{"id":4822,"date":"2018-12-11T14:06:48","date_gmt":"2018-12-11T19:06:48","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/graphing-systems-of-linear-inequalities\/"},"modified":"2018-12-11T14:09:16","modified_gmt":"2018-12-11T19:09:16","slug":"graphing-systems-of-linear-inequalities","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/graphing-systems-of-linear-inequalities\/","title":{"raw":"Graphing Systems of Linear Inequalities","rendered":"Graphing Systems of Linear Inequalities"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\">\r\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Determine whether an ordered pair is a solution of a system of linear inequalities<\/li>\r\n \t<li>Solve a system of linear inequalities by graphing<\/li>\r\n \t<li>Solve applications of systems of inequalities<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829586823\" class=\"be-prepared\">\r\n<p id=\"fs-id1167836535609\">Before you get started, take this readiness quiz.<\/p>\r\n\r\n<ol id=\"fs-id1167829905390\" type=\"1\">\r\n \t<li>Solve the inequality \\(2a&lt;5a+12.\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835324646\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>Determine whether the ordered pair \\(\\left(3,\\frac{1}{2}\\right)\\) is a solution to the system \\(y&gt;2x+3.\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/f15d09b6-aae4-4ad3-a9e6-c9d1ac2436e3#fs-id1167835253977\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836521500\">\r\n<h3 data-type=\"title\">Determine whether an ordered pair is a solution of a system of linear inequalities<\/h3>\r\n<p id=\"fs-id1167836550209\">The definition of a <span data-type=\"term\">system of linear inequalities<\/span> is very similar to the definition of a system of linear equations.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836791176\">\r\n<div data-type=\"title\">System of Linear Inequalities<\/div>\r\n<p id=\"fs-id1167833329231\">Two or more linear inequalities grouped together form a system of linear inequalities.<\/p>\r\n\r\n<\/div>\r\nA system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.\r\n<div data-type=\"equation\" id=\"fs-id1167836441020\" class=\"unnumbered\" data-label=\"\">\\(\\left\\{\\begin{array}{c}x+4y\\ge 10\\hfill \\\\ 3x-2y&lt;12\\hfill \\end{array}\\)<\/div>\r\n<p id=\"fs-id1167829597513\">To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs \\(\\left(x,y\\right)\\) that make both inequalities true.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167829717259\">\r\n<div data-type=\"title\">Solutions of a System of Linear Inequalities<\/div>\r\n<p id=\"fs-id1167826170280\">Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.<\/p>\r\nThe solution of a system of linear inequalities is shown as a shaded region in the <em data-effect=\"italics\">x, y<\/em> coordinate system that includes all the points whose ordered pairs make the inequalities true.\r\n\r\n<\/div>\r\nTo determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.\r\n<div data-type=\"example\" id=\"fs-id1167829650722\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167826132435\">\r\n<div data-type=\"problem\" id=\"fs-id1167829717048\">\r\n<p id=\"fs-id1167836549293\">Determine whether the ordered pair is a solution to the system \\(\\left\\{\\begin{array}{c}x+4y\\ge 10\\hfill \\\\ 3x-2y&lt;12\\hfill \\end{array}.\\)<\/p>\r\n<p id=\"fs-id1167836663929\"><span class=\"token\">\u24d0<\/span>\\(\\left(-2,4\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(3,1\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833338505\">\r\n<p id=\"fs-id1167836737853\"><span class=\"token\">\u24d0<\/span> Is the ordered pair \\(\\left(-2,4\\right)\\) a solution?<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167832935757\" data-alt=\"We substitute x equal to negative 2 and y equal to 4 into both inequalities. First inequality is x plus 4 times y greater than or equal to 10. So negative 2 plus 4 open parentheses 4 close parenthesis is greater than or equal to 10 or not. 14 is greater than or equal to 10 is true. Second inequality, 3 times x minus 2 times y is less than 12. Three open parentheses negative 2 close parentheses minus two open parentheses 4 close parentheses is less than 12 or not. Negative 14 is less than 12 is true.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"We substitute x equal to negative 2 and y equal to 4 into both inequalities. First inequality is x plus 4 times y greater than or equal to 10. So negative 2 plus 4 open parentheses 4 close parenthesis is greater than or equal to 10 or not. 14 is greater than or equal to 10 is true. Second inequality, 3 times x minus 2 times y is less than 12. Three open parentheses negative 2 close parentheses minus two open parentheses 4 close parentheses is less than 12 or not. Negative 14 is less than 12 is true.\" \/><\/span>\r\n<p id=\"fs-id1167836688862\">The ordered pair \\(\\left(-2,4\\right)\\) made both inequalities true. Therefore \\(\\left(-2,4\\right)\\) is a solution to this system.<\/p>\r\n<p id=\"fs-id1167833018011\"><span class=\"token\">\u24d1<\/span> Is the ordered pair \\(\\left(3,1\\right)\\) a solution?<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836621274\" data-alt=\"We substitute x equal to three and y equal to one into both inequalities. First inequality is x plus four times y greater than or equal to ten. So three plus four open parentheses one close parenthesis is greater than or equal to ten or not. Seven greater than or equal to ten is false. Second inequality, three times x minus two times y is less than twelve. Three open parentheses three close parentheses minus two open parentheses one close parentheses is less than twelve or not. Seven less than 12 holds true.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_002_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"We substitute x equal to three and y equal to one into both inequalities. First inequality is x plus four times y greater than or equal to ten. So three plus four open parentheses one close parenthesis is greater than or equal to ten or not. Seven greater than or equal to ten is false. Second inequality, three times x minus two times y is less than twelve. Three open parentheses three close parentheses minus two open parentheses one close parentheses is less than twelve or not. Seven less than 12 holds true.\" \/><\/span>\r\n<p id=\"fs-id1167833138779\">The ordered pair \\(\\left(3,1\\right)\\) made one inequality true, but the other one false. Therefore \\(\\left(3,1\\right)\\) is not a solution to this system.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836515280\">\r\n<div data-type=\"problem\" id=\"fs-id1167836544143\">\r\n\r\nDetermine whether the ordered pair is a solution to the system: \\(\\left\\{\\begin{array}{c}x-5y&gt;10\\hfill \\\\ 2x+3y&gt;-2\\hfill \\end{array}.\\)\r\n<p id=\"fs-id1167833060161\"><span class=\"token\">\u24d0<\/span>\\(\\left(3,-1\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(6,-3\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829750280\">\r\n\r\n<span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829580320\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167832981652\">\r\n<div data-type=\"problem\" id=\"fs-id1167833364848\">\r\n\r\nDetermine whether the ordered pair is a solution to the system: \\(\\left\\{\\begin{array}{c}y&gt;4x-2\\hfill \\\\ 4x-y&lt;20\\hfill \\end{array}.\\)\r\n<p id=\"fs-id1167836554187\"><span class=\"token\">\u24d0<\/span>\\(\\left(-2,1\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(4,-1\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836534637\">\r\n<p id=\"fs-id1167833023095\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836568053\">\r\n<h3 data-type=\"title\">Solve a System of Linear Inequalities by Graphing<\/h3>\r\n<p id=\"fs-id1167829906617\">The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167836792628\" class=\"textbox textbox--examples\">\r\n<div data-type=\"title\">How to Solve a System of Linear Inequalities by Graphing<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833053710\">\r\n<div data-type=\"problem\" id=\"fs-id1167829719558\">\r\n<p id=\"fs-id1167833138397\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y\\ge 2x-1\\hfill \\\\ y&lt;x+1\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836558351\"><span data-type=\"media\" id=\"fs-id1167833328924\" data-alt=\"Step 1. Graph the first inequality. We graph y less than 2x minus 1. Graph the boundary line y equal to 2x minus 1. It is a solid line because the inequality sign is less than. Shade in the side of the boundary line where the inequality is true. We choose 0, 0 as a test point. It is a solution to the equation, so we shade in above the boundary line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_003a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1. Graph the first inequality. We graph y less than 2x minus 1. Graph the boundary line y equal to 2x minus 1. It is a solid line because the inequality sign is less than. Shade in the side of the boundary line where the inequality is true. We choose 0, 0 as a test point. It is a solution to the equation, so we shade in above the boundary line.