{"id":16523,"date":"2019-10-03T17:08:46","date_gmt":"2019-10-03T17:08:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-determine-whether-an-ordered-pair-is-a-solution-to-an-inequality\/"},"modified":"2024-04-30T23:15:23","modified_gmt":"2024-04-30T23:15:23","slug":"read-or-watch-determine-whether-an-ordered-pair-is-a-solution-to-an-inequality","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-determine-whether-an-ordered-pair-is-a-solution-to-an-inequality\/","title":{"raw":"Identifying Solutions to Inequalities","rendered":"Identifying Solutions to Inequalities"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Classify solutions and graphs as equations or inequalities<\/li>\r\n \t<li>Identify whether an ordered pair is in the solution set of a linear inequality<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"Identify the difference between the graph of a linear equation and linear inequality\">Identify the difference between the graph of a linear equation and linear inequality<\/h2>\r\nRecall that solutions to linear inequalities are whole sets of numbers,\u00a0rather than just one number, like you find with solutions to equalities (equations).\r\n\r\nHere is an example from the section on solving linear inequalities:\r\n\r\nSolve for p. [latex]4p+5&lt;29[\/latex]\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}4p+5&lt;\\,\\,\\,29\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,-5\\,\\,\\,\\,\\,\\,\\,-5}\\\\\\underline{4p}\\,\\,\\,\\,\\,\\,\\,\\,&lt;\\,\\,\\underline{24}\\,\\,\\\\4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,p&lt;6\\end{array}[\/latex]<\/p>\r\nYou can interpret the solution as p can be any number less than six. Now recall that we can graph equations of lines by defining the outputs, [latex]y[\/latex], and the inputs, [latex]x[\/latex], and writing an equation.\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-gr\u2026ar-relationships\/\">Previously, we showed how to graph the line described by th equation\u00a0 [latex]y=2x+3[\/latex]<\/a>\u00a0<b><strong>\u00a0<\/strong><\/b>and found that we can construct a never-ending table of values that make points on the line\u2014these are some of the solutions to the equation [latex]y=2x+3[\/latex].\r\n<table>\r\n<thead>\r\n<tr>\r\n<th><em>x<\/em> values<\/th>\r\n<th>[latex]2x+3[\/latex]<\/th>\r\n<th><em>y<\/em> values<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]2(0)+3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2(1)+3[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]2(2)+3[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]2(3)+3[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAdditionally, we learned how to graph the line that represents all the points that make\u00a0[latex]y=2x+3[\/latex] a true statement.\r\n\r\n<img class=\"aligncenter wp-image-1398\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/07211017\/Putting-It-Together-Graphs.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A line labeled y=2x+3 is drawn in the coordinate plane.\" width=\"476\" height=\"367\" \/>\r\n\r\nWhat if we combined these two ideas\u2014linear inequalities and graphs of lines? First translate the line, [latex]y=2x+3[\/latex], into words:\r\n\r\nYou get y by multiplying [latex]x[\/latex] by two and adding three. [latex]y=2x+3[\/latex]\r\n\r\nHow would you translate this inequality into words? [latex]y&lt;2x+3[\/latex]\r\n\r\nFor what values of [latex]x[\/latex] will you get an output, [latex]y[\/latex], that is less than [latex]2[\/latex] times [latex]x[\/latex] plus three?\r\n\r\nWOW, that may\u00a0seem confusing, but keep reading, we'll help you figure it out.\r\n\r\nLinear inequalities are different than linear equations, although you can apply what you know about equations to help you understand inequalities. Inequalities and equations are both math statements that compare two values. Equations use the symbol = ; recall that inequalities are\u00a0represented by the symbols &lt; , \u2264 , &gt; , and \u2265.\r\n\r\nOne way to visualize two-variable inequalities is to plot them on a coordinate plane. Here is what the inequality\u00a0[latex]x&gt;y[\/latex]\u00a0looks like. The solution is a region, which is shaded. This region is made up of lots and lots of ordered pairs that all make the statement\u00a0[latex]x&gt;y[\/latex] true.\r\n\r\n<img class=\"aligncenter wp-image-2875 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19172339\/Screen-Shot-2016-04-19-at-10.23.13-AM.png\" alt=\"Dotted upward-sloping line. Everything below the dotted line is shaded and is labeled x is greater than y. Everything above the line is unshaded and is labeled x equals y.\" width=\"550\" height=\"553\" \/>\r\n\r\n&nbsp;\r\n\r\nThere are a few things to notice here. First, look at the dashed red boundary line: this is the graph of the related linear equation\u00a0[latex]x=y[\/latex]. Next, look at the light red region that is to the right of the line. This region (excluding the line [latex]x=y[\/latex]) represents the entire set of solutions for the inequality [latex]x&gt;y[\/latex]. Remember how all points on a <em>line<\/em> are solutions to the linear equation of the line? Well, all points in a <em>region<\/em> are solutions to the <b>linear inequality<\/b> representing that region.