{"id":16524,"date":"2019-10-03T17:08:47","date_gmt":"2019-10-03T17:08:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-graph-an-inequality-in-two-variables\/"},"modified":"2024-04-30T23:15:16","modified_gmt":"2024-04-30T23:15:16","slug":"read-or-watch-graph-an-inequality-in-two-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-graph-an-inequality-in-two-variables\/","title":{"raw":"Graphing an Inequality in Two Variables","rendered":"Graphing an Inequality in Two Variables"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph an inequality in two variables<\/li>\r\n<\/ul>\r\n<\/div>\r\nSo how do you get from the algebraic form of an inequality, like [latex]y&gt;3x+1[\/latex], to a graph of that inequality? Plotting inequalities is fairly straightforward if you follow a couple steps.\r\n<div class=\"textbox shaded\">\r\n<h3>Graphing Inequalities<\/h3>\r\nTo graph an inequality:\r\n<ul>\r\n \t<li>Graph the related boundary line. Replace the &lt;, &gt;, \u2264 or \u2265 sign in the inequality with = to find the equation of the boundary line.<\/li>\r\n \t<li>Identify at least one ordered pair on either side of the boundary line and substitute those [latex](x,y)[\/latex] values into the inequality. Shade the region that contains the ordered pairs that make the inequality a true statement.<b>\u00a0<\/b><\/li>\r\n \t<li>If points on the boundary line are solutions, then use a solid line for drawing the boundary line. This will happen for \u2264 or \u2265 inequalities.<\/li>\r\n \t<li>If points on the boundary line aren\u2019t solutions, then use a dotted line for the boundary line. This will happen for &lt; or &gt; inequalities.<\/li>\r\n<\/ul>\r\n<\/div>\r\nLet\u2019s graph the inequality [latex]x+4y\\leq4[\/latex].\r\n\r\nTo graph the boundary line, find at least two values that lie on the line [latex]x+4y=4[\/latex]. You can use the <i>x<\/i>- and <i>y<\/i>-intercepts for this equation by substituting 0 in for <i>x<\/i> first and finding the value of <i>y<\/i>; then substitute 0 in for <i>y<\/i> and find <i>x<\/i>.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the points [latex](0,1)[\/latex] and [latex](4,0)[\/latex], and draw a line through these two points for the boundary line. The line is solid because \u2264 means \u201cless than or equal to,\u201d so all ordered pairs along the line are included in the solution set.\r\n\r\n<img class=\"aligncenter size-full wp-image-2936\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19230042\/Screen-Shot-2016-04-19-at-4.00.26-PM.png\" alt=\"Solid downward-sloping line that crosses the points (0,1) and (4,0). The point (-1,3) and the point (2,0) are also plotted.\" width=\"417\" height=\"419\" \/>\r\n\r\n&nbsp;\r\n\r\nThe next step is to find the region that contains the solutions. Is it above or below the boundary line? To identify the region where the inequality holds true, you can test a couple of ordered pairs, one on each side of the boundary line.\r\n\r\nIf you substitute [latex](\u22121,3)[\/latex] into\u00a0[latex]x+4y\\leq4[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\u22121+4\\left(3\\right)\\leq4\\\\\u22121+12\\leq4\\\\11\\leq4\\end{array}[\/latex]<\/p>\r\nThis is a false statement, since [latex]11[\/latex] is not less than or equal to [latex]4[\/latex].\r\n\r\nOn the other hand, if you substitute [latex](2,0)[\/latex] into\u00a0[latex]x+4y\\leq4[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2+4\\left(0\\right)\\leq4\\\\2+0\\leq4\\\\2\\leq4\\end{array}[\/latex]<\/p>\r\nThis is true! The region that includes [latex](2,0)[\/latex] should be shaded, as this is the region of solutions.\r\n\r\n<img class=\"aligncenter size-full wp-image-2934\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19225534\/Screen-Shot-2016-04-19-at-3.54.55-PM.png\" alt=\"Solid downward-sloping line marked x+4y=4. The region below the line is shaded and is labeled x+4y is less than or equal to 4.\" width=\"413\" height=\"419\" \/>\r\n\r\n&nbsp;\r\n\r\nAnd there you have it\u2014the graph of the set of solutions for [latex]x+4y\\leq4[\/latex].\r\n<h2>Graphing Linear Inequalities in Two Variables<\/h2>\r\nhttps:\/\/youtu.be\/2VgFg2ztspI\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGraph the inequality [latex]2y&gt;4x\u20136[\/latex].