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829745782\" data-alt=\"Step 2. On the same grid, graph the second inequality y less than x plus 1. Graph the boundary line y equal to x plus 1. It is a dashed line because the inequality sign is less than. Shade in the side of that boundary line where the inequality is true. Again, we use 0, 0 as a test point. It is a solution so we shade below the line y equals x plus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_003b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2. On the same grid, graph the second inequality y less than x plus 1. Graph the boundary line y equal to x plus 1. It is a dashed line because the inequality sign is less than. Shade in the side of that boundary line where the inequality is true. Again, we use 0, 0 as a test point. It is a solution so we shade below the line y equals x plus 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829930090\" data-alt=\"Step 3. The solution is the region where the shading overlaps. The point where the boundary lines intersect is not a solution because it is not a solution to y less than x plus 1. The solution is all points in the area bounded by the lines on the bottom left.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_003c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3. The solution is the region where the shading overlaps. The point where the boundary lines intersect is not a solution because it is not a solution to y less than x plus 1. The solution is all points in the area bounded by the lines on the bottom left.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167833186666\" data-alt=\"Step 4. Check by choosing a test point. We use minus 1, minus 1. Substituting in the inequality y less than 2x minus 1, we get minus 1 less than minus 3 which is true. Hence, it is a solution. Similarly, it is also true for the other inequality. The region containing minus 1, minus 1 is the solution to this system.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_003d_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. Check by choosing a test point. We use minus 1, minus 1. Substituting in the inequality y less than 2x minus 1, we get minus 1 less than minus 3 which is true. Hence, it is a solution. Similarly, it is also true for the other inequality. The region containing minus 1, minus 1 is the solution to this system.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167833022166\" class=\"try\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167836299481\">\r\n<p id=\"fs-id1167836609707\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y&lt;3x+2\\hfill \\\\ y&gt;\\text{\u2212}x-1\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836543963\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836600273\" data-alt=\"The figure shows a graph plotted for the inequalities y less than three times x plus two and y greater than minus x minus one. Two lines intersect each other on the graph. An area to the right of both the lines is colored in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_301_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph plotted for the inequalities y less than three times x plus two and y greater than minus x minus one. Two lines intersect each other on the graph. An area to the right of both the lines is colored in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836523171\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836667050\">\r\n<div data-type=\"problem\" id=\"fs-id1167825913897\">\r\n<p id=\"fs-id1167833382178\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y&lt;-\\frac{1}{2}x+3\\hfill \\\\ y&lt;3x-4\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167833207697\" data-alt=\"The figure shows the graph plotted for the inequalities y less than minus half of x plus three and y less than three times x minus four. Two intersecting lines are shown on the graph. The area bound by the two lines to the bottom right is shown in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_302_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph plotted for the inequalities y less than minus half of x plus three and y less than three times x minus four. Two intersecting lines are shown on the graph. The area bound by the two lines to the bottom right is shown in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"howto\">\r\n<div data-type=\"title\">Solve a system of linear inequalities by graphing.<\/div>\r\n<ol id=\"fs-id1167836481249\" class=\"stepwise\" type=\"1\">\r\n \t<li>Graph the first inequality.\r\n<ul data-bullet-style=\"bullet\">\r\n \t<li>Graph the boundary line.<\/li>\r\n \t<li>Shade in the side of the boundary line where the inequality is true.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>On the same grid, graph the second inequality.\r\n<ul id=\"fs-id1167829811762\" data-bullet-style=\"bullet\">\r\n \t<li>Graph the boundary line.<\/li>\r\n \t<li>Shade in the side of that boundary line where the inequality is true.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The solution is the region where the shading overlaps.<\/li>\r\n \t<li>Check by choosing a test point.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167836300564\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836418956\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167824740437\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x-y&gt;3\\hfill \\\\ y&lt;-\\frac{1}{5}x+4\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<table id=\"fs-id1167829908693\" class=\"unnumbered unstyled\" summary=\"The figure shows the graph of the inequality x minus y greater than three. It is plotted by graphing x minus y equal to three and testing a point. The intercepts are x equal to three and y equal to minus three and the boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality false so shade the side that does not contain this point in red.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\left\\{\\begin{array}{c}x-y&gt;3\\hfill \\\\ y&lt;-\\frac{1}{5}x+4\\hfill \\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph <em data-effect=\"italics\">x<\/em> \u2212 <em data-effect=\"italics\">y<\/em> &gt; 3, by graphing <em data-effect=\"italics\">x<\/em> \u2212 <em data-effect=\"italics\">y<\/em> = 3\r\n<div data-type=\"newline\"><\/div>\r\nand testing a point.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nThe intercepts are <em data-effect=\"italics\">x<\/em> = 3 and <em data-effect=\"italics\">y<\/em> = \u22123 and the\r\n<div data-type=\"newline\"><\/div>\r\nboundary line will be dashed.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nTest (0, 0) which makes the inequality false so shade\r\n<div data-type=\"newline\"><\/div>\r\n(red) the side that does not contain (0, 0).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836531569\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_004a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"fs-id1167836329255\" class=\"unnumbered unstyled\" summary=\"The figure shows the making of graph of y less than minus one fifth x plus four by graphing y equal to minus one fifth x plus four using the slope m equal to minus one by five and y intercept b equal to four. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality true so the side that contains this point is shaded blue.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph \\(y&lt;-\\frac{1}{5}x+4\\) by graphing \\(y=-\\frac{1}{5}x+4\\)\r\n<div data-type=\"newline\"><\/div>\r\nusing the slope \\(m=-\\frac{1}{5}\\) and <em data-effect=\"italics\">y<\/em>-intercept <em data-effect=\"italics\">b<\/em> = 4.\r\n<div data-type=\"newline\"><\/div>\r\nThe boundary line will be dashed\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nTest (0, 0) which makes the inequality true, so\r\n<div data-type=\"newline\"><\/div>\r\nshade (blue) the side that contains (0, 0).\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nChoose a test point in the solution and verify that it is a solution to both inequalties.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836531590\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_005a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167836515907\">The point of intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice\u2014which appears as the darkest shaded region.