{"id":374,"date":"2019-07-15T22:44:33","date_gmt":"2019-07-15T22:44:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/graph-solutions-to-systems-of-linear-inequalities\/"},"modified":"2019-07-15T22:44:33","modified_gmt":"2019-07-15T22:44:33","slug":"graph-solutions-to-systems-of-linear-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/graph-solutions-to-systems-of-linear-inequalities\/","title":{"raw":"Graph Solutions to Systems of Linear Inequalities","rendered":"Graph Solutions to Systems of Linear Inequalities"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Solve systems of linear inequalities by graphing the solution region<\/li>\n \t<li>Graph solutions to a system that contains a compound inequality<\/li>\n<\/ul>\n<\/div>\nWe will continue to practice graphing the solution region for systems of linear inequalities. We will also&nbsp;graph the solutions to a system that includes a compound inequality.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nShade the region of the graph that represents solutions for both inequalities. [latex]x+y\\geq1[\/latex] and [latex]y\u2013x\\geq5[\/latex].\n\n[reveal-answer q=\"873537\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"873537\"]\n\nGraph one inequality. First graph the boundary line using a table of values, intercepts, or any other method you prefer. The boundary line for [latex]x+y\\geq1[\/latex] is [latex]x+y=1[\/latex], or [latex]y=\u2212x+1[\/latex]. Since the equal sign is included with the greater than sign, the boundary line is solid.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064406\/image017-2.jpg\" alt=\"A downward-sloping solid line labeled x+y is greater than 1.\" width=\"370\" height=\"370\">\n\nFind an ordered pair on either side of the boundary line. Insert the <i>x<\/i>&nbsp;and <i>y<\/i>-values into the inequality [latex]x+y\\geq1[\/latex] and see which ordered pair results in a true statement.\n\n[latex]\\begin{array}{r}\\text{Test }1:\\left(\u22123,0\\right)\\\\x+y\\geq1\\\\\u22123+0\\geq1\\\\\u22123\\geq1\\\\\\text{FALSE}\\end{array}[\/latex]\n\n[latex]\\begin{array}{r}\\text{Test }2:\\left(4,1\\right)\\\\x+y\\geq1\\\\4+1\\geq1\\\\5\\geq1\\\\\\text{TRUE}\\end{array}[\/latex]\n\nSince&nbsp;[latex](4, 1)[\/latex] results in a true statement, the region that includes&nbsp;[latex](4, 1)[\/latex] should be shaded.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064409\/image018.gif\" alt=\"A solid downward-sloping line with the region above it shaded and labeled x+y is greater than or equal to 1. The point (4,1) is in the shaded region. The point (-3,0) is not.\" width=\"345\" height=\"342\">\n\nDo the same with the second inequality. Graph the boundary line, then test points to find which region is the solution to the inequality. In this case, the boundary line is [latex]y\u2013x=5\\left(\\text{or }y=x+5\\right)[\/latex] and is solid. Test point&nbsp;[latex](\u22123, 0)[\/latex] is not a solution of [latex]y\u2013x\\geq5[\/latex] and test point&nbsp;[latex](0, 6)[\/latex] is a solution.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064410\/image019.gif\" alt=\"A solid blue line with the region above it shaded and labeled y-x is greater than or equal to 5. A solid red line with the region above it shaded and labeled x+y is greater than 1. The point (-3,0) is not in any shaded region. The point (0,6) is in the overlapping shaded region.\" width=\"337\" height=\"334\">\n\nThe purple region in this graph shows the set of all solutions of the system.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064412\/image020-2.jpg\" alt=\"The previous graph, with the purple overlapping shaded region labeled x+y is greater than or equal to 1 and y-x is greater than or equal to 5.\" width=\"329\" height=\"325\">\n\n[\/hidden-answer]\n\n<\/div>\nThe videos that follow show more&nbsp;examples of graphing the solution set of a system of linear inequalities.\n\nhttps:\/\/youtu.be\/ACTxJv1h2_c\n\nhttps:\/\/youtu.be\/cclH2h1NurM\n\nThe system in our last example includes a compound inequality. &nbsp;We will see that you can treat a compound inequality like two lines when you are graphing them.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nFind the solution to the system [latex] 3x + 2y &lt; 12 [\/latex]&nbsp;and [latex] -1 \u2264 y \u2264 5 [\/latex].