\r\n\r\nLet\u2019s think about it for a moment\u2014if [latex]x&gt;y[\/latex], then a graph of [latex]x&gt;y[\/latex]\u00a0will show all ordered pairs [latex](x,y)[\/latex] for which the <i>x-<\/i>coordinate is greater than the <i>y-<\/i>coordinate.\r\n<h2>Identify ordered pairs that are the solution set of a linear inequality<\/h2>\r\nThe graph below shows the region [latex]x&gt;y[\/latex]\u00a0as well as some ordered pairs on the coordinate plane. Look at each ordered pair. Is the <i>x-<\/i>coordinate greater than the <i>y-<\/i>coordinate? Does the ordered pair sit inside or outside of the shaded region?\r\n\r\n<img class=\"aligncenter wp-image-2874 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19172149\/Screen-Shot-2016-04-19-at-10.21.21-AM.png\" alt=\"Dotted upward-sloping line. Everything below the dotted line is shaded and is labeled x is greater than y. Everything above the line is unshaded and is labeled x equals y. The points (-3,3) and (2,3) are in the unshaded region. The points (4,0) and (0,-3) are in the shaded region. The point (-2,-2) is on the dotted line.\" width=\"417\" height=\"419\" \/>\r\n\r\n&nbsp;\r\n\r\nThe ordered pairs [latex](4,0)[\/latex] and [latex](0,\u22123)[\/latex] lie inside the shaded region. In these ordered pairs, the <i>x-<\/i>coordinate is larger than the <i>y-<\/i>coordinate. These ordered pairs are in the solution set of the equation [latex]x&gt;y[\/latex].\r\n\r\nThe ordered pairs [latex](\u22123,3)[\/latex] and [latex](2,3)[\/latex] are outside of the shaded area. In these ordered pairs, the <i>x-<\/i>coordinate is <i>smaller<\/i> than the <i>y-<\/i>coordinate, so they are not included in the set of solutions for the inequality.\r\n\r\nThe ordered pair [latex](\u22122,\u22122)[\/latex] is on the boundary line. It is not a solution as [latex]\u22122[\/latex] is not greater than [latex]\u22122[\/latex]. However, had the inequality been [latex]x\\geq y[\/latex]\u00a0(read as \u201c<i>x<\/i> is greater than or equal to <i>y<\/i>\u201d), then [latex](\u22122,\u22122)[\/latex] would have been included (and the line would have been represented by a solid line, not a dashed line).\r\n<p class=\"yt watch-title-container\"><span id=\"eow-title\" class=\"watch-title\" dir=\"ltr\" title=\"The Difference Between a Linear Equation and Linear Inequality (Two Variables)\">Watch the video below for another explanation of the difference between a linear equation and a linear inequality.<\/span><\/p>\r\nhttps:\/\/youtu.be\/EcrLbRJ2zV0\r\n\r\nLet\u2019s take a look at one more example: the inequality [latex]3x+2y\\leq6[\/latex]. The graph below shows the region of values that makes this inequality true (shaded red), the boundary line [latex]3x+2y=6[\/latex], as well as a handful of ordered pairs. The boundary line is solid this time, because points on the boundary line [latex]3x+2y=6[\/latex]\u00a0will make the inequality [latex]3x+2y\\leq6[\/latex]\u00a0true.\r\n\r\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\" \/>\r\n\r\n&nbsp;\r\n\r\nAs you did with the previous example, you can substitute the [latex]x[\/latex]- and [latex]y[\/latex]-values in each of the [latex](x,y)[\/latex] ordered pairs into the inequality to find solutions. While you may have been able to do this in your head for the inequality\u00a0[latex]x&gt;y[\/latex], sometimes making a table of values makes sense for more complicated inequalities.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Ordered Pair<\/th>\r\n<th>Makes the inequality\r\n\r\n[latex]3x+2y\\leq6[\/latex]\r\n\r\na true statement<\/th>\r\n<th>Makes the inequality\r\n\r\n[latex]3x+2y\\leq6[\/latex]\r\n\r\na false statement<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex](\u22125, 5)[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{r}3\\left(\u22125\\right)+2\\left(5\\right)\\leq6\\\\\u221215+10\\leq6\\\\\u22125\\leq6\\end{array}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex](\u22122,\u22122)[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{r}3\\left(\u22122\\right)+2\\left(\u20132\\right)\\leq6\\\\\u22126+\\left(\u22124\\right)\\leq6\\\\\u201310\\leq6\\end{array}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex](2,3)[\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\begin{array}{r}3\\left(2\\right)+2\\left(3\\right)\\leq6\\\\6+6\\leq6\\\\12\\leq6\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex](2,0)[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{r}3\\left(2\\right)+2\\left(0\\right)\\leq6\\\\6+0\\leq6\\\\6\\leq6\\end{array}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex](4,\u22121)[\/latex]<\/td>\r\n<td><\/td>\r\n<td>[latex]\\begin{array}{r}3\\left(4\\right)+2\\left(\u22121\\right)\\leq6\\\\12+\\left(\u22122\\right)\\leq6\\\\10\\leq6\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\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.