{"id":16602,"date":"2019-10-03T19:34:49","date_gmt":"2019-10-03T19:34:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-graph-solutions-to-systems-of-linear-inequalities\/"},"modified":"2021-01-19T12:17:17","modified_gmt":"2021-01-19T12:17:17","slug":"read-graph-solutions-to-systems-of-linear-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-graph-solutions-to-systems-of-linear-inequalities\/","title":{"raw":"10.4.a - Graphing Solutions to Systems of Linear Inequalities","rendered":"10.4.a &#8211; Graphing 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>Graph systems of linear inequalities<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2 id=\"title2\">Graph a System of Two Inequalities<\/h2>\r\nRemember from the module on graphing that the graph of a single linear inequality splits the <b>coordinate plane<\/b> into two regions.\u00a0On one side lie all the solutions to the inequality. On the other side, there are no solutions. Consider the graph of the inequality [latex]y&lt;2x+5[\/latex].\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\/04064400\/image012.gif\" alt=\"An upward-sloping dotted line with the region below it shaded. The shaded region is labeled y is less than 2x+5. A is equal to (-1,1). B is equal to (3,1).\" width=\"346\" height=\"343\" \/>\r\n\r\nThe dashed line is [latex]y=2x+5[\/latex]. Every ordered pair in the shaded\u00a0area below the line is a solution to [latex]y&lt;2x+5[\/latex], as all of the points below the line will make the inequality true. If you doubt that, try substituting the <i>x<\/i> and <i>y<\/i> coordinates of Points A and B into the inequality; you will see that they work. So, the shaded area shows all of the solutions for this inequality.\r\n\r\nThe boundary line divides the coordinate plane in half. In this case, it is shown as a dashed line as the points on the line do not satisfy the inequality. If the inequality had been [latex]y\\leq2x+5[\/latex], then the boundary line would have been solid.\r\n\r\nNow graph another inequality: [latex]y&gt;\u2212x[\/latex]. You can check a couple of points to determine which side of the boundary line to shade. Checking points M and N yield true statements. So, we shade the area above the line. The line is dashed as points on the line are not true.\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\/04064401\/image013.gif\" alt=\"Downward-sloping dotted line with the region above it shaded. The shaded region is y is greater than negative x. Point M=(-2,3). Point N=(4,-1).\" width=\"327\" height=\"324\" \/>\r\n\r\nTo create a system of inequalities, you need to graph two or more inequalities together. Let us use\u00a0[latex]y&lt;2x+5[\/latex] and [latex]y&gt;\u2212x[\/latex] since we have already graphed each of them.\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\/04064403\/image014.gif\" alt=\"The two previous graphs combined. A blue dotted line with the region above shaded and labeled y is greater than negative x. A red dotted line with the region below it shaded and labeled y is less than 2x+5. The region where the shaded areas overlap is labeled y is greater than negative x and y is less than 2x+5. The point M equals (-2,3) and is in the blue shaded region. The point A equals (-1,-1) and is in the red shaded region. The point B equals (3,1) and is in the purple overlapping region. The point N equals (4,-1) and is also in the purple overlapping region.\" width=\"318\" height=\"315\" \/>\r\n\r\nThe purple area shows where the solutions of the two inequalities overlap. This area is the solution to the system of inequalities. Any point within this purple region will be true for both [latex]y&gt;\u2212x[\/latex] and [latex]y&lt;2x+5[\/latex].\r\n\r\nAs shown above, finding the solutions of a system of inequalities can be done by graphing each inequality and identifying the region they share. The general steps are outlined below:\r\n<ul>\r\n \t<li>Graph each inequality as a line and determine whether it will be solid or dashed<\/li>\r\n \t<li>Determine which side of each boundary line represents solutions to the inequality by testing a point on each side<\/li>\r\n \t<li>Shade the region\u00a0that represents solutions for both inequalities<\/li>\r\n<\/ul>\r\nWe will continue to practice graphing the solution region for systems of linear inequalities. We will also\u00a0graph the solutions to a system that includes a compound inequality.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nShade the region of the graph that represents solutions for both inequalities. [latex]x+y\\geq1[\/latex] and [latex]y\u2013x\\geq5[\/latex].\r\n\r\n[reveal-answer q=\"873537\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"873537\"]\r\n\r\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.\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\/04064406\/image017-2.jpg\" alt=\"A downward-sloping solid line labeled x+y is greater than 1.\" width=\"370\" height=\"370\" \/>\r\n\r\nFind an ordered pair on either side of the boundary line. Insert the <i>x<\/i>\u00a0and <i>y<\/i>-values into the inequality [latex]x+y\\geq1[\/latex] and see which ordered pair results in a true statement.