\n[reveal-answer q=\"163187\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"163187\"]\n\nGraph one inequality. First graph the boundary line, then test points.\n\nRemember, because the inequality [latex] 3x + 2y &lt; 12 [\/latex]&nbsp;does not include the equal sign, draw a dashed border line.\n\nTesting a point like&nbsp;[latex](0, 0)[\/latex] will show that the area below the line is the solution to this inequality.\n\n<img class=\"alignnone size-medium wp-image-2427\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223640\/image021-300x297.gif\" alt=\"image021\" width=\"300\" height=\"297\">\n\nThe inequality [latex] -1 \u2264 y \u2264 5[\/latex] is actually two inequalities:&nbsp;[latex]\u22121 \u2264 y[\/latex], and&nbsp;[latex]y \u2264 5[\/latex]. Another way to think of this is y must be between&nbsp;[latex]\u22121[\/latex] and&nbsp;[latex]5[\/latex]. The border lines for both are horizontal. The region between those two lines contains the solutions of [latex] -1 \u2264 y \u2264 5[\/latex]. We make the lines solid because we also want to include&nbsp;[latex]y = \u22121 [\/latex] and [latex] y = 5[\/latex].\n\nGraph this region on the same axes as the other inequality.\n\n<img class=\"alignnone size-medium wp-image-2428\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223642\/image022-300x298.gif\" alt=\"image022\" width=\"300\" height=\"298\">\n\nThe purple region shows the set of all solutions of the system.\n\n<img class=\"alignnone size-medium wp-image-2429\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223644\/image023-300x297.jpg\" alt=\"image023\" width=\"300\" height=\"297\">\n\n[\/hidden-answer]\n\n<\/div>\nIn the video that follows, we show how to solve another system of inequalities that contains a compound inequality.\n\nhttps:\/\/youtu.be\/ACTxJv1h2_c\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve systems of linear inequalities by graphing the solution region<\/li>\n<li>Graph solutions to a system that contains a compound inequality<\/li>\n<\/ul>\n<\/div>\n<p>We will continue to practice graphing the solution region for systems of linear inequalities. We will also&nbsp;graph the solutions to a system that includes a compound inequality.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Shade the region of the graph that represents solutions for both inequalities. [latex]x+y\\geq1[\/latex] and [latex]y\u2013x\\geq5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q873537\">Show Solution<\/span><\/p>\n<div id=\"q873537\" class=\"hidden-answer\" style=\"display: none\">\n<p>Graph one inequality. First graph the boundary line using a table of values, intercepts, or any other method you prefer. The boundary line for [latex]x+y\\geq1[\/latex] is [latex]x+y=1[\/latex], or [latex]y=\u2212x+1[\/latex]. Since the equal sign is included with the greater than sign, the boundary line is solid.<\/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\/04064406\/image017-2.jpg\" alt=\"A downward-sloping solid line labeled x+y is greater than 1.\" width=\"370\" height=\"370\" \/><\/p>\n<p>Find an ordered pair on either side of the boundary line. Insert the <i>x<\/i>&nbsp;and <i>y<\/i>-values into the inequality [latex]x+y\\geq1[\/latex] and see which ordered pair results in a true statement.<\/p>\n<p>[latex]\\begin{array}{r}\\text{Test }1:\\left(\u22123,0\\right)\\\\x+y\\geq1\\\\\u22123+0\\geq1\\\\\u22123\\geq1\\\\\\text{FALSE}\\end{array}[\/latex]<\/p>\n<p>[latex]\\begin{array}{r}\\text{Test }2:\\left(4,1\\right)\\\\x+y\\geq1\\\\4+1\\geq1\\\\5\\geq1\\\\\\text{TRUE}\\end{array}[\/latex]<\/p>\n<p>Since&nbsp;[latex](4, 1)[\/latex] results in a true statement, the region that includes&nbsp;[latex](4, 1)[\/latex] should be shaded.<\/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\/04064409\/image018.gif\" alt=\"A solid downward-sloping line with the region above it shaded and labeled x+y is greater than or equal to 1. The point (4,1) is in the shaded region. The point (-3,0) is not.\" width=\"345\" height=\"342\" \/><\/p>\n<p>Do the same with the second inequality. Graph the boundary line, then test points to find which region is the solution to the inequality. In this case, the boundary line is [latex]y\u2013x=5\\left(\\text{or }y=x+5\\right)[\/latex] and is solid. Test point&nbsp;[latex](\u22123, 0)[\/latex] is not a solution of [latex]y\u2013x\\geq5[\/latex] and test point&nbsp;[latex](0, 6)[\/latex] is a solution.