\r\n\r\n[reveal-answer q=\"138506\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"138506\"]\r\n\r\nSolve for [latex]y[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}2y&gt;4x-6\\\\\\\\\\frac{2y}{2}&gt;\\frac{4x}{2}-\\frac{6}{2}\\\\\\\\y&gt;2x-3\\\\\\end{array}[\/latex]<\/p>\r\nCreate a table of values to find two points on the line [latex] \\displaystyle y=2x-3[\/latex], or graph it based on the slope-intercept method, the b value of the y-intercept is [latex]-3[\/latex] and the slope is [latex]2[\/latex].\r\n\r\nPlot the points, and graph the line. The line is dotted because the sign in the inequality is &gt;, not \u2265 and therefore points on the line are not solutions to the inequality.\r\n\r\n<img class=\"aligncenter size-full wp-image-2937\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19230258\/Screen-Shot-2016-04-19-at-4.02.07-PM.png\" alt=\"Dotted upward-sloping line that crosses the points (2,1) and (0,-3). The points (-3,1) and (4,1) are also plotted.\" width=\"423\" height=\"422\" \/>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle y=2x-3[\/latex]<\/p>\r\n\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\u22123[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFind an ordered pair on either side of the boundary line. Insert the <i>x<\/i>- and <i>y<\/i>-values into the inequality\r\n[latex]2y&gt;4x\u20136[\/latex] and see which ordered pair results in a true statement. Since [latex](\u22123,1)[\/latex] results in a true statement, the region that includes [latex](\u22123,1)[\/latex] should be shaded.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}2y&gt;4x\u20136\\\\\\\\\\text{Test }1:\\left(\u22123,1\\right)\\\\2\\left(1\\right)&gt;4\\left(\u22123\\right)\u20136\\\\\\,\\,\\,\\,\\,\\,\\,2&gt;\u201312\u20136\\\\\\,\\,\\,\\,\\,\\,\\,2&gt;\u221218\\\\\\text{TRUE}\\\\\\\\\\text{Test }2:\\left(4,1\\right)\\\\2(1)&gt;4\\left(4\\right)\u2013 6\\\\\\,\\,\\,\\,\\,\\,2&gt;16\u20136\\\\\\,\\,\\,\\,\\,\\,2&gt;10\\\\\\text{FALSE}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe graph of the inequality [latex]2y&gt;4x\u20136[\/latex] is:\r\n\r\n<img class=\"aligncenter wp-image-2935 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19225738\/Screen-Shot-2016-04-19-at-3.56.57-PM.png\" alt=\"The dotted upward-sloping line of 2y=4x-6, with the region above the line shaded.\" width=\"387\" height=\"391\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nA quick note about the problem above\u2014notice that you can use the points [latex](0,\u22123)[\/latex] and [latex](2,1)[\/latex] to graph the boundary line, but that these points are not included in the region of solutions, since the region does not include the boundary line!\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nMake sure you click <em>Try Another Version of This Question<\/em> so that you can get lots of practice graphing each type of inequality!\r\n\r\n[ohm_question]149925[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<p id=\"video2\">Below is a video about how to graph inequalities with two variables when the equation is in what is known as slope-intercept form.<\/p>\r\nhttps:\/\/youtu.be\/Hzxc4HASygU\r\n<h3>Summary<\/h3>\r\nWhen inequalities are graphed on a coordinate plane, the solutions are located in a region of the coordinate plane, which is represented as a shaded area on the plane. The boundary line for the inequality is drawn as a solid line if the points on the line itself do satisfy the inequality, as in the cases of \u2264 and \u2265. It is drawn as a dashed line if the points on the line do not satisfy the inequality, as in the cases of &lt; and &gt;. You can tell which region to shade by testing some points in the inequality. Using a coordinate plane is especially helpful for visualizing the region of solutions for inequalities with two variables.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph an inequality in two variables<\/li>\n<\/ul>\n<\/div>\n<p>So how do you get from the algebraic form of an inequality, like [latex]y>3x+1[\/latex], to a graph of that inequality? Plotting inequalities is fairly straightforward if you follow a couple steps.