{"id":16601,"date":"2019-10-03T19:34:49","date_gmt":"2019-10-03T19:34:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-define-solutions-to-systems-of-linear-inequalities\/"},"modified":"2024-05-01T19:02:53","modified_gmt":"2024-05-01T19:02:53","slug":"read-define-solutions-to-systems-of-linear-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-define-solutions-to-systems-of-linear-inequalities\/","title":{"raw":"Defining Solutions to Systems of Linear Inequalities","rendered":"Defining Solutions to Systems of Linear Inequalities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify solutions to systems of linear inequalities<\/li>\r\n<\/ul>\r\n<\/div>\r\n<a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-ordered-pai\u2026tions-to-systems\/\">Previously, we learned how we can plug an ordered pair into a system of equations to determine whether it is a solution to the system<\/a>.\u00a0 We can use this same method to determine whether a point is a solution to a system of linear inequalities.\r\n<h2 id=\"title4\">Determine whether an ordered pair is a solution to a system of linear inequalities<\/h2>\r\n<img class=\"size-medium wp-image-398 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064403\/image014-300x297.gif\" alt=\"image014\" width=\"300\" height=\"297\" \/>\r\n\r\nOn the graph above, you can see that the points B and N are solutions for the system because their coordinates will make both inequalities true statements.\r\n\r\nIn contrast, points M and A both lie outside the solution region (purple). While point M is a solution for the inequality [latex]y&gt;\u2212x[\/latex] and point A is a solution for the inequality [latex]y&lt;2x+5[\/latex], neither point is a solution for the system. The following example shows how to test a point to see whether it is a solution to a system of inequalities.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIs the point [latex](2, 1)[\/latex] a solution of the system [latex]x+y&gt;1[\/latex] and [latex]2x+y&lt;8[\/latex]?\r\n\r\n[reveal-answer q=\"84880\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"84880\"]Check the point with each of the inequalities. Substitute [latex]2[\/latex] for [latex]x[\/latex] and [latex]1[\/latex] for [latex]y[\/latex]. Is the point a solution of both inequalities?\r\n\r\n[latex]\\begin{array}{r}x+y&gt;1\\\\2+1&gt;1\\\\3&gt;1\\\\\\text{TRUE}\\end{array}[\/latex]\r\n\r\n[latex](2, 1)[\/latex] is a solution for [latex]x+y&gt;1[\/latex].\r\n\r\n[latex]\\begin{array}{r}2x+y&lt;8\\\\2\\left(2\\right)+1&lt;8\\\\4+1&lt;8\\\\5&lt;8\\\\\\text{TRUE}\\end{array}[\/latex]\r\n\r\n[latex](2, 1)[\/latex] is a solution for [latex]2x+y&lt;8.[\/latex]\r\n\r\nSince [latex](2, 1)[\/latex] is a solution of each inequality, it is also a solution of the system of inequalities.\r\n<h4>Answer<\/h4>\r\nThe point [latex](2, 1)[\/latex] is a solution of the system [latex]x+y&gt;1[\/latex] and [latex]2x+y&lt;8[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere is a graph of the system in the example above. Notice that [latex](2, 1)[\/latex] lies in the purple area, which is the overlapping area for the two inequalities.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064404\/image015.gif\" alt=\"Two dotted lines, one red and one blue. The region below the blue dotted line is shaded and labeled 2x+y is less than 8. The region above the dotted red line is shaded and labeled x+y is greater than 1. The overlapping shaded region is purple and is labeled x+y is greater than 1 and 2x+y is less than 8. The point (2,1) is in the overlapping purple region.\" width=\"321\" height=\"317\" \/>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIs the point [latex](2, 1)[\/latex] a solution of the system [latex]x+y&gt;1[\/latex] and [latex]3x+y&lt;4[\/latex]?\r\n\r\n[reveal-answer q=\"833522\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"833522\"]\r\n\r\nCheck the point with each of the inequalities. Substitute [latex]2[\/latex] for [latex]x[\/latex] and [latex]1[\/latex] for [latex]y[\/latex]. Is the point a solution of both inequalities?\r\n\r\n[latex]\\begin{array}{r}x+y&gt;1\\\\2+1&gt;1\\\\3&gt;1\\\\\\text{TRUE}\\end{array}[\/latex]\r\n\r\n[latex](2, 1)[\/latex] is a solution for [latex]x+y&gt;1[\/latex].\r\n\r\n[latex]\\begin{array}{r}3x+y&lt;4\\\\3\\left(2\\right)+1&lt;4\\\\6+1&lt;4\\\\7&lt;4\\\\\\text{FALSE}\\end{array}[\/latex]\r\n\r\n[latex](2, 1)[\/latex] is not a solution for [latex]3x+y&lt;4[\/latex].\r\n\r\nSince [latex](2, 1)[\/latex] is not a solution of one of the inequalities, it is not a solution of the system.