\r\n\r\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]\r\n\r\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]\r\n\r\nSince\u00a0[latex](4, 1)[\/latex] results in a true statement, the region that includes\u00a0[latex](4, 1)[\/latex] should be shaded.\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\/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\" \/>\r\n\r\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\u00a0[latex](\u22123, 0)[\/latex] is not a solution of [latex]y\u2013x\\geq5[\/latex] and test point\u00a0[latex](0, 6)[\/latex] is a solution.\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\/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\" \/>\r\n\r\nThe purple region in this graph shows the set of all solutions of the system.\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\/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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe videos that follow show more\u00a0examples of graphing the solution set of a system of linear inequalities.\r\n\r\nhttps:\/\/youtu.be\/ACTxJv1h2_c\r\n\r\nhttps:\/\/youtu.be\/cclH2h1NurM\r\n\r\nThe system in our next example includes a compound inequality. \u00a0We will see that you can treat a compound inequality like two lines when you are graphing them.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the solution to the system [latex] 3x + 2y &lt; 12 [\/latex]\u00a0and [latex] -1 \u2264 y \u2264 5 [\/latex].\r\n[reveal-answer q=\"163187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"163187\"]\r\n\r\nGraph one inequality. First graph the boundary line, then test points.\r\n\r\nRemember, because the inequality [latex] 3x + 2y &lt; 12 [\/latex]\u00a0does not include the equal sign, draw a dashed border line.\r\n\r\nTesting a point like\u00a0[latex](0, 0)[\/latex] will show that the area below the line is the solution to this inequality.\r\n\r\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\" \/>\r\n\r\nThe inequality [latex] -1 \u2264 y \u2264 5[\/latex] is actually two inequalities:\u00a0[latex]\u22121 \u2264 y[\/latex], and\u00a0[latex]y \u2264 5[\/latex]. Another way to think of this is y must be between\u00a0[latex]\u22121[\/latex] and\u00a0[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\u00a0[latex]y = \u22121 [\/latex] and [latex] y = 5[\/latex].\r\n\r\nGraph this region on the same axes as the other inequality.\r\n\r\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\" \/>\r\n\r\nThe purple region shows the set of all solutions of the system.\r\n\r\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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video that follows, we show how to solve another system of inequalities that contains a compound inequality.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]93186[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Systems with No Solutions<\/h2>\r\nIn the next\u00a0example, we will show the\u00a0solution to\u00a0a system of two inequalities whose boundary lines are parallel to each other. \u00a0When the graphs of a system of two linear equations are parallel to each other, we found that there was no solution to the system. \u00a0We will get a similar result for the following system of linear inequalities.\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nGraph the system\u00a0[latex]\\begin{array}{c}y\\ge2x+1\\\\y\\lt2x-3\\end{array}[\/latex]\r\n[reveal-answer q=\"780322\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"780322\"]\r\n\r\nThe boundary lines for this system\u00a0are parallel to each other. Note how they have the same slopes.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}y=2x+1\\\\y=2x-3\\end{array}[\/latex]<\/p>\r\nPlotting the boundary lines will give the graph below. Note\u00a0that the inequality [latex]y\\lt2x-3[\/latex] requires that we draw a dashed line, while the inequality [latex]y\\ge2x+1[\/latex] requires a solid line.\r\n\r\n<img class=\"wp-image-4148 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183205\/Screen-Shot-2016-05-13-at-1.56.45-PM-300x300.png\" alt=\"y=2x+1\" width=\"410\" height=\"410\" \/>\r\n\r\nNow we need to shade the regions that represent the inequalities. \u00a0For the inequality [latex]y\\ge2x+1[\/latex], we can test a point on either side of the line to see which region to shade. Test [latex]\\left(0,0\\right)[\/latex] to make it easy.\r\n\r\nSubstitute\u00a0[latex]\\left(0,0\\right)[\/latex] into\u00a0[latex]y\\ge2x+1[\/latex]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}y\\ge2x+1\\\\0\\ge2\\left(0\\right)+1\\\\0\\ge{1}\\end{array}[\/latex]<\/p>\r\nThis is not true, so we know that we need to shade the other side of the boundary line for the inequality\u00a0\u00a0[latex]y\\ge2x+1[\/latex]. The graph will now look like this:\r\n\r\n<img class=\"wp-image-4149 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183206\/Screen-Shot-2016-05-13-at-2.02.49-PM-300x300.png\" alt=\"y=2x+1\" width=\"355\" height=\"355\" \/>\r\n\r\nNow shade the region that shows the solutions to the inequality [latex]y\\lt2x-3[\/latex]. \u00a0Again, we can pick\u00a0[latex]\\left(0,0\\right)[\/latex] to test, because it makes easy algebra.