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836532492\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829688088\">\r\n<div data-type=\"problem\" id=\"fs-id1167836615921\">\r\n<p id=\"fs-id1167833379505\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x+y\\le 2\\hfill \\\\ y\\ge \\frac{2}{3}x-1\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836547471\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836399021\" data-alt=\"The figure shows the graph for the inequalities x plus y less than or equal to two and y greater than or equal to two by three of x minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_303_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph for the inequalities x plus y less than or equal to two and y greater than or equal to two by three of x minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829715839\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167826130667\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167836613306\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}3x-2y\\le 6\\hfill \\\\ y&gt;\\text{\u2212}\\frac{1}{4}x+5\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836390184\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836540392\" data-alt=\"The figure shows graph for the inequalities three times x minus two times y less than or equal to six and y greater than or equal to minus one by four of x plus five. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_304_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows graph for the inequalities three times x minus two times y less than or equal to six and y greater than or equal to minus one by four of x plus five. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167836355790\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167836800431\">\r\n<p id=\"fs-id1167836506374\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x-2y&lt;5\\hfill \\\\ y&gt;-4\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836322831\">\r\n<table id=\"fs-id1167836507012\" class=\"unnumbered unstyled\" summary=\"The figure shows the making of graph of x minus two times y less than five by graphing x minus two times y equal to five and testing a point The intercepts are x equal to five and y equal to minus two point five. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality true so the side that contains this point is shaded red.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{2em}{0ex}}\\left\\{\\begin{array}{c}x-2y&lt;5\\hfill \\\\ y&gt;-4\\hfill \\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph \\(x-2y&lt;5,\\) by graphing \\(x-2y=5\\)\r\n<div data-type=\"newline\"><\/div>\r\nand testing a point. The intercepts are\r\n<div data-type=\"newline\"><\/div>\r\n<em data-effect=\"italics\">x<\/em> = 5 and <em data-effect=\"italics\">y<\/em> = \u22122.5 and the\r\n<div data-type=\"newline\"><\/div>\r\nboundary line will be dashed.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nTest (0, 0) which makes the inequality true, so shade\r\n<div data-type=\"newline\"><\/div>\r\n(red) the side that contains (0, 0).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836662872\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_006a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"fs-id1167836330926\" class=\"unnumbered unstyled can-break\" summary=\"The figure shows the making of graph of y greater than minus four by graphing y equal to minus four and recognizing that it is a horizontal line through y equal to minus four. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality true so the side that contains this point is shaded blue.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph \\(y&gt;-4,\\) by graphing \\(y=-4\\) and\r\n<div data-type=\"newline\"><\/div>\r\nrecognizing that it is a horizontal line\r\n<div data-type=\"newline\"><\/div>\r\nthrough \\(y=-4.\\) The boundary line will\r\n<div data-type=\"newline\"><\/div>\r\nbe dashed.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nTest (0, 0) which makes the inequality\r\n<div data-type=\"newline\"><\/div>\r\ntrue so shade (blue) the side that contains (0, 0).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829713444\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_007a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167836521734\">The point \\(\\left(0,0\\right)\\) is in the solution and we have already found it to be a solution of each inequality. The point of intersection of the two lines is not included as both boundary lines were dashed.<\/p>\r\n<p id=\"fs-id1167836319295\">The solution is the area shaded twice\u2014which appears as the darkest shaded region.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836527766\">\r\n<div data-type=\"problem\" id=\"fs-id1167836508780\">\r\n<p id=\"fs-id1167833086952\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y\\ge 3x-2\\hfill \\\\ y&lt;-1\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836571625\" data-alt=\"The figure shows graph for the inequalities y greater than or equal to three times x minus two and y less than minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_305_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows graph for the inequalities y greater than or equal to three times x minus two and y less than minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829746816\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167833378448\">\r\n<div data-type=\"problem\" id=\"fs-id1167832939943\">\r\n<p id=\"fs-id1167833397126\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x&gt;-4\\hfill \\\\ x-2y\\ge -4\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833369871\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836510670\" data-alt=\"The figure shows graph for the inequalities x greater than or equal to minus four and x minus two times y greater than minus four. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_306_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows graph for the inequalities x greater than or equal to minus four and x minus two times y greater than minus four. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836549970\">Systems of linear inequalities where the boundary lines are parallel might have no solution. We\u2019ll see this in the next example.<\/p>\r\n\r\n<div data-type=\"example\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836423663\">\r\n<div data-type=\"problem\" id=\"fs-id1167829783800\">\r\n<p id=\"fs-id1167829609167\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}4x+3y\\ge 12\\hfill \\\\ y&lt;-\\frac{4}{3}x+1\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836379111\">\r\n<table id=\"fs-id1167836367953\" class=\"unnumbered unstyled can-break\" summary=\"The figure shows the making of two graphs. First the graph of four times x plus three times y greater than or equal to twelve is made by graphing four times x plus three times y equal to twelve. The intercepts are x equal to three and y equal to four. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality false so the side that contains this point is shaded blue. Next the graph of y less than minus four by three of x plus one is made by graphing y equal to minus four by three of x plus one using the slope m equal to minus four by three and y intercept b equal to one. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality true so the side that contains this point is shaded red.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\left\\{\\begin{array}{c}4x+3y\\ge 12\\hfill \\\\ y&lt;-\\frac{4}{3}x+1\\hfill \\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph \\(4x+3y\\ge 12,\\) by graphing \\(4x+3y=12\\)\r\n<div data-type=\"newline\"><\/div>\r\nand testing a point. The intercepts are <em data-effect=\"italics\">x<\/em> = 3\r\n<div data-type=\"newline\"><\/div>\r\nand <em data-effect=\"italics\">y<\/em> = 4 and the boundary line will be solid.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nTest (0, 0) which makes the inequality false, so\r\n<div data-type=\"newline\"><\/div>\r\nshade (red) the side that does not contain (0, 0).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826206421\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_008a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph \\(y&lt;-\\frac{4}{3}x+1\\) by graphing \\(y=-\\frac{4}{3}x+1\\)\r\n<div data-type=\"newline\"><\/div>\r\nusing the slope \\(m=-\\frac{4}{3}\\) and <em data-effect=\"italics\">y<\/em>-intercept\r\n<div data-type=\"newline\"><\/div>\r\n<em data-effect=\"italics\">b<\/em> = 1. The boundary line will be dashed.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nTest (0, 0) which makes the inequality true, so\r\n<div data-type=\"newline\"><\/div>\r\nshade (blue) the side that contains (0, 0).