\r\n<h4>Answer<\/h4>\r\n<span lang=\"X-NONE\">The point [latex](2, 1)[\/latex] is not a solution of the system [latex]x+y&gt;1[\/latex]<\/span>\u00a0and [latex]3x+y&lt;4[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere is a graph of this system. Notice that [latex](2, 1)[\/latex] is not in the purple area, which is the overlapping area; it is a solution for one inequality (the red region), but it is not a solution for the second inequality (the blue region).\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064405\/image016.gif\" alt=\"A downward-sloping bue dotted line with the region below shaded and labeled 3x+y is less than 4. A downward-sloping red dotted line with the region above it shaded and labeled x+y is greater than 1. An overlapping purple shaded region is labeled x+y is greater than 1 and 3x+y is less than 4. A point (2,1) is in the red shaded region, but not the blue or overlapping purple shaded region.\" width=\"346\" height=\"342\" \/>\r\n\r\nYou can watch the video below for another example of how to verify whether an ordered pair is a solution to a system of linear inequalities.\r\n\r\nhttps:\/\/youtu.be\/o9hTFJEBcXs\r\n<h2>Summary<\/h2>\r\n<ul>\r\n \t<li>Solutions to systems of linear inequalities are entire regions of points.<\/li>\r\n \t<li>You can verify whether a point is a solution to a system of linear inequalities in the same way you verify whether a point is a solution to a system of equations.<\/li>\r\n<\/ul>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify solutions to systems of linear inequalities<\/li>\n<\/ul>\n<\/div>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-ordered-pai\u2026tions-to-systems\/\">Previously, we learned how we can plug an ordered pair into a system of equations to determine whether it is a solution to the system<\/a>.\u00a0 We can use this same method to determine whether a point is a solution to a system of linear inequalities.<\/p>\n<h2 id=\"title4\">Determine whether an ordered pair is a solution to a system of linear inequalities<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-398 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064403\/image014-300x297.gif\" alt=\"image014\" width=\"300\" height=\"297\" \/><\/p>\n<p>On the graph above, you can see that the points B and N are solutions for the system because their coordinates will make both inequalities true statements.<\/p>\n<p>In contrast, points M and A both lie outside the solution region (purple). While point M is a solution for the inequality [latex]y>\u2212x[\/latex] and point A is a solution for the inequality [latex]y<2x+5[\/latex], neither point is a solution for the system. The following example shows how to test a point to see whether it is a solution to a system of inequalities.\n\n\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Is the point [latex](2, 1)[\/latex] a solution of the system [latex]x+y>1[\/latex] and [latex]2x+y<8[\/latex]?\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q84880\">Show Solution<\/span><\/p>\n<div id=\"q84880\" class=\"hidden-answer\" style=\"display: none\">Check the point with each of the inequalities. Substitute [latex]2[\/latex] for [latex]x[\/latex] and [latex]1[\/latex] for [latex]y[\/latex]. Is the point a solution of both inequalities?<\/p>\n<p>[latex]\\begin{array}{r}x+y>1\\\\2+1>1\\\\3>1\\\\\\text{TRUE}\\end{array}[\/latex]<\/p>\n<p>[latex](2, 1)[\/latex] is a solution for [latex]x+y>1[\/latex].<\/p>\n<p>[latex]\\begin{array}{r}2x+y<8\\\\2\\left(2\\right)+1<8\\\\4+1<8\\\\5<8\\\\\\text{TRUE}\\end{array}[\/latex]\n\n[latex](2, 1)[\/latex] is a solution for [latex]2x+y<8.[\/latex]\n\nSince [latex](2, 1)[\/latex] is a solution of each inequality, it is also a solution of the system of inequalities.\n\n\n<h4>Answer<\/h4>\n<p>The point [latex](2, 1)[\/latex] is a solution of the system [latex]x+y>1[\/latex] and [latex]2x+y<8[\/latex].\n\n<\/div>\n<\/div>\n<\/div>\n<p>Here is a graph of the system in the example above. Notice that [latex](2, 1)[\/latex] lies in the purple area, which is the overlapping area for the two inequalities.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064404\/image015.gif\" alt=\"Two dotted lines, one red and one blue. The region below the blue dotted line is shaded and labeled 2x+y is less than 8. The region above the dotted red line is shaded and labeled x+y is greater than 1. The overlapping shaded region is purple and is labeled x+y is greater than 1 and 2x+y is less than 8. The point (2,1) is in the overlapping purple region.