>\n<td style=\"text-align: left\">[latex]\\begin{array}{r}3\\left(\u22122\\right)+2\\left(\u20132\\right)\\leq6\\\\\u22126+\\left(\u22124\\right)\\leq6\\\\\u201310\\leq6\\end{array}[\/latex]<\/td>\n<td style=\"text-align: left\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left\">[latex](2,3)[\/latex]<\/td>\n<td style=\"text-align: left\"><\/td>\n<td style=\"text-align: left\">[latex]\\begin{array}{r}3\\left(2\\right)+2\\left(3\\right)\\leq6\\\\6+6\\leq6\\\\12\\leq6\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left\">[latex](2,0)[\/latex]<\/td>\n<td style=\"text-align: left\">[latex]\\begin{array}{r}3\\left(2\\right)+2\\left(0\\right)\\leq6\\\\6+0\\leq6\\\\6\\leq6\\end{array}[\/latex]<\/td>\n<td style=\"text-align: left\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left\">[latex](4,\u22121)[\/latex]<\/td>\n<td style=\"text-align: left\"><\/td>\n<td style=\"text-align: left\">[latex]\\begin{array}{r}3\\left(4\\right)+2\\left(\u22121\\right)\\leq6\\\\12+\\left(\u22122\\right)\\leq6\\\\10\\leq6\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If substituting [latex](x,y)[\/latex] into the inequality yields a true statement, then the ordered pair is a solution to the inequality, and the point will be plotted within the shaded region or the point will be part of a solid boundary line. A false statement means that the ordered pair is not a solution, and the point will graph outside the shaded region, or the point will be part of a dotted boundary line.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use the graph to determine which ordered pairs plotted below are solutions of the inequality&nbsp;[latex]x\u2013y<3[\/latex].\n\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2876 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19172536\/Screen-Shot-2016-04-19-at-10.25.12-AM.png\" alt=\"Upward-sloping dotted line. The region above the line is shaded and labeled x-y&lt;3. The points (4,0) and (3,-2) are in the unshaded region. The point (1,-2) is on the dotted line. The points (-1,1) and (-2,-2) are in the shaded region.\" width=\"410\" height=\"415\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q840389\">Show Solution<\/span><\/p>\n<div id=\"q840389\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solutions will be located in the shaded region. Since this is a \u201cless than\u201d problem, ordered pairs on the boundary line are not included in the solution set.<\/p>\n<p>The values below are located in the shaded region so they are solutions. When substituted into the inequality&nbsp;[latex]x\u2013y<3[\/latex], they produce true statements.\n\n\n<p style=\"text-align: center\">[latex](\u22121,1)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](\u22122,\u22122)[\/latex]<\/p>\n<p>These values are not located in the shaded region, so are not solutions. When substituted into the inequality [latex]x-y<3[\/latex], they produce false statements.\n\n\n<p style=\"text-align: center\">[latex](1,\u22122)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](3,\u22122)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](4,0)[\/latex]<\/p>\n<p>[latex](\u22121,1)[\/latex] and [latex](\u22122,\u22122)[\/latex] are the ordered pairs indicated in the graph that are solutions to the inequality.<\/p><\/div>\n<\/div>\n<\/div>\n<p>The following video shows an example of determining whether an ordered pair is a solution to an inequality.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Use a Graph Determine Ordered Pair Solutions of a Linear Inequality in Two Variable\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GQVdDRVq5_o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Is [latex](2,\u22123)[\/latex] a solution of the inequality [latex]y<\u22123x+1[\/latex]?\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q746731\">Show Solution<\/span><\/p>\n<div id=\"q746731\" class=\"hidden-answer\" style=\"display: none\">\n<p>If [latex](2,\u22123)[\/latex] is a solution, then it will yield a true statement when substituted into the inequality&nbsp;[latex]y<\u22123x+1[\/latex].\n\n\n<p style=\"text-align: center\">[latex]y<\u22123x+1[\/latex]<\/p>\n<p>Substitute&nbsp;[latex]x=2[\/latex] and [latex]y=\u22123[\/latex]&nbsp;into inequality.<\/p>\n<p style=\"text-align: center\">[latex]\u22123<\u22123\\left(2\\right)+1[\/latex]<\/p>\n<p>Evaluate.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\u22123<\u22126+1\\\\\u22123<\u22125\\end{array}[\/latex]<\/p>\n<p>This statement is <b>not <\/b>true, so the ordered pair [latex](2,\u22123)[\/latex] is <b>not <\/b>a solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows another example of determining whether an ordered pair is a solution to an inequality.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Determine if Ordered Pairs Satisfy a Linear Inequality\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/-x-zt_yM0RM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-372\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Use a Graph Determine Ordered Pair Solutions of a Linear Inequalty in Two Variable. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/GQVdDRVq5_o\">https:\/\/youtu.be\/GQVdDRVq5_o<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Determine if Ordered Pairs Satisfy a Linear Inequality. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/-x-zt_yM0RM\">https:\/\/youtu.be\/-x-zt_yM0RM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 13: Graphing, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/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":17533,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Use a Graph Determine Ordered Pair Solutions of a Linear Inequalty in Two Variable\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/GQVdDRVq5_o\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Determine if Ordered Pairs Satisfy a Linear Inequality\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/-x-zt_yM0RM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 13: Graphing, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-372","chapter","type-chapter","status-publish","hentry"],"part":368,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/372","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/372\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/parts\/368"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapters\/372\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/media?parent=372"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=372"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/contributor?post=372"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/wp-json\/wp\/v2\/license?post=372"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}