{"id":773,"date":"2016-06-01T20:49:11","date_gmt":"2016-06-01T20:49:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=773"},"modified":"2019-08-06T18:14:35","modified_gmt":"2019-08-06T18:14:35","slug":"read-cost-and-revenue-problems-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-cost-and-revenue-problems-2\/","title":{"raw":"Applications of Systems of Linear Inequalities","rendered":"Applications of Systems of Linear Inequalities"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Interpret graphs and solutions to systems of linear inequalities<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn our first\u00a0example we will show how to write and graph a system of linear inequalities that models the amount of sales needed to obtain a specific amount of money.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCathy is selling ice cream cones at a school fundraiser. She is selling two sizes: small (which has\u00a0[latex]1[\/latex] scoop) and large (which has\u00a0[latex]2[\/latex] scoops). She knows that she can get a maximum of\u00a0[latex]70[\/latex] scoops of ice cream out of her supply. She charges\u00a0[latex]$3[\/latex] for a small cone and\u00a0[latex]$5[\/latex] for a large cone.\r\n\r\nCathy wants to earn at least\u00a0[latex]$120[\/latex] to give back to the school. Write and graph a system of inequalities that models this situation.\r\n[reveal-answer q=\"737192\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"737192\"]\r\n\r\nFirst, identify the variables. There are two variables: the number of small cones and the number of large cones.\r\n<p style=\"text-align: center;\"><i>s <\/i>= small cone<\/p>\r\n<p style=\"text-align: center;\"><i>l <\/i>= large cone<\/p>\r\n<p style=\"text-align: left;\">Write the first equation: the maximum number of scoops she can give out. The scoops she has available\u00a0[latex](70)[\/latex] must be greater than or equal to the number of scoops for the small cones (<i>s<\/i>) and the large cones\u00a0[latex](2[\/latex]<i>l<\/i>) she sells.<\/p>\r\n<p style=\"text-align: center;\">[latex]s+2l\\le70[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Write the second equation: the amount of money she raises. She wants the total amount of money earned from small cones\u00a0[latex](3<i>s<\/i>)[\/latex] and large cones\u00a0[latex](5<i>l<\/i>)[\/latex] to be at least\u00a0[latex]$120[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]3s+5l\\ge120[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Write the system.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{cases}s+2l\\le70\\\\3s+5l\\ge120\\\\s&gt;=0\\\\l&gt;=0\\end{cases}[\/latex]<\/p>\r\nNow graph the system. The variables <i>x<\/i> and <i>y<\/i> have been replaced by <i>s<\/i> and <i>l<\/i>; graph <i>s<\/i> along the <i>x<\/i>-axis and <i>l<\/i> along the <i>y<\/i>-axis.\r\n\r\nFirst graph the region [latex]s + 2l \u2264 70[\/latex]. Graph the boundary line and then test individual points to see which region to shade. We only shade the regions that also satisfy [latex]x\\ge0[\/latex], [latex]y\\ge0[\/latex]. The graph is shown below.\r\n\r\n<img class=\"alignnone size-medium wp-image-2433\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223652\/image027-300x203.jpg\" alt=\"image027\" width=\"300\" height=\"203\" \/>\r\n\r\nNow graph the region [latex]3s+5l\\ge120[\/latex]\u00a0Graph the boundary line and then test individual points to see which region to shade. The graph is shown below.\r\n\r\n<img class=\"alignnone size-medium wp-image-2434\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223654\/image028-300x203.jpg\" alt=\"image028\" width=\"300\" height=\"203\" \/>\r\n\r\nGraphing the regions together, you find the following:\r\n\r\n<img class=\"alignnone size-medium wp-image-2435\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223656\/image029-300x203.jpg\" alt=\"image029\" width=\"300\" height=\"203\" \/>\r\n\r\nLooking at just the overlapping region, you have:\r\n\r\n<img class=\"alignnone size-medium wp-image-2436\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223658\/image030-300x203.jpg\" alt=\"image030\" width=\"300\" height=\"203\" \/>\r\n\r\nThe region in purple is the solution. As long as the combination of small cones and large cones that Cathy sells can be mapped in the purple region, she will have earned at least\u00a0[latex]$120[\/latex] and not used more than\u00a0[latex]70[\/latex] scoops of ice cream.