\r\n\r\nSubstitute\u00a0[latex]\\left(0,0\\right)[\/latex] into\u00a0[latex]y\\lt2x-3[\/latex]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}y\\lt2x-3\\\\0\\lt2\\left(0,\\right)x-3\\\\0\\lt{-3}\\end{array}[\/latex]<\/p>\r\nThis is not true, so we know that we need to shade the other side of the boundary line for the inequality [latex]y\\lt2x-3[\/latex]. The graph will now look like this:\r\n\r\n<img class=\"wp-image-4150 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183208\/Screen-Shot-2016-05-13-at-2.07.01-PM-297x300.png\" alt=\"y=2x+1\" width=\"394\" height=\"398\" \/>\r\n\r\nThis system of inequalities has no points in common so has no solution.\r\n\r\nWhat would the graph look like if the system had looked like this?\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}y\\ge2x+1\\\\y\\gt2x-3\\end{array}[\/latex].<\/p>\r\nTesting the point [latex]\\left(0,0\\right)[\/latex] would return a positive result for the inequality [latex]y\\gt2x-3[\/latex], and the graph would then look like this:\r\n\r\n<img class=\" wp-image-4157 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/13212029\/Screen-Shot-2016-05-13-at-2.19.42-PM-297x300.png\" alt=\"y&gt;2x-3 and y&gt;=2x+1\" width=\"388\" height=\"392\" \/>\r\n\r\nThe pink region is the region of overlap for both inequalities.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph systems of linear inequalities<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2 id=\"title2\">Graph a System of Two Inequalities<\/h2>\n<p>Remember from the module on graphing that the graph of a single linear inequality splits the <b>coordinate plane<\/b> into two regions.\u00a0On one side lie all the solutions to the inequality. On the other side, there are no solutions. Consider the graph of the inequality [latex]y<2x+5[\/latex].\n\n<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\/04064400\/image012.gif\" alt=\"An upward-sloping dotted line with the region below it shaded. The shaded region is labeled y is less than 2x+5. A is equal to (-1,1). B is equal to (3,1).\" width=\"346\" height=\"343\" \/><\/p>\n<p>The dashed line is [latex]y=2x+5[\/latex]. Every ordered pair in the shaded\u00a0area below the line is a solution to [latex]y<2x+5[\/latex], as all of the points below the line will make the inequality true. If you doubt that, try substituting the <i>x<\/i> and <i>y<\/i> coordinates of Points A and B into the inequality; you will see that they work. So, the shaded area shows all of the solutions for this inequality.<\/p>\n<p>The boundary line divides the coordinate plane in half. In this case, it is shown as a dashed line as the points on the line do not satisfy the inequality. If the inequality had been [latex]y\\leq2x+5[\/latex], then the boundary line would have been solid.<\/p>\n<p>Now graph another inequality: [latex]y>\u2212x[\/latex]. You can check a couple of points to determine which side of the boundary line to shade. Checking points M and N yield true statements. So, we shade the area above the line. The line is dashed as points on the line are not true.<\/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\/04064401\/image013.gif\" alt=\"Downward-sloping dotted line with the region above it shaded. The shaded region is y is greater than negative x. Point M=(-2,3). Point N=(4,-1).\" width=\"327\" height=\"324\" \/><\/p>\n<p>To create a system of inequalities, you need to graph two or more inequalities together. Let us use\u00a0[latex]y<2x+5[\/latex] and [latex]y>\u2212x[\/latex] since we have already graphed each of them.<\/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\/04064403\/image014.gif\" alt=\"The two previous graphs combined. A blue dotted line with the region above shaded and labeled y is greater than negative x. A red dotted line with the region below it shaded and labeled y is less than 2x+5. The region where the shaded areas overlap is labeled y is greater than negative x and y is less than 2x+5. The point M equals (-2,3) and is in the blue shaded region. The point A equals (-1,-1) and is in the red shaded region. The point B equals (3,1) and is in the purple overlapping region. The point N equals (4,-1) and is also in the purple overlapping region.\" width=\"318\" height=\"315\" \/><\/p>\n<p>The purple area shows where the solutions of the two inequalities overlap. This area is the solution to the system of inequalities. Any point within this purple region will be true for both [latex]y>\u2212x[\/latex] and [latex]y<2x+5[\/latex].\n\nAs shown above, finding the solutions of a system of inequalities can be done by graphing each inequality and identifying the region they share. The general steps are outlined below:\n\n\n<ul>\n<li>Graph each inequality as a line and determine whether it will be solid or dashed<\/li>\n<li>Determine which side of each boundary line represents solutions to the inequality by testing a point on each side<\/li>\n<li>Shade the region\u00a0that represents solutions for both inequalities<\/li>\n<\/ul>\n<p>We will continue to practice graphing the solution region for systems of linear inequalities. We will also\u00a0graph 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>\u00a0and <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\u00a0[latex](4, 1)[\/latex] results in a true statement, the region that includes\u00a0[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\u00a0[latex](\u22123, 0)[\/latex] is not a solution of [latex]y\u2013x\\geq5[\/latex] and test point\u00a0[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\u00a0examples 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 next example includes a compound inequality. \u00a0We 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]\u00a0and [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]\u00a0does not include the equal sign, draw a dashed border line.