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832981583\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_008b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167833407721\">There is no point in both shaded regions, so the system has no solution.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836378100\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836418859\">\r\n<div data-type=\"problem\" id=\"fs-id1167836630482\">\r\n<p id=\"fs-id1167836353188\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}3x-2y\\ge 12\\hfill \\\\ y\\ge \\frac{3}{2}x+1\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836518216\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167829688838\" data-alt=\"The graph of three times x minus two times y greater than or equal to twelve and y greater than or equal to three by two of x plus one is shown. Two intersecting lines are shown. The inequalities do not have a solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_307_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of three times x minus two times y greater than or equal to twelve and y greater than or equal to three by two of x plus one is shown. Two intersecting lines are shown. The inequalities do not have a solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nNo solution.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836352956\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836306660\">\r\n<div data-type=\"problem\" id=\"fs-id1167836699421\">\r\n<p id=\"fs-id1167836378871\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x+3y&gt;8\\hfill \\\\ y&lt;-\\frac{1}{3}x-2\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836602905\" data-alt=\"The graph of x plus three times y greater than eight and y less than minus one by three of x minus two is shown. Two intersecting lines are shown. The inequalities do not have a solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_308_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of x plus three times y greater than eight and y less than minus one by three of x minus two is shown. Two intersecting lines are shown. The inequalities do not have a solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nNo solution.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1168756714670\">Some systems of linear inequalities where the boundary lines are parallel will have a solution. We\u2019ll see this in the next example.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167836596568\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829785685\">\r\n<div data-type=\"problem\" id=\"fs-id1167829746913\">\r\n<p id=\"fs-id1167836378180\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y&gt;\\frac{1}{2}x-4\\hfill \\\\ x-2y&lt;-4\\hfill \\end{array}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829830048\">\r\n<table id=\"fs-id1167836390685\" class=\"unnumbered unstyled can-break\" summary=\"The figure shows the making of two graphs. First the graph of y greater than half of x minus four is made by graphing y equal to half of x minus four using the slope m equal to half and y intercept b equal to minus 4. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality so the side that contains this point is shaded red. Next the graph of x minus two times y less than minus four is made by graphing x minus two times y equal to minus four. The intercepts are x equal to minus four and y equal to two. The boundary line is dashed.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{2em}{0ex}}\\left\\{\\begin{array}{c}y&gt;\\frac{1}{2}x-4\\hfill \\\\ x-2y&lt;-4\\hfill \\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph \\(y&gt;\\frac{1}{2}x-4\\) by graphing \\(y=\\frac{1}{2}x-4\\)\r\n<div data-type=\"newline\"><\/div>\r\nusing the slope \\(m=\\frac{1}{2}\\) and the intercept\r\n<div data-type=\"newline\"><\/div>\r\n<em data-effect=\"italics\">b<\/em> = \u22124. The boundary line will be dashed.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nTest (0, 0) which makes the inequality true, so\r\n<div data-type=\"newline\"><\/div>\r\nshade (red) the side that contains (0, 0).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836440079\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Graph \\(x-2y&lt;-4\\) by graphing \\(x-2y=-4\\)\r\n<div data-type=\"newline\"><\/div>\r\nand testing a point. The intercepts are\r\n<div data-type=\"newline\"><\/div>\r\n<em data-effect=\"italics\">x<\/em> = \u22124 and <em data-effect=\"italics\">y<\/em> = 2 and the boundary line will be dashed.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nChoose a test point in the solution and verify\r\n<div data-type=\"newline\"><\/div>\r\nthat it is a solution to both inequalties.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nTest (0, 0) which makes the inequality false, so\r\n<div data-type=\"newline\"><\/div>\r\nshade (blue) the side that does not contain (0, 0).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824734677\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167833346510\">No point on the boundary lines is included in the solution as both lines are dashed.<\/p>\r\n<p id=\"fs-id1167829749840\">The solution is the region that is shaded twice which is also the solution to \\(x-2y&lt;-4.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836537024\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836297393\">\r\n<div data-type=\"problem\" id=\"fs-id1167833048343\">\r\n\r\nSolve the system by graphing: \\(\\left\\{\\begin{array}{c}y\\ge 3x+1\\hfill \\\\ -3x+y\\ge -4\\hfill \\end{array}.\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836789202\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836341685\" data-alt=\"The figure shows the graph of the inequalities y greater than or equal to three times x plus one and minus three times x plus y greater than or equal to minus four. Two parallel lines are shown and the region to the left of both is colored in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_309_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y greater than or equal to three times x plus one and minus three times x plus y greater than or equal to minus four. Two parallel lines are shown and the region to the left of both is colored in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829744169\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836611939\">\r\n<div data-type=\"problem\" id=\"fs-id1167836523596\">\r\n\r\nSolve the system by graphing: \\(\\left\\{\\begin{array}{c}y\\le -\\frac{1}{4}x+2\\hfill \\\\ x+4y\\le 4\\hfill \\end{array}.\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167833058054\" data-alt=\"The figure shows the graph of the inequalities y less than or equal to minus one fourth of x plus 2 and x plus four times y less than or equal to four. Two parallel lines are shown and the region to the bottom of both is colored in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_310_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y less than or equal to minus one fourth of x plus 2 and x plus four times y less than or equal to four. Two parallel lines are shown and the region to the bottom of both is colored in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829744348\">\r\n<h3 data-type=\"title\">Solve Applications of Systems of Inequalities<\/h3>\r\n<p id=\"fs-id1167829843274\">The first thing we\u2019ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so we add inequalities to the system as additional requirements.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167836556132\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829620350\">\r\n<div data-type=\"problem\" id=\"fs-id1167836510446\">\r\n<p id=\"fs-id1167829695726\">Christy sells her photographs at a booth at a street fair. At the start of the day, she wants to have at least 25 photos to display at her booth. Each small photo she displays costs her ?4 and each large photo costs her ?10. She doesn\u2019t want to spend more than ?200 on photos to display.<\/p>\r\n<p id=\"fs-id1167829694291\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Could she display 10 small and 20 large photos?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Could she display 20 large and 10 small photos?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836731946\">\r\n<p id=\"fs-id1167836390111\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cc}\\text{Let}\\hfill &amp; x=\\text{the number of small photos}.\\hfill \\\\ &amp; y=\\text{the number of large photos}\\hfill \\end{array}\\)\r\n<p id=\"fs-id1167836649828\">To find the system of equations translate the information.<\/p>\r\n<p id=\"fs-id1167836693211\">\\(\\begin{array}{}\\\\ \\\\ \\\\ \\phantom{\\rule{3em}{0ex}}\\text{She wants to have at least 25 photos.