<\/p>\n<div class=\"textbox shaded\">\n<h3>Graphing Inequalities<\/h3>\n<p>To graph an inequality:<\/p>\n<ul>\n<li>Graph the related boundary line. Replace the &lt;, &gt;, \u2264 or \u2265 sign in the inequality with = to find the equation of the boundary line.<\/li>\n<li>Identify at least one ordered pair on either side of the boundary line and substitute those [latex](x,y)[\/latex] values into the inequality. Shade the region that contains the ordered pairs that make the inequality a true statement.<b>\u00a0<\/b><\/li>\n<li>If points on the boundary line are solutions, then use a solid line for drawing the boundary line. This will happen for \u2264 or \u2265 inequalities.<\/li>\n<li>If points on the boundary line aren\u2019t solutions, then use a dotted line for the boundary line. This will happen for &lt; or &gt; inequalities.<\/li>\n<\/ul>\n<\/div>\n<p>Let\u2019s graph the inequality [latex]x+4y\\leq4[\/latex].<\/p>\n<p>To graph the boundary line, find at least two values that lie on the line [latex]x+4y=4[\/latex]. You can use the <i>x<\/i>&#8211; and <i>y<\/i>-intercepts for this equation by substituting 0 in for <i>x<\/i> first and finding the value of <i>y<\/i>; then substitute 0 in for <i>y<\/i> and find <i>x<\/i>.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the points [latex](0,1)[\/latex] and [latex](4,0)[\/latex], and draw a line through these two points for the boundary line. The line is solid because \u2264 means \u201cless than or equal to,\u201d so all ordered pairs along the line are included in the solution set.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2936\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19230042\/Screen-Shot-2016-04-19-at-4.00.26-PM.png\" alt=\"Solid downward-sloping line that crosses the points (0,1) and (4,0). The point (-1,3) and the point (2,0) are also plotted.\" width=\"417\" height=\"419\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>The next step is to find the region that contains the solutions. Is it above or below the boundary line? To identify the region where the inequality holds true, you can test a couple of ordered pairs, one on each side of the boundary line.<\/p>\n<p>If you substitute [latex](\u22121,3)[\/latex] into\u00a0[latex]x+4y\\leq4[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\u22121+4\\left(3\\right)\\leq4\\\\\u22121+12\\leq4\\\\11\\leq4\\end{array}[\/latex]<\/p>\n<p>This is a false statement, since [latex]11[\/latex] is not less than or equal to [latex]4[\/latex].<\/p>\n<p>On the other hand, if you substitute [latex](2,0)[\/latex] into\u00a0[latex]x+4y\\leq4[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2+4\\left(0\\right)\\leq4\\\\2+0\\leq4\\\\2\\leq4\\end{array}[\/latex]<\/p>\n<p>This is true! The region that includes [latex](2,0)[\/latex] should be shaded, as this is the region of solutions.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2934\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19225534\/Screen-Shot-2016-04-19-at-3.54.55-PM.png\" alt=\"Solid downward-sloping line marked x+4y=4. The region below the line is shaded and is labeled x+4y is less than or equal to 4.\" width=\"413\" height=\"419\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>And there you have it\u2014the graph of the set of solutions for [latex]x+4y\\leq4[\/latex].<\/p>\n<h2>Graphing Linear Inequalities in Two Variables<\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 2:  Graphing Linear Inequalities in Two Variables (Standard Form)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/2VgFg2ztspI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph the inequality [latex]2y>4x\u20136[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q138506\">Show Solution<\/span><\/p>\n<div id=\"q138506\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}2y>4x-6\\\\\\\\\\frac{2y}{2}>\\frac{4x}{2}-\\frac{6}{2}\\\\\\\\y>2x-3\\\\\\end{array}[\/latex]<\/p>\n<p>Create a table of values to find two points on the line [latex]\\displaystyle y=2x-3[\/latex], or graph it based on the slope-intercept method, the b value of the y-intercept is [latex]-3[\/latex] and the slope is [latex]2[\/latex].<\/p>\n<p>Plot the points, and graph the line. The line is dotted because the sign in the inequality is &gt;, not \u2265 and therefore points on the line are not solutions to the inequality.