\r\n\r\nYou can watch the following video to see more examples of how to determine whether an ordered pair satisfies a linear inequality.\r\n\r\nhttps:\/\/youtu.be\/-x-zt_yM0RM\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nUse the graph to determine which ordered pairs plotted below are solutions of the inequality\u00a0[latex]x\u2013y&lt;3[\/latex].\r\n\r\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&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\" \/>\r\n\r\n[reveal-answer q=\"840389\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"840389\"]\r\n\r\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.\r\n\r\nThese values are located in the shaded region, so are solutions. (When substituted into the inequality\u00a0[latex]x\u2013y&lt;3[\/latex], they produce true statements.)\r\n<p style=\"text-align: center;\">[latex](\u22121,1)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex](\u22122,\u22122)[\/latex]<\/p>\r\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.)\r\n<p style=\"text-align: center;\">[latex](1,\u22122)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex](3,\u22122)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex](4,0)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex](\u22121,1)\\,\\,\\,(\u22122,\u22122)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the video below to see an example of how to use a graph to determine whether an ordered pair is a solutions of a linear inequality in two variables\r\n\r\nhttps:\/\/youtu.be\/GQVdDRVq5_o\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIs [latex](2,\u22123)[\/latex] a solution of the inequality [latex]y&lt;\u22123x+1[\/latex]?\r\n\r\n[reveal-answer q=\"746731\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"746731\"]\r\n\r\nIf [latex](2,\u22123)[\/latex] is a solution, then it will yield a true statement when substituted into the inequality\u00a0[latex]y&lt;\u22123x+1[\/latex].\r\n<p style=\"text-align: center;\">[latex]y&lt;\u22123x+1[\/latex]<\/p>\r\nSubstitute\u00a0[latex]x=2[\/latex] and [latex]y=\u22123[\/latex]\u00a0into inequality.\r\n<p style=\"text-align: center;\">[latex]\u22123&lt;\u22123\\left(2\\right)+1[\/latex]<\/p>\r\nEvaluate.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\u22123&lt;\u22126+1\\\\\u22123&lt;\u22125\\end{array}[\/latex]<\/p>\r\nThis statement is <b>not <\/b>true, so the ordered pair [latex](2,\u22123)[\/latex] is <b>not <\/b>a solution.\r\n<h4>Answer<\/h4>\r\n[latex](2,\u22123)[\/latex] is not a solution.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]3401[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Classify solutions and graphs as equations or inequalities<\/li>\n<li>Identify whether an ordered pair is in the solution set of a linear inequality<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"Identify the difference between the graph of a linear equation and linear inequality\">Identify the difference between the graph of a linear equation and linear inequality<\/h2>\n<p>Recall that solutions to linear inequalities are whole sets of numbers,\u00a0rather than just one number, like you find with solutions to equalities (equations).<\/p>\n<p>Here is an example from the section on solving linear inequalities:<\/p>\n<p>Solve for p. [latex]4p+5<29[\/latex]\n\n\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}4p+5<\\,\\,\\,29\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,-5\\,\\,\\,\\,\\,\\,\\,-5}\\\\\\underline{4p}\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,\\underline{24}\\,\\,\\\\4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,p<6\\end{array}[\/latex]<\/p>\n<p>You can interpret the solution as p can be any number less than six. Now recall that we can graph equations of lines by defining the outputs, [latex]y[\/latex], and the inputs, [latex]x[\/latex], and writing an equation.<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-gr\u2026ar-relationships\/\">Previously, we showed how to graph the line described by th equation\u00a0 [latex]y=2x+3[\/latex]<\/a>\u00a0<b><strong>\u00a0<\/strong><\/b>and found that we can construct a never-ending table of values that make points on the line\u2014these are some of the solutions to the equation [latex]y=2x+3[\/latex].<\/p>\n<table>\n<thead>\n<tr>\n<th><em>x<\/em> values<\/th>\n<th>[latex]2x+3[\/latex]<\/th>\n<th><em>y<\/em> values<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]2(0)+3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2(1)+3[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]2(2)+3[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]2(3)+3[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Additionally, we learned how to graph the line that represents all the points that make\u00a0[latex]y=2x+3[\/latex] a true statement.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1398\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/07211017\/Putting-It-Together-Graphs.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A line labeled y=2x+3 is drawn in the coordinate plane.