}\\hfill \\\\ \\phantom{\\rule{3em}{0ex}}\\text{The number of small plus the number of large should be at least 25.}\\hfill \\\\ \\hfill x+y\\ge 25\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{3em}{0ex}}\\text{?4 for each small and ?10 for each large must be no more than ?200}\\hfill \\\\ \\hfill 4x+10y\\le 200\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{3em}{0ex}}\\text{The number of small photos must be greater than or equal to 0.}\\hfill \\\\ \\hfill x\\ge 0\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{3em}{0ex}}\\text{The number of large photos must be greater than or equal to 0.}\\hfill \\\\ \\hfill y\\ge 0\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167829645186\">We have our system of equations.<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1169146740783\" class=\"unnumbered\" data-label=\"\">\\(\\phantom{\\rule{1em}{0ex}}\\left\\{\\begin{array}{c}x+y\\ge 25\\hfill \\\\ 4x+10y\\le 200\\hfill \\\\ x\\ge 0\\hfill \\\\ y\\ge 0\\hfill \\end{array}\\)<\/div>\r\n<p id=\"fs-id1167829693372\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\nSince \\(x\\ge 0\\) and \\(y\\ge 0\\) (both are greater than or equal to) all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167833245815\" class=\"unnumbered unstyled can-break\" summary=\"The figure shows the making of two graphs. First the inequality x plus y greater than or equal to twenty five is made by graphing x plus y equal to twenty five as a solid line. The point with coordinates zero and zero does not make the inequality true so the side that does not include this point is shaded red. Next the inequality four times x plus ten times y greater than or equal to two hundred is made by graphing four times x plus ten times y equal to two hundred. The point with coordinates zero and zero does make the inequality true so the side which contains this point is shaded blue.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">To graph \\(x+y\\ge 25,\\) graph \\(x+y=25\\) as a solid line.\r\n<div data-type=\"newline\"><\/div>\r\nChoose (0, 0) as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point (0, 0).\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nTo graph \\(4x+10y\\le 200,\\) graph \\(4x+10y=200\\) as a solid line.\r\n<div data-type=\"newline\"><\/div>\r\nChoose (0, 0) as a test point. Since it does make the inequality true, shade (blue) the side that include the point (0, 0).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829614356\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_010a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167836434591\">The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly-shaded section are included in the solution as are the points on the <em data-effect=\"italics\">x<\/em>-axis from (25, 0) to (55, 0).<\/p>\r\n<span class=\"token\">\u24d2<\/span> To determine if 10 small and 20 large photos would work, we look at the graph to see if the point (10, 20) is in the solution region. We could also test the point to see if it is a solution of both equations.\r\n<p id=\"fs-id1167836378351\">It is not, Christy would not display 10 small and 20 large photos.<\/p>\r\n<p id=\"fs-id1167833224670\"><span class=\"token\">\u24d3<\/span> To determine if 20 small and 10 large photos would work, we look at the graph to see if the point (20, 10) is in the solution region. We could also test the point to see if it is a solution of both equations.<\/p>\r\n<p id=\"fs-id1167833050449\">It is, so Christy could choose to display 20 small and 10 large photos.<\/p>\r\n<p id=\"fs-id1167836537096\">Notice that we could also test the possible solutions by substituting the values into each inequality.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836706462\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167822971721\">\r\n<div data-type=\"problem\" id=\"fs-id1167836688596\">\r\n<p id=\"fs-id1167829718513\">A trailer can carry a maximum weight of 160 pounds and a maximum volume of 15 cubic feet. A microwave oven weighs 30 pounds and has 2 cubic feet of volume, while a printer weighs 20 pounds and has 3 cubic feet of space.<\/p>\r\n<p id=\"fs-id1167836327777\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Could 4 microwaves and 2 printers be carried on this trailer?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Could 7 microwaves and 3 printers be carried on this trailer?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829685153\">\r\n<p id=\"fs-id1167836287444\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}30m+20p\\le 160\\hfill \\\\ 2m+3p\\le 15\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167836300551\" data-alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_311_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> yes\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> no\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829745732\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167833316799\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167836312119\">Mary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least 5 more than the number of pencils. The pencils cost ?2 and the answer sheets cost ?1. Mary\u2019s budget for these supplies allows for a maximum cost of ?400.<\/p>\r\n<p id=\"fs-id1167836392171\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Could Mary purchase 100 pencils and 100 answer sheets?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Could Mary purchase 150 pencils and 150 answer sheets?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826102750\">\r\n<p id=\"fs-id1167836689366\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}a\\ge p+5\\hfill \\\\ a+2p\\le 400\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167836610639\" data-alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_312_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> no\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> no\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1168756762423\">When we use variables other than <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> to define an unknown quantity, we must change the names of the axes of the graph as well.<\/p>\r\n\r\n<div data-type=\"example\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836507645\">\r\n<div data-type=\"problem\" id=\"fs-id1167836628288\">\r\n<p id=\"fs-id1167836295269\">Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers and cookies, and he doesn\u2019t want to spend more than ?5. At the hamburger restaurant near his college, each hamburger has 240 calories and costs ?1.40. Each cookie has 160 calories and costs ?0.50.<\/p>\r\n<p id=\"fs-id1167833396849\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Could he eat 3 hamburgers and 1 cookie?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Could he eat 2 hamburgers and 4 cookies?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836518697\">\r\n<p id=\"fs-id1167836329245\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cc}\\text{Let}\\hfill &amp; h=\\text{the number of hamburgers}.\\hfill \\\\ &amp; c=\\text{the number of cookies}\\hfill \\end{array}\\)\r\n<p id=\"fs-id1167833021820\">To find the system of equations translate the information.<\/p>\r\n<p id=\"fs-id1167836539916\">The calories from hamburgers at 240 calories each, plus the calories from cookies at 160 calories each must be more that 800.<\/p>\r\n<p id=\"fs-id1167836621351\">\\(\\begin{array}{c}\\hfill 240h+160c\\ge 800\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{3em}{0ex}}\\text{The amount spent on hamburgers at ?1.40 each, plus the amount spent on cookies}\\hfill \\\\ \\phantom{\\rule{3em}{0ex}}\\text{at ?0.50 each must be no more than ?5.00.}\\hfill \\\\ \\hfill 1.40h+0.50c\\le 5\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{3em}{0ex}}\\text{The number of hamburgers must be greater than or equal to 0.}\\hfill \\\\ \\hfill h\\ge 0\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{3em}{0ex}}\\text{The number of cookies must be greater than or equal to 0.}\\hfill \\\\ \\hfill c\\ge 0\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1171792683972\">\\(\\begin{array}{cccc}\\text{We have our system of equations.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left\\{\\begin{array}{c}240h+160c\\ge 800\\hfill \\\\ 1.40h+0.50c\\le 5\\hfill \\\\ h\\ge 0\\hfill \\\\ c\\ge 0\\hfill \\end{array}\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167833310128\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\nSince \\(h&gt;=0\\) and \\(c&gt;=0\\) (both are greater than or equal to) all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1167833136536\" class=\"unnumbered unstyled can-break\" summary=\"The graph of inequality two forty times h plus one sixty times c greater than or equal to eight hundred is shown. It is made by graphing two forty h plus one sixty c equal to eight hundred as a solid line. The point with coordinates zero and zero does not make the inequality true so the side which does not include this point is shaded red.