<\/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\/04064410\/image019.gif\" alt=\"A solid blue line with the region above it shaded and labeled y-x is greater than or equal to 5. A solid red line with the region above it shaded and labeled x+y is greater than 1. The point (-3,0) is not in any shaded region. The point (0,6) is in the overlapping shaded region.\" width=\"337\" height=\"334\" \/><\/p>\n<p>The purple region in this graph shows the set of all solutions of the system.<\/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\/04064412\/image020-2.jpg\" alt=\"The previous graph, with the purple overlapping shaded region labeled x+y is greater than or equal to 1 and y-x is greater than or equal to 5.\" width=\"329\" height=\"325\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The videos that follow show more&nbsp;examples of graphing the solution set of a system of linear inequalities.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Graph a System of Linear Inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ACTxJv1h2_c?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 2:  Graph a System of Linear Inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/cclH2h1NurM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The system in our last example includes a compound inequality. &nbsp;We will see that you can treat a compound inequality like two lines when you are graphing them.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the solution to the system [latex]3x + 2y < 12[\/latex]&nbsp;and [latex]-1 \u2264 y \u2264 5[\/latex].\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q163187\">Show Solution<\/span><\/p>\n<div id=\"q163187\" class=\"hidden-answer\" style=\"display: none\">\n<p>Graph one inequality. First graph the boundary line, then test points.<\/p>\n<p>Remember, because the inequality [latex]3x + 2y < 12[\/latex]&nbsp;does not include the equal sign, draw a dashed border line.\n\nTesting a point like&nbsp;[latex](0, 0)[\/latex] will show that the area below the line is the solution to this inequality.\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2427\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223640\/image021-300x297.gif\" alt=\"image021\" width=\"300\" height=\"297\" \/><\/p>\n<p>The inequality [latex]-1 \u2264 y \u2264 5[\/latex] is actually two inequalities:&nbsp;[latex]\u22121 \u2264 y[\/latex], and&nbsp;[latex]y \u2264 5[\/latex]. Another way to think of this is y must be between&nbsp;[latex]\u22121[\/latex] and&nbsp;[latex]5[\/latex]. The border lines for both are horizontal. The region between those two lines contains the solutions of [latex]-1 \u2264 y \u2264 5[\/latex]. We make the lines solid because we also want to include&nbsp;[latex]y = \u22121[\/latex] and [latex]y = 5[\/latex].<\/p>\n<p>Graph this region on the same axes as the other inequality.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2428\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223642\/image022-300x298.gif\" alt=\"image022\" width=\"300\" height=\"298\" \/><\/p>\n<p>The purple region shows the set of all solutions of the system.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2429\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223644\/image023-300x297.jpg\" alt=\"image023\" width=\"300\" height=\"297\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, we show how to solve another system of inequalities that contains a compound inequality.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 1:  Graph a System of Linear Inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ACTxJv1h2_c?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-374\">\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>Ex 1: Graph a System of Linear Inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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>Determine the Solution to a System of Inequalities (Compound). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/D-Cnt6m8l18\">https:\/\/youtu.be\/D-Cnt6m8l18<\/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":17533,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"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\":\"original\",\"description\":\"Ex 1: Graph a System of Linear 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