\" width=\"476\" height=\"367\" \/><\/p>\n<p>What if we combined these two ideas\u2014linear inequalities and graphs of lines? First translate the line, [latex]y=2x+3[\/latex], into words:<\/p>\n<p>You get y by multiplying [latex]x[\/latex] by two and adding three. [latex]y=2x+3[\/latex]<\/p>\n<p>How would you translate this inequality into words? [latex]y<2x+3[\/latex]\n\nFor what values of [latex]x[\/latex] will you get an output, [latex]y[\/latex], that is less than [latex]2[\/latex] times [latex]x[\/latex] plus three?\n\nWOW, that may\u00a0seem confusing, but keep reading, we&#8217;ll help you figure it out.\n\nLinear inequalities are different than linear equations, although you can apply what you know about equations to help you understand inequalities. Inequalities and equations are both math statements that compare two values. Equations use the symbol = ; recall that inequalities are\u00a0represented by the symbols &lt; , \u2264 , &gt; , and \u2265.\n\nOne way to visualize two-variable inequalities is to plot them on a coordinate plane. Here is what the inequality\u00a0[latex]x>y[\/latex]\u00a0looks like. The solution is a region, which is shaded. This region is made up of lots and lots of ordered pairs that all make the statement\u00a0[latex]x>y[\/latex] true.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2875 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19172339\/Screen-Shot-2016-04-19-at-10.23.13-AM.png\" alt=\"Dotted upward-sloping line. Everything below the dotted line is shaded and is labeled x is greater than y. Everything above the line is unshaded and is labeled x equals y.\" width=\"550\" height=\"553\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>There are a few things to notice here. First, look at the dashed red boundary line: this is the graph of the related linear equation\u00a0[latex]x=y[\/latex]. Next, look at the light red region that is to the right of the line. This region (excluding the line [latex]x=y[\/latex]) represents the entire set of solutions for the inequality [latex]x>y[\/latex]. Remember how all points on a <em>line<\/em> are solutions to the linear equation of the line? Well, all points in a <em>region<\/em> are solutions to the <b>linear inequality<\/b> representing that region.<\/p>\n<p>Let\u2019s think about it for a moment\u2014if [latex]x>y[\/latex], then a graph of [latex]x>y[\/latex]\u00a0will show all ordered pairs [latex](x,y)[\/latex] for which the <i>x-<\/i>coordinate is greater than the <i>y-<\/i>coordinate.<\/p>\n<h2>Identify ordered pairs that are the solution set of a linear inequality<\/h2>\n<p>The graph below shows the region [latex]x>y[\/latex]\u00a0as well as some ordered pairs on the coordinate plane. Look at each ordered pair. Is the <i>x-<\/i>coordinate greater than the <i>y-<\/i>coordinate? Does the ordered pair sit inside or outside of the shaded region?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2874 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19172149\/Screen-Shot-2016-04-19-at-10.21.21-AM.png\" alt=\"Dotted upward-sloping line. Everything below the dotted line is shaded and is labeled x is greater than y. Everything above the line is unshaded and is labeled x equals y. The points (-3,3) and (2,3) are in the unshaded region. The points (4,0) and (0,-3) are in the shaded region. The point (-2,-2) is on the dotted line.\" width=\"417\" height=\"419\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>The ordered pairs [latex](4,0)[\/latex] and [latex](0,\u22123)[\/latex] lie inside the shaded region. In these ordered pairs, the <i>x-<\/i>coordinate is larger than the <i>y-<\/i>coordinate. These ordered pairs are in the solution set of the equation [latex]x>y[\/latex].<\/p>\n<p>The ordered pairs [latex](\u22123,3)[\/latex] and [latex](2,3)[\/latex] are outside of the shaded area. In these ordered pairs, the <i>x-<\/i>coordinate is <i>smaller<\/i> than the <i>y-<\/i>coordinate, so they are not included in the set of solutions for the inequality.<\/p>\n<p>The ordered pair [latex](\u22122,\u22122)[\/latex] is on the boundary line. It is not a solution as [latex]\u22122[\/latex] is not greater than [latex]\u22122[\/latex]. However, had the inequality been [latex]x\\geq y[\/latex]\u00a0(read as \u201c<i>x<\/i> is greater than or equal to <i>y<\/i>\u201d), then [latex](\u22122,\u22122)[\/latex] would have been included (and the line would have been represented by a solid line, not a dashed line).<\/p>\n<p class=\"yt watch-title-container\"><span id=\"eow-title\" class=\"watch-title\" dir=\"ltr\" title=\"The Difference Between a Linear Equation and Linear Inequality (Two Variables)\">Watch the video below for another explanation of the difference between a linear equation and a linear inequality.