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">To graph \\(240h+160c\\ge 800,\\) graph \\(240h+160c=800\\) as a solid line.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\nChoose (0, 0) as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point (0, 0).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833136523\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_011a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167833239251\">Graph \\(1.40h+0.50c\\le 5.\\) The boundary line is \\(1.40h+0.50c=5.\\) We test (0, 0) and it makes the inequality true. We shade the side of the line that includes (0, 0).<\/p>\r\n<p id=\"fs-id1167836512397\">The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly shaded section are included in the solution as are the points on the <em data-effect=\"italics\">x<\/em>-axis from (5, 0) to (10, 0).<\/p>\r\n<p id=\"fs-id1167833135690\"><span class=\"token\">\u24d2<\/span> To determine if 3 hamburgers and 2 cookies would meet Omar\u2019s criteria, we see if the point (3, 2) is in the solution region. It is, so Omar might choose to eat 3 hamburgers and 2 cookies.<\/p>\r\n<p id=\"fs-id1167829942426\"><span class=\"token\">\u24d3<\/span> To determine if 2 hamburgers and 4 cookies would meet Omar\u2019s criteria, we see if the point (2, 4) is in the solution region. It is, Omar might choose to eat 2 hamburgers and 4 cookies.<\/p>\r\n<p id=\"fs-id1167836559981\">We could also test the possible solutions by substituting the values into each inequality.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829619410\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829718986\">\r\n<div data-type=\"problem\" id=\"fs-id1167836706434\">\r\n<p id=\"fs-id1167836391752\">Tension needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only ?25 to spend on the extra food he needs and will spend it on ?0.75 donuts which have 360 calories each and ?2 energy drinks which have 110 calories.<\/p>\r\n<p id=\"fs-id1167836516260\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities that models this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Can he buy 8 donuts and 4 energy drinks and satisfy his caloric needs?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Can he buy 1 donut and 3 energy drinks and satisfy his caloric needs?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836493823\">\r\n<p id=\"fs-id1167836705878\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}0.75d+2e\\le 25\\hfill \\\\ 360d+110e\\ge 1000\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829579782\" data-alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_313_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> yes\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> no\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167833338266\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167833407364\">\r\n<div data-type=\"problem\" id=\"fs-id1167829619467\">\r\n<p id=\"fs-id1167829924727\">Philip\u2019s doctor tells him he should add at least 1,000 more calories per day to his usual diet. Philip wants to buy protein bars that cost ?1.80 each and have 140 calories and juice that costs ?1.25 per bottle and have 125 calories. He doesn\u2019t want to spend more than ?12.<\/p>\r\n<p id=\"fs-id1167833086963\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities that models this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Can he buy 3 protein bars and 5 bottles of juice?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Can he buy 5 protein bars and 3 bottles of juice?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836282522\">\r\n<p id=\"fs-id1167836386502\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}140p+125j\\ge 1000\\hfill \\\\ 1.80p+1.25j\\le 12\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167836522128\" data-alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_314_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> yes\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> no\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"media-2\">\r\n<p id=\"fs-id1167832980753\">Access these online resources for additional instruction and practice with solving systems of linear inequalities by graphing.<\/p>\r\n\r\n<ul id=\"fs-id1167836690512\" data-display=\"block\">\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/37sysineqgraph\">Solving Systems of Linear Inequalities by Graphing<\/a><\/li>\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/37sysineqgraph2\">Systems of Linear Inequalities<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836622961\">\r\n<h3 data-type=\"title\">Key Concepts<\/h3>\r\n<ul id=\"fs-id1167836610518\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Solutions of a System of Linear Inequalities:<\/strong> Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the <em data-effect=\"italics\">x, y<\/em> coordinate system that includes all the points whose ordered pairs make the inequalities true.<\/li>\r\n \t<li><strong data-effect=\"bold\">How to solve a system of linear inequalities by graphing.<\/strong>\r\n<ol id=\"fs-id1167833310565\" class=\"stepwise\" type=\"1\">\r\n \t<li>Graph the first inequality.\r\n<div data-type=\"newline\"><\/div>\r\nGraph the boundary line.\r\n<div data-type=\"newline\"><\/div>\r\nShade in the side of the boundary line where the inequality is true.<\/li>\r\n \t<li>On the same grid, graph the second inequality.\r\n<div data-type=\"newline\"><\/div>\r\nGraph the boundary line.\r\n<div data-type=\"newline\"><\/div>\r\nShade in the side of that boundary line where the inequality is true.<\/li>\r\n \t<li>The solution is the region where the shading overlaps.<\/li>\r\n \t<li>Check by choosing a test point.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836551834\">\r\n<h3 data-type=\"title\">Section Exercises<\/h3>\r\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829907198\">\r\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\r\n<strong data-effect=\"bold\">Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities<\/strong>\r\n<p id=\"fs-id1167836521359\">In the following exercises, determine whether each ordered pair is a solution to the system.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836608480\">\r\n<div data-type=\"problem\" id=\"fs-id1167836701180\">\r\n<p id=\"fs-id1167836697073\">\\(\\left\\{\\begin{array}{c}3x+y&gt;5\\hfill \\\\ 2x-y\\le 10\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167829624252\"><span class=\"token\">\u24d0<\/span>\\(\\left(3,-3\\right)\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(7,1\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836333949\">\r\n<div data-type=\"problem\" id=\"fs-id1167836349238\">\r\n<p id=\"fs-id1167836334178\">\\(\\left\\{\\begin{array}{c}4x-y&lt;10\\hfill \\\\ -2x+2y&gt;-8\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167836294884\"><span class=\"token\">\u24d0<\/span>\\(\\left(5,-2\\right)\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(-1,3\\right)\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167825703167\">\r\n\r\n<span class=\"token\">\u24d0<\/span> false <span class=\"token\">\u24d1<\/span> true\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833017814\">\r\n<div data-type=\"problem\" id=\"fs-id1167836334160\">\r\n<p id=\"fs-id1167833412555\">\\(\\left\\{\\begin{array}{c}y&gt;\\frac{2}{3}x-5\\hfill \\\\ x+\\frac{1}{2}y\\le 4\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167836743453\"><span class=\"token\">\u24d0<\/span>\\(\\text{(6, \u22124)}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\text{(3, 0)}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836688087\">\r\n<div data-type=\"problem\" id=\"fs-id1167836574907\">\r\n<p id=\"fs-id1167836619659\">\\(\\left\\{\\begin{array}{c}y&lt;\\frac{3}{2}x+3\\hfill \\\\ \\frac{3}{4}x-2y&lt;5\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167829749517\"><span class=\"token\">\u24d0<\/span>\\(\\left(-4,-1\\right)\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\text{(8, 3)}\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836514155\">\r\n<p id=\"fs-id1167833310849\"><span class=\"token\">\u24d0<\/span> false <span class=\"token\">\u24d1<\/span> true<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829713662\">\r\n<div data-type=\"problem\" id=\"fs-id1167836288276\">\r\n<p id=\"fs-id1167836557972\">\\(\\left\\{\\begin{array}{c}7x+2y&gt;14\\hfill \\\\ 5x-y\\le 8\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167829905595\"><span class=\"token\">\u24d0<\/span>\\(\\text{(2, 3)}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\text{(7, \u22121)}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836408854\">\r\n<div data-type=\"problem\" id=\"fs-id1167833224335\">\r\n<p id=\"fs-id1167833138577\">\\(\\left\\{\\begin{array}{c}6x-5y&lt;20\\hfill \\\\ -2x+7y&gt;-8\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167836738279\"><span class=\"token\">\u24d0<\/span>\\(\\text{(1, \u22123)}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\text{(\u22124, 4)}\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829691536\">\r\n<p id=\"fs-id1167836538843\"><span class=\"token\">\u24d0<\/span> false <span class=\"token\">\u24d1<\/span> true<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167833071549\"><strong data-effect=\"bold\">Solve a System of Linear Inequalities by Graphing<\/strong><\/p>\r\n<p id=\"fs-id1167836622335\">In the following exercises, solve each system by graphing.