\" width=\"321\" height=\"317\" \/><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Is the point [latex](2, 1)[\/latex] a solution of the system [latex]x+y>1[\/latex] and [latex]3x+y<4[\/latex]?\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q833522\">Show Solution<\/span><\/p>\n<div id=\"q833522\" class=\"hidden-answer\" style=\"display: none\">\n<p>Check the point with each of the inequalities. Substitute [latex]2[\/latex] for [latex]x[\/latex] and [latex]1[\/latex] for [latex]y[\/latex]. Is the point a solution of both inequalities?<\/p>\n<p>[latex]\\begin{array}{r}x+y>1\\\\2+1>1\\\\3>1\\\\\\text{TRUE}\\end{array}[\/latex]<\/p>\n<p>[latex](2, 1)[\/latex] is a solution for [latex]x+y>1[\/latex].<\/p>\n<p>[latex]\\begin{array}{r}3x+y<4\\\\3\\left(2\\right)+1<4\\\\6+1<4\\\\7<4\\\\\\text{FALSE}\\end{array}[\/latex]\n\n[latex](2, 1)[\/latex] is not a solution for [latex]3x+y<4[\/latex].\n\nSince [latex](2, 1)[\/latex] is not a solution of one of the inequalities, it is not a solution of the system.\n\n\n<h4>Answer<\/h4>\n<p><span lang=\"X-NONE\">The point [latex](2, 1)[\/latex] is not a solution of the system [latex]x+y>1[\/latex]<\/span>\u00a0and [latex]3x+y<4[\/latex].\n\n<\/div>\n<\/div>\n<\/div>\n<p>Here is a graph of this system. Notice that [latex](2, 1)[\/latex] is not in the purple area, which is the overlapping area; it is a solution for one inequality (the red region), but it is not a solution for the second inequality (the blue region).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064405\/image016.gif\" alt=\"A downward-sloping bue dotted line with the region below shaded and labeled 3x+y is less than 4. A downward-sloping red dotted line with the region above it shaded and labeled x+y is greater than 1. An overlapping purple shaded region is labeled x+y is greater than 1 and 3x+y is less than 4. A point (2,1) is in the red shaded region, but not the blue or overlapping purple shaded region.\" width=\"346\" height=\"342\" \/><\/p>\n<p>You can watch the video below for another example of how to verify whether an ordered pair is a solution to a system of linear inequalities.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine if an Ordered Pair is a Solution to a System of Linear Inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/o9hTFJEBcXs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<ul>\n<li>Solutions to systems of linear inequalities are entire regions of points.<\/li>\n<li>You can verify whether a point is a solution to a system of linear inequalities in the same way you verify whether a point is a solution to a system of equations.<\/li>\n<\/ul>\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-16601\">\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>Ex 1: Graph a System of Linear Inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ACTxJv1h2_c\">https:\/\/youtu.be\/ACTxJv1h2_c<\/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 2: Graph a System of Linear Inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/cclH2h1NurM\">https:\/\/youtu.be\/cclH2h1NurM<\/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 14: Systems of Equations and Inequalities, 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":169554,"menu_order":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Graph a System of Linear Inequalities\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ACTxJv1h2_c\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2: Graph a System of Linear Inequalities\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/cclH2h1NurM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 14: Systems of Equations and Inequalities, 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":"08fed06aff724cbab7096a4a463cbae9, 5764dba018d641a291470af6400625ee","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16601","chapter","type-chapter","status-publish","hentry"],"part":16192,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16601","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16601\/revisions"}],"predecessor-version":[{"id":19073,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16601\/revisions\/19073"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/16192"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16601\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=16601"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=16601"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=16601"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=16601"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}