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"The Difference Between a Linear Equation and Linear Inequality (Two Variables)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/EcrLbRJ2zV0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Let\u2019s take a look at one more example: the inequality [latex]3x+2y\\leq6[\/latex]. The graph below shows the region of values that makes this inequality true (shaded red), the boundary line [latex]3x+2y=6[\/latex], as well as a handful of ordered pairs. The boundary line is solid this time, because points on the boundary line [latex]3x+2y=6[\/latex]\u00a0will make the inequality [latex]3x+2y\\leq6[\/latex]\u00a0true.<\/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>&nbsp;<\/p>\n<p>As you did with the previous example, you can substitute the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-values in each of the [latex](x,y)[\/latex] ordered pairs into the inequality to find solutions. While you may have been able to do this in your head for the inequality\u00a0[latex]x>y[\/latex], sometimes making a table of values makes sense for more complicated inequalities.<\/p>\n<table>\n<thead>\n<tr>\n<th>Ordered Pair<\/th>\n<th>Makes the inequality<\/p>\n<p>[latex]3x+2y\\leq6[\/latex]<\/p>\n<p>a true statement<\/th>\n<th>Makes the inequality<\/p>\n<p>[latex]3x+2y\\leq6[\/latex]<\/p>\n<p>a false statement<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex](\u22125, 5)[\/latex]<\/td>\n<td>[latex]\\begin{array}{r}3\\left(\u22125\\right)+2\\left(5\\right)\\leq6\\\\\u221215+10\\leq6\\\\\u22125\\leq6\\end{array}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex](\u22122,\u22122)[\/latex]<\/td>\n<td>[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><\/td>\n<\/tr>\n<tr>\n<td>[latex](2,3)[\/latex]<\/td>\n<td><\/td>\n<td>[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>[latex](2,0)[\/latex]<\/td>\n<td>[latex]\\begin{array}{r}3\\left(2\\right)+2\\left(0\\right)\\leq6\\\\6+0\\leq6\\\\6\\leq6\\end{array}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex](4,\u22121)[\/latex]<\/td>\n<td><\/td>\n<td>[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<p>You can watch the following video to see more examples of how to determine whether an ordered pair satisfies a linear 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<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\u00a0[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>These values are located in the shaded region, so are solutions. (When substituted into the inequality\u00a0[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<h4>Answer<\/h4>\n<p>[latex](\u22121,1)\\,\\,\\,(\u22122,\u22122)[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>Watch the video below to see an example of how to use a graph to determine whether an ordered pair is a solutions of a linear inequality in two variables<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" 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\u00a0[latex]y<\u22123x+1[\/latex].\n\n\n<p style=\"text-align: center;\">[latex]y<\u22123x+1[\/latex]<\/p>\n<p>Substitute\u00a0[latex]x=2[\/latex] and [latex]y=\u22123[\/latex]\u00a0into 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<h4>Answer<\/h4>\n<p>[latex](2,\u22123)[\/latex] is not a solution.<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3401\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3401&theme=oea&iframe_resize_id=ohm3401&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-16523\">\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>The Difference Between a Linear Equation and Linear Inequality (Two Variables). <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/EcrLbRJ2zV0\">https:\/\/youtu.be\/EcrLbRJ2zV0<\/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>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><strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/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":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"The Difference Between a Linear Equation and Linear Inequality (Two Variables)\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/EcrLbRJ2zV0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"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\":\"\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology\",\"url\":\" http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"305947b72e4d461ea5bda8516ea1aa9a, 7a14ce052e6640979319f7589ee8ce67","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16523","chapter","type-chapter","status-publish","hentry"],"part":8524,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16523","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":10,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16523\/revisions"}],"predecessor-version":[{"id":20471,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16523\/revisions\/20471"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/8524"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16523\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=16523"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=16523"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=16523"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=16523"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}