<\/p>\r\n\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167836508300\">\r\n<p id=\"fs-id1167836408694\">\\(\\left\\{\\begin{array}{c}y\\le 3x+2\\hfill \\\\ y&gt;x-1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836536081\">\r\n<div data-type=\"problem\" id=\"fs-id1167830123178\">\r\n<p id=\"fs-id1167836694111\">\\(\\left\\{\\begin{array}{c}y&lt;-2x+2\\hfill \\\\ y\\ge -x-1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826132859\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167824773950\" data-alt=\"The figure shows the graph of inequalities y less than minus two times x plus two and y greater than or equal to minus x minus one. Two intersecting lines are shown, one in red and the other in blue. The area bound by the two lines is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_203_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of inequalities y less than minus two times x plus two and y greater than or equal to minus x minus one. Two intersecting lines are shown, one in red and the other in blue. The area bound by the two lines is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833048440\">\r\n<div data-type=\"problem\" id=\"fs-id1167833128530\">\r\n<p id=\"fs-id1167833056596\">\\(\\left\\{\\begin{array}{c}y&lt;2x-1\\hfill \\\\ y\\le -\\frac{1}{2}x+4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836607397\">\r\n<div data-type=\"problem\" id=\"fs-id1167836692537\">\r\n<p id=\"fs-id1167829908011\">\\(\\left\\{\\begin{array}{c}y\\ge -\\frac{2}{3}x+2\\hfill \\\\ y&gt;2x-3\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836538932\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836509816\" data-alt=\"The figure shows the graph of the inequalities y greater than or equal to minus two by three x plus two and y greater than two times x minus three. Two intersecting lines, one in red and the other in blue, are shown. The region bound by them is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_205_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y greater than or equal to minus two by three x plus two and y greater than two times x minus three. Two intersecting lines, one in red and the other in blue, are shown. The region bound by them is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833055904\">\r\n<div data-type=\"problem\" id=\"fs-id1167833350378\">\r\n<p id=\"fs-id1167836512297\">\\(\\left\\{\\begin{array}{c}x-y&gt;1\\hfill \\\\ y&lt;-\\frac{1}{4}x+\\hfill \\end{array}3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836727811\">\r\n<div data-type=\"problem\" id=\"fs-id1167829743662\">\r\n<p id=\"fs-id1167836629307\">\\(\\left\\{\\begin{array}{c}x+2y&lt;4\\hfill \\\\ y&lt;x-2\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836629128\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836685960\" data-alt=\"The figure shows the graph of the inequalities x minus two times y less than four and y less than x minus two. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_207_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities x minus two times y less than four and y less than x minus two. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836705535\">\r\n<div data-type=\"problem\" id=\"fs-id1167829755866\">\r\n<p id=\"fs-id1167829755868\">\\(\\left\\{\\begin{array}{c}3x-y\\ge 6\\hfill \\\\ y\\ge -\\frac{1}{2}x\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836448216\">\r\n<div data-type=\"problem\" id=\"fs-id1167836448218\">\r\n<p id=\"fs-id1167836322981\">\\(\\left\\{\\begin{array}{c}2x+4y\\ge 8\\hfill \\\\ y\\le \\frac{3}{4}x\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836598601\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167833191635\" data-alt=\"The figure shows the graph of the inequalities two times x plus four times y greater than or equal to eight and y less than or equal to minus three fourth of x. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_209_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities two times x plus four times y greater than or equal to eight and y less than or equal to minus three fourth of x. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167836391716\">\r\n<p id=\"fs-id1167836391718\">\\(\\left\\{\\begin{array}{c}2x-5y&lt;10\\hfill \\\\ 3x+4y\\ge 12\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833271854\">\r\n<div data-type=\"problem\" id=\"fs-id1167836535416\">\r\n<p id=\"fs-id1167836535418\">\\(\\left\\{\\begin{array}{c}3x-2y\\le 6\\hfill \\\\ -4x-2y&gt;8\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836615038\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167824764655\" data-alt=\"The figure shows the graph of the inequalities three times x minus two times y less than or equal to six and minus four times x minus two times y greater than eight. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_211_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities three times x minus two times y less than or equal to six and minus four times x minus two times y greater than eight. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824773992\">\r\n<div data-type=\"problem\" id=\"fs-id1167833031765\">\r\n<p id=\"fs-id1167833031767\">\\(\\left\\{\\begin{array}{c}2x+2y&gt;-4\\hfill \\\\ \\text{\u2212}x+3y\\ge 9\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836508671\">\r\n<div data-type=\"problem\" id=\"fs-id1167833387018\">\r\n<p id=\"fs-id1167833387020\">\\(\\left\\{\\begin{array}{c}2x+y&gt;-6\\hfill \\\\ \\text{\u2212}x+2y\\ge -4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833056781\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836614346\" data-alt=\"The figure shows the graph of the inequalities two times x plus y greater than minus six and minus x plus two times y greater than or equal to minus four. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_213_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities two times x plus y greater than minus six and minus x plus two times y greater than or equal to minus four. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836717278\">\r\n<div data-type=\"problem\" id=\"fs-id1167836717280\">\r\n<p id=\"fs-id1167833350465\">\\(\\left\\{\\begin{array}{c}x-2y&lt;3\\hfill \\\\ y\\le 1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833053554\">\r\n<div data-type=\"problem\" id=\"fs-id1167833053556\">\r\n<p id=\"fs-id1167836477283\">\\(\\left\\{\\begin{array}{c}x-3y&gt;4\\hfill \\\\ y\\le -1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833018419\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167833018422\" data-alt=\"The figure shows the graph of the inequalities x minus three times y greater than four and y less than or equal to minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_215_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities x minus three times y greater than four and y less than or equal to minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836484677\">\r\n<div data-type=\"problem\" id=\"fs-id1167836484679\">\r\n<p id=\"fs-id1167833057051\">\\(\\left\\{\\begin{array}{c}y\\ge -\\frac{1}{2}x-3\\hfill \\\\ x\\le 2\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836596407\">\r\n<div data-type=\"problem\" id=\"fs-id1167836596409\">\r\n\r\n\\(\\left\\{\\begin{array}{c}y\\le -\\frac{2}{3}x+5\\hfill \\\\ x\\ge 3\\hfill \\end{array}\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836606749\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836318353\" data-alt=\"The figure shows the graph of the inequality y less than or equal to minus two by three times x plus five and x greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_217_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequality y less than or equal to minus two by