\n\nTesting a point like\u00a0[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:\u00a0[latex]\u22121 \u2264 y[\/latex], and\u00a0[latex]y \u2264 5[\/latex]. Another way to think of this is y must be between\u00a0[latex]\u22121[\/latex] and\u00a0[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\u00a0[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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm93186\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93186&theme=oea&iframe_resize_id=ohm93186&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Systems with No Solutions<\/h2>\n<p>In the next\u00a0example, we will show the\u00a0solution to\u00a0a system of two inequalities whose boundary lines are parallel to each other. \u00a0When the graphs of a system of two linear equations are parallel to each other, we found that there was no solution to the system. \u00a0We will get a similar result for the following system of linear inequalities.<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Graph the system\u00a0[latex]\\begin{array}{c}y\\ge2x+1\\\\y\\lt2x-3\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q780322\">Show Solution<\/span><\/p>\n<div id=\"q780322\" class=\"hidden-answer\" style=\"display: none\">\n<p>The boundary lines for this system\u00a0are parallel to each other. Note how they have the same slopes.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}y=2x+1\\\\y=2x-3\\end{array}[\/latex]<\/p>\n<p>Plotting the boundary lines will give the graph below. Note\u00a0that the inequality [latex]y\\lt2x-3[\/latex] requires that we draw a dashed line, while the inequality [latex]y\\ge2x+1[\/latex] requires a solid line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4148 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183205\/Screen-Shot-2016-05-13-at-1.56.45-PM-300x300.png\" alt=\"y=2x+1\" width=\"410\" height=\"410\" \/><\/p>\n<p>Now we need to shade the regions that represent the inequalities. \u00a0For the inequality [latex]y\\ge2x+1[\/latex], we can test a point on either side of the line to see which region to shade. Test [latex]\\left(0,0\\right)[\/latex] to make it easy.<\/p>\n<p>Substitute\u00a0[latex]\\left(0,0\\right)[\/latex] into\u00a0[latex]y\\ge2x+1[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}y\\ge2x+1\\\\0\\ge2\\left(0\\right)+1\\\\0\\ge{1}\\end{array}[\/latex]<\/p>\n<p>This is not true, so we know that we need to shade the other side of the boundary line for the inequality\u00a0\u00a0[latex]y\\ge2x+1[\/latex]. The graph will now look like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4149 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183206\/Screen-Shot-2016-05-13-at-2.02.49-PM-300x300.png\" alt=\"y=2x+1\" width=\"355\" height=\"355\" \/><\/p>\n<p>Now shade the region that shows the solutions to the inequality [latex]y\\lt2x-3[\/latex]. \u00a0Again, we can pick\u00a0[latex]\\left(0,0\\right)[\/latex] to test, because it makes easy algebra.<\/p>\n<p>Substitute\u00a0[latex]\\left(0,0\\right)[\/latex] into\u00a0[latex]y\\lt2x-3[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}y\\lt2x-3\\\\0\\lt2\\left(0,\\right)x-3\\\\0\\lt{-3}\\end{array}[\/latex]<\/p>\n<p>This is not true, so we know that we need to shade the other side of the boundary line for the inequality [latex]y\\lt2x-3[\/latex]. The graph will now look like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4150 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183208\/Screen-Shot-2016-05-13-at-2.07.01-PM-297x300.png\" alt=\"y=2x+1\" width=\"394\" height=\"398\" \/><\/p>\n<p>This system of inequalities has no points in common so has no solution.<\/p>\n<p>What would the graph look like if the system had looked like this?<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}y\\ge2x+1\\\\y\\gt2x-3\\end{array}[\/latex].<\/p>\n<p>Testing the point [latex]\\left(0,0\\right)[\/latex] would return a positive result for the inequality [latex]y\\gt2x-3[\/latex], and the graph would then look like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4157 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/13212029\/Screen-Shot-2016-05-13-at-2.19.42-PM-297x300.png\" alt=\"y&gt;2x-3 and y&gt;=2x+1\" width=\"388\" height=\"392\" \/><\/p>\n<p>The pink region is the region of overlap for both inequalities.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\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-16602\">\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) . <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":169554,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 2: Graph a System of Linear Inequalities\",\"author\":\"James Sousa (Mathispower4u.com) 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