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2937\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19230258\/Screen-Shot-2016-04-19-at-4.02.07-PM.png\" alt=\"Dotted upward-sloping line that crosses the points (2,1) and (0,-3). The points (-3,1) and (4,1) are also plotted.\" width=\"423\" height=\"422\" \/><\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle y=2x-3[\/latex]<\/p>\n<table>\n<thead>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Find an ordered pair on either side of the boundary line. Insert the <i>x<\/i>&#8211; and <i>y<\/i>-values into the inequality<br \/>\n[latex]2y>4x\u20136[\/latex] and see which ordered pair results in a true statement. Since [latex](\u22123,1)[\/latex] results in a true statement, the region that includes [latex](\u22123,1)[\/latex] should be shaded.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}2y>4x\u20136\\\\\\\\\\text{Test }1:\\left(\u22123,1\\right)\\\\2\\left(1\\right)>4\\left(\u22123\\right)\u20136\\\\\\,\\,\\,\\,\\,\\,\\,2>\u201312\u20136\\\\\\,\\,\\,\\,\\,\\,\\,2>\u221218\\\\\\text{TRUE}\\\\\\\\\\text{Test }2:\\left(4,1\\right)\\\\2(1)>4\\left(4\\right)\u2013 6\\\\\\,\\,\\,\\,\\,\\,2>16\u20136\\\\\\,\\,\\,\\,\\,\\,2>10\\\\\\text{FALSE}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The graph of the inequality [latex]2y>4x\u20136[\/latex] is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2935 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/04\/19225738\/Screen-Shot-2016-04-19-at-3.56.57-PM.png\" alt=\"The dotted upward-sloping line of 2y=4x-6, with the region above the line shaded.\" width=\"387\" height=\"391\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>A quick note about the problem above\u2014notice that you can use the points [latex](0,\u22123)[\/latex] and [latex](2,1)[\/latex] to graph the boundary line, but that these points are not included in the region of solutions, since the region does not include the boundary line!<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Make sure you click <em>Try Another Version of This Question<\/em> so that you can get lots of practice graphing each type of inequality!<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm149925\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=149925&theme=oea&iframe_resize_id=ohm149925&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"video2\">Below is a video about how to graph inequalities with two variables when the equation is in what is known as slope-intercept form.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1:  Graphing Linear Inequalities in Two Variables (Slope Intercept Form)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Hzxc4HASygU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Summary<\/h3>\n<p>When inequalities are graphed on a coordinate plane, the solutions are located in a region of the coordinate plane, which is represented as a shaded area on the plane. The boundary line for the inequality is drawn as a solid line if the points on the line itself do satisfy the inequality, as in the cases of \u2264 and \u2265. It is drawn as a dashed line if the points on the line do not satisfy the inequality, as in the cases of &lt; and &gt;. You can tell which region to shade by testing some points in the inequality. Using a coordinate plane is especially helpful for visualizing the region of solutions for inequalities with two variables.<\/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-16524\">\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><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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 2: Graphing Linear Inequalities in Two Variables (Standard Form). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/2VgFg2ztspI\">https:\/\/youtu.be\/2VgFg2ztspI<\/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 1: Graphing Linear Inequalities in Two Variables (Slope Intercept Form). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Hzxc4HASygU\">https:\/\/youtu.be\/Hzxc4HASygU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: 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