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn a previous example for finding a solution to a system of linear equations, we introduced a manufacturer\u2019s\u00a0cost and revenue equations:\r\n\r\nCost: [latex]y=0.85x+35,000[\/latex]\r\n\r\nRevenue: [latex]y=1.55x[\/latex]\r\n\r\n[latex]x\\ge0,y\\ge0[\/latex]\r\n\r\nThe cost equation is shown in blue in the graph below, and the revenue equation is graphed in orange. The point at which the two lines intersect is called the break-even point. We learned that this is the solution to the system of linear equations that cause the cost and revenue equations to equal each other. Note how the lines shown only represent where [latex]x\\ge0, y\\ge0[\/latex]. It is easy to forget to include this part in the graph.\r\n\r\nThe shaded region to the right of the break-even point represents quantities where the company makes a profit. The region to the left represents quantities where the company suffers a loss.\r\n\r\nIn the next example, you will see how the information you learned about systems of linear inequalities can be applied to answering questions about cost and revenue.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183256\/CNX_Precalc_Figure_09_01_0102.jpg\" alt=\"A graph showing money in dollars on the y axis and quantity on the x axis. A line representing cost and a line representing revenue cross at the break-even point of fifty thousand, seventy-seven thousand five hundred. The cost line's equation is C(x)=0.85x+35,000. The revenue line's equation is R(x)=1.55x. The shaded space between the two lines to the right of the break-even point is labeled profit.\" width=\"487\" height=\"390\" \/>\r\n\r\nNote how the blue shaded region between the cost and revenue equations is labeled profit. This is the \"sweet spot\" that the company wants to achieve\u00a0where they\u00a0produce enough bike frames at a minimal enough cost to\u00a0make money. They do not want more money going out than coming in!\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDefine the profit region for the skateboard\u00a0manufacturing business using inequalities given the system of linear equations:\r\n\r\nCost: [latex]y=0.85x+35,000[\/latex]\r\n\r\nRevenue: [latex]y=1.55x[\/latex]\r\n\r\n[latex]x\\ge0, y\\ge0[\/latex]\r\n[reveal-answer q=\"563864\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"563864\"]\r\n\r\nWe know that graphically, solutions to\u00a0linear inequalities are entire regions, and we learned how to graph systems of linear inequalities earlier in this module. Based on the graph below and the equations that define cost and revenue, we can use inequalities to define the region where the skateboard\u00a0manufacturer will make a profit.\u00a0 Again, note how only the region for [latex]x\\ge0, y\\ge0[\/latex] is included.\r\n\r\n<img class=\"wp-image-4210 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183258\/Screen-Shot-2016-05-18-at-3.33.11-PM-300x280.png\" alt=\"Cost\/ Revenue with Profit\" width=\"358\" height=\"334\" \/>\r\n\r\nStart with the revenue equation. We know that the break-even point is at\u00a0[latex](50,000, 77,500)[\/latex], and the profit region is\u00a0the blue area. \u00a0If we choose a point in the region and test it like we did for finding solution regions for inequalities, we will know which kind of inequality sign to use.\r\n\r\nTest the point [latex]\\left(6500,100000\\right)[\/latex] in both equations to determine which inequality sign to use.\r\n\r\nCost:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y=0.85x+{35,000}\\\\{100,000}\\text{ ? }0.85\\left(65,000\\right)+35,000\\\\100,000\\text{ ? }90,250\\end{array}[\/latex]<\/p>\r\nWe need to use &gt; because\u00a0[latex]100,000[\/latex] is greater than\u00a0[latex]90,250[\/latex]\r\n\r\nThe cost inequality that will ensure the company makes profit, not just break-even, is\u00a0[latex]y&gt;0.85x+35,000[\/latex]\r\n\r\nNow test the point in the revenue equation:\r\n\r\nRevenue:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y=1.55x\\\\100,000\\text{ ? }1.55\\left(65,000\\right)\\\\100,000\\text{ ? }100,750\\end{array}[\/latex]<\/p>\r\nWe need to use &lt; because\u00a0[latex]100,000[\/latex] is less than\u00a0[latex]100,750[\/latex]\r\n\r\nThe revenue inequality that will ensure the company makes profit, not just break-even, is\u00a0[latex]y&lt;1.55x[\/latex]\r\n\r\nThe system of inequalities that defines the profit region for the bike manufacturer is:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y&gt;0.85x+35,000\\\\y&lt;1.55x\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see an example of how to find the break-even point for a small sno-cone business.\r\n\r\nhttps:\/\/youtu.be\/qey3FmE8saQ\r\n\r\nBelow is one more video example about solving an application\u00a0using a system of linear inequalities.\r\n\r\nhttps:\/\/youtu.be\/gbHl6K-dJ8o\r\n\r\nWe have seen that systems of linear equations and inequalities can help to define market behaviors that are very helpful to businesses. The intersection of cost and revenue equations gives the break-even point and also helps define the region where a company will make a profit.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Interpret graphs and solutions to systems of linear inequalities<\/li>\n<\/ul>\n<\/div>\n<p>In our first\u00a0example we will show how to write and graph a system of linear inequalities that models the amount of sales needed to obtain a specific amount of money.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Cathy is selling ice cream cones at a school fundraiser. She is selling two sizes: small (which has\u00a0[latex]1[\/latex] scoop) and large (which has\u00a0[latex]2[\/latex] scoops). She knows that she can get a maximum of\u00a0[latex]70[\/latex] scoops of ice cream out of her supply. She charges\u00a0[latex]$3[\/latex] for a small cone and\u00a0[latex]$5[\/latex] for a large cone.<\/p>\n<p>Cathy wants to earn at least\u00a0[latex]$120[\/latex] to give back to the school. Write and graph a system of inequalities that models this situation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q737192\">Show Solution<\/span><\/p>\n<div id=\"q737192\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, identify the variables. There are two variables: the number of small cones and the number of large cones.<\/p>\n<p style=\"text-align: center;\"><i>s <\/i>= small cone<\/p>\n<p style=\"text-align: center;\"><i>l <\/i>= large cone<\/p>\n<p style=\"text-align: left;\">Write the first equation: the maximum number of scoops she can give out. The scoops she has available\u00a0[latex](70)[\/latex] must be greater than or equal to the number of scoops for the small cones (<i>s<\/i>) and the large cones\u00a0[latex](2[\/latex]<i>l<\/i>) she sells.<\/p>\n<p style=\"text-align: center;\">[latex]s+2l\\le70[\/latex]<\/p>\n<p style=\"text-align: left;\">Write the second equation: the amount of money she raises. She wants the total amount of money earned from small cones\u00a0[latex](3<i>s<\/i>)[\/latex] and large cones\u00a0[latex](5<i>l<\/i>)[\/latex] to be at least\u00a0[latex]$120[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3s+5l\\ge120[\/latex]<\/p>\n<p style=\"text-align: left;\">Write the system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{cases}s+2l\\le70\\\\3s+5l\\ge120\\\\s>=0\\\\l>=0\\end{cases}[\/latex]<\/p>\n<p>Now graph the system. The variables <i>x<\/i> and <i>y<\/i> have been replaced by <i>s<\/i> and <i>l<\/i>; graph <i>s<\/i> along the <i>x<\/i>-axis and <i>l<\/i> along the <i>y<\/i>-axis.<\/p>\n<p>First graph the region [latex]s + 2l \u2264 70[\/latex]. Graph the boundary line and then test individual points to see which region to shade. We only shade the regions that also satisfy [latex]x\\ge0[\/latex], [latex]y\\ge0[\/latex]. The graph is shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2433\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223652\/image027-300x203.jpg\" alt=\"image027\" width=\"300\" height=\"203\" \/><\/p>\n<p>Now graph the region [latex]3s+5l\\ge120[\/latex]\u00a0Graph the boundary line and then test individual points to see which region to shade. The graph is shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2434\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223654\/image028-300x203.jpg\" alt=\"image028\" width=\"300\" height=\"203\" \/><\/p>\n<p>Graphing the regions together, you find the following:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2435\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223656\/image029-300x203.jpg\" alt=\"image029\" width=\"300\" height=\"203\" \/><\/p>\n<p>Looking at just the overlapping region, you have:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2436\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/11223658\/image030-300x203.jpg\" alt=\"image030\" width=\"300\" height=\"203\" \/><\/p>\n<p>The region in purple is the solution. As long as the combination of small cones and large cones that Cathy sells can be mapped in the purple region, she will have earned at least\u00a0[latex]$120[\/latex] and not used more than\u00a0[latex]70[\/latex] scoops of ice cream.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In a previous example for finding a solution to a system of linear equations, we introduced a manufacturer\u2019s\u00a0cost and revenue equations:<\/p>\n<p>Cost: [latex]y=0.85x+35,000[\/latex]<\/p>\n<p>Revenue: [latex]y=1.55x[\/latex]<\/p>\n<p>[latex]x\\ge0,y\\ge0[\/latex]<\/p>\n<p>The cost equation is shown in blue in the graph below, and the revenue equation is graphed in orange. The point at which the two lines intersect is called the break-even point. We learned that this is the solution to the system of linear equations that cause the cost and revenue equations to equal each other. Note how the lines shown only represent where [latex]x\\ge0, y\\ge0[\/latex]. It is easy to forget to include this part in the graph.<\/p>\n<p>The shaded region to the right of the break-even point represents quantities where the company makes a profit. The region to the left represents quantities where the company suffers a loss.<\/p>\n<p>In the next example, you will see how the information you learned about systems of linear inequalities can be applied to answering questions about cost and revenue.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183256\/CNX_Precalc_Figure_09_01_0102.jpg\" alt=\"A graph showing money in dollars on the y axis and quantity on the x axis. A line representing cost and a line representing revenue cross at the break-even point of fifty thousand, seventy-seven thousand five hundred. The cost line's equation is C(x)=0.85x+35,000. The revenue line's equation is R(x)=1.55x. The shaded space between the two lines to the right of the break-even point is labeled profit.\" width=\"487\" height=\"390\" \/><\/p>\n<p>Note how the blue shaded region between the cost and revenue equations is labeled profit. This is the &#8220;sweet spot&#8221; that the company wants to achieve\u00a0where they\u00a0produce enough bike frames at a minimal enough cost to\u00a0make money. They do not want more money going out than coming in!<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Define the profit region for the skateboard\u00a0manufacturing business using inequalities given the system of linear equations:<\/p>\n<p>Cost: [latex]y=0.85x+35,000[\/latex]<\/p>\n<p>Revenue: [latex]y=1.55x[\/latex]<\/p>\n<p>[latex]x\\ge0, y\\ge0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q563864\">Show Solution<\/span><\/p>\n<div id=\"q563864\" class=\"hidden-answer\" style=\"display: none\">\n<p>We know that graphically, solutions to\u00a0linear inequalities are entire regions, and we learned how to graph systems of linear inequalities earlier in this module. Based on the graph below and the equations that define cost and revenue, we can use inequalities to define the region where the skateboard\u00a0manufacturer will make a profit.\u00a0 Again, note how only the region for [latex]x\\ge0, y\\ge0[\/latex] is included.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4210 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183258\/Screen-Shot-2016-05-18-at-3.33.11-PM-300x280.png\" alt=\"Cost\/ Revenue with Profit\" width=\"358\" height=\"334\" \/><\/p>\n<p>Start with the revenue equation. We know that the break-even point is at\u00a0[latex](50,000, 77,500)[\/latex], and the profit region is\u00a0the blue area. \u00a0If we choose a point in the region and test it like we did for finding solution regions for inequalities, we will know which kind of inequality sign to use.<\/p>\n<p>Test the point [latex]\\left(6500,100000\\right)[\/latex] in both equations to determine which inequality sign to use.<\/p>\n<p>Cost:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y=0.85x+{35,000}\\\\{100,000}\\text{ ? }0.85\\left(65,000\\right)+35,000\\\\100,000\\text{ ? }90,250\\end{array}[\/latex]<\/p>\n<p>We need to use &gt; because\u00a0[latex]100,000[\/latex] is greater than\u00a0[latex]90,250[\/latex]<\/p>\n<p>The cost inequality that will ensure the company makes profit, not just break-even, is\u00a0[latex]y>0.85x+35,000[\/latex]<\/p>\n<p>Now test the point in the revenue equation:<\/p>\n<p>Revenue:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y=1.55x\\\\100,000\\text{ ? }1.55\\left(65,000\\right)\\\\100,000\\text{ ? }100,750\\end{array}[\/latex]<\/p>\n<p>We need to use &lt; because\u00a0[latex]100,000[\/latex] is less than\u00a0[latex]100,750[\/latex]<\/p>\n<p>The revenue inequality that will ensure the company makes profit, not just break-even, is\u00a0[latex]y<1.55x[\/latex]\n\nThe system of inequalities that defines the profit region for the bike manufacturer is:\n\n\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y>0.85x+35,000\\\\y<1.55x\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you will see an example of how to find the break-even point for a small sno-cone business.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"System of Equations App:  Break-Even Point\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/qey3FmE8saQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Below is one more video example about solving an application\u00a0using a system of linear inequalities.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Linear Inequality in Two Variables Application Problem  (Phone Cost:  Day and Night Minutes)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/gbHl6K-dJ8o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We have seen that systems of linear equations and inequalities can help to define market behaviors that are very helpful to businesses. The intersection of cost and revenue equations gives the break-even point and also helps define the region where a company will make a profit.<\/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-773\">\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>System of Equations App: Break-Even Point. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/qey3FmE8saQ\">https:\/\/youtu.be\/qey3FmE8saQ<\/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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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>College Algebra. <strong>Authored by<\/strong>: Jay Abrams, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstaxcollege.org\/textbooks\/college-algebra.\">https:\/\/openstaxcollege.org\/textbooks\/college-algebra.<\/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 and Education. <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><li>Ex: Linear Inequality in Two Variables Application Problem (Phone Cost: Day and Night Minutes). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/gbHl6K-dJ8o\">https:\/\/youtu.be\/gbHl6K-dJ8o<\/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":21,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"System of Equations App: Break-Even Point\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/qey3FmE8saQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Jay Abrams, et al.\",\"organization\":\"OpenStax\",\"url\":\" https:\/\/openstaxcollege.org\/textbooks\/college-algebra.\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 14: Systems of Equations and Inequalities, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Linear Inequality in Two Variables Application Problem (Phone Cost: Day and Night Minutes)\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/gbHl6K-dJ8o\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"d27b41ce-5731-444f-bad7-0826cbf1eb77","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-773","chapter","type-chapter","status-publish","hentry"],"part":2069,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/773","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":18,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/773\/revisions"}],"predecessor-version":[{"id":5548,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/773\/revisions\/5548"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/parts\/2069"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/773\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/media?parent=773"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=773"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/contributor?post=773"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/license?post=773"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}