three times x plus five and x greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836663874\">\r\n<div data-type=\"problem\" id=\"fs-id1167836663876\">\r\n<p id=\"fs-id1167836663878\">\\(\\left\\{\\begin{array}{c}y\\ge \\frac{3}{4}x-2\\hfill \\\\ y&lt;2\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836363505\">\r\n<div data-type=\"problem\" id=\"fs-id1167836363507\">\r\n<p id=\"fs-id1167836664850\">\\(\\left\\{\\begin{array}{c}y\\le -\\frac{1}{2}x+3\\hfill \\\\ y&lt;1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836506503\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836607100\" data-alt=\"The figure shows the graph of the inequalities y less than or equal to minus half x plus three and y less than one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_219_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y less than or equal to minus half x plus three and y less than one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836506581\">\r\n<div data-type=\"problem\" id=\"fs-id1167836506583\">\r\n<p id=\"fs-id1167829877906\">\\(\\left\\{\\begin{array}{c}3x-4y&lt;8\\hfill \\\\ x&lt;1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833060370\">\r\n<div data-type=\"problem\" id=\"fs-id1167833060372\">\r\n<p id=\"fs-id1167833025692\">\\(\\left\\{\\begin{array}{c}-3x+5y&gt;10\\hfill \\\\ x&gt;-1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836729905\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836729908\" data-alt=\"The figure shows the graph of the inequalities minus three times x plus five times y greater than ten and x greater than minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_221_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities minus three times x plus five times y greater than ten and x greater than minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836792647\">\r\n<div data-type=\"problem\" id=\"fs-id1167836620418\">\r\n<p id=\"fs-id1167836620420\">\\(\\left\\{\\begin{array}{c}x\\ge 3\\hfill \\\\ y\\le 2\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829890995\">\r\n<div data-type=\"problem\" id=\"fs-id1167829890997\">\r\n<p id=\"fs-id1167836621261\">\\(\\left\\{\\begin{array}{c}x\\le -1\\hfill \\\\ y\\ge 3\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829742976\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167833364682\" data-alt=\"The figure shows the graph of the inequalities x less than or equal to minus one and y greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_223_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities x less than or equal to minus one and y greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836554483\">\r\n<div data-type=\"problem\" id=\"fs-id1167836554485\">\r\n<p id=\"fs-id1167836375864\">\\(\\left\\{\\begin{array}{c}2x+4y&gt;4\\hfill \\\\ y\\le -\\frac{1}{2}x-2\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833058265\">\r\n<div data-type=\"problem\" id=\"fs-id1167836535017\">\r\n<p id=\"fs-id1167836535019\">\\(\\left\\{\\begin{array}{c}x-3y\\ge 6\\hfill \\\\ y&gt;\\frac{1}{3}x+1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829745862\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836531493\" data-alt=\"The figure shows the graph of the inequalities x minus three times y greater than or equal to six and y greater than one third of x plus one. Two non intersecting lines, one in blue and the other in red, are shown.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_225_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities x minus three times y greater than or equal to six and y greater than one third of x plus one. Two non intersecting lines, one in blue and the other in red, are shown.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nNo solution.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833250778\">\r\n<div data-type=\"problem\" id=\"fs-id1167833250780\">\r\n<p id=\"fs-id1167833382481\">\\(\\left\\{\\begin{array}{c}-2x+6y&lt;0\\hfill \\\\ 6y&gt;2x+4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836697594\">\r\n<div data-type=\"problem\" id=\"fs-id1167836697596\">\r\n<p id=\"fs-id1167836697598\">\\(\\left\\{\\begin{array}{c}-3x+6y&gt;12\\hfill \\\\ 4y\\le 2x-4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833197311\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167829754453\" data-alt=\"The figure shows the graph of the inequalities minus three times x plus six times y greater than twelve and four times y less than or equal to two times x minus four. Two non intersecting lines, one in blue and the other in red, are shown.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_227_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities minus three times x plus six times y greater than twelve and four times y less than or equal to two times x minus four. Two non intersecting lines, one in blue and the other in red, are shown.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nNo solution.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167833056019\">\r\n<p id=\"fs-id1167833056021\">\\(\\left\\{\\begin{array}{c}y\\ge -3x+2\\hfill \\\\ 3x+y&gt;5\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836608976\">\r\n<div data-type=\"problem\" id=\"fs-id1167833186286\">\r\n<p id=\"fs-id1167833186288\">\\(\\left\\{\\begin{array}{c}y\\ge \\frac{1}{2}x-1\\hfill \\\\ -2x+4y\\ge 4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829586638\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836547001\" data-alt=\"The figure shows the graph of the inequalities y greater than or equal to minus half x minus one and minus two times x plus four times y greater than or equal to four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_229_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y greater than or equal to minus half x minus one and minus two times x plus four times y greater than or equal to four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833346852\">\r\n<div data-type=\"problem\" id=\"fs-id1167826182813\">\r\n<p id=\"fs-id1167826182815\">\\(\\left\\{\\begin{array}{c}y\\le -\\frac{1}{4}x-2\\hfill \\\\ x+4y&lt;6\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833066673\">\r\n<div data-type=\"problem\" id=\"fs-id1167825913939\">\r\n<p id=\"fs-id1167825913941\">\\(\\left\\{\\begin{array}{c}y\\ge 3x-1\\hfill \\\\ -3x+y&gt;-4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836440537\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167833272751\" data-alt=\"The figure shows the graph of the inequalities y greater than or equal to three times x minus one and minus three times x plus y greater than minus four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_231_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y greater than or equal to three times x minus one and minus three times x plus y greater than minus four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836700468\">\r\n<div data-type=\"problem\" id=\"fs-id1167836356885\">\r\n<p id=\"fs-id1167836356887\">\\(\\left\\{\\begin{array}{c}3y&gt;x+2\\hfill \\\\ -2x+6y&gt;8\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833008893\">\r\n<div data-type=\"problem\" id=\"fs-id1167833008895\">\r\n<p id=\"fs-id1167836701006\">\\(\\left\\{\\begin{array}{c}y&lt;\\frac{3}{4}x-2\\hfill \\\\ -3x+4y&lt;7\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836523933\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836663644\" data-alt=\"The figure shows the graph of the inequalities y less than three by fourth x minus two and minus three x plus four y less than seven. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_233_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y less than three by fourth x minus two and minus three x plus four y less than seven. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836596247\"><strong data-effect=\"bold\">Solve Applications of Systems of Inequalities<\/strong><\/p>\r\n<p id=\"fs-id1167836609964\">In the following exercises, translate to a system of inequalities and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836609967\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836611857\">\r\n<p id=\"fs-id1167836611859\">Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for ?15 and the landscapes for ?10. She needs to sell at least ?800 worth of drawings in order to earn a profit.<\/p>\r\n<p id=\"fs-id1167833326352\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Will she make a profit if she sells 20 portraits and 35 landscapes?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Will she make a profit if she sells 50 portraits and 20 landscapes?\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836523534\">\r\n<p id=\"fs-id1167836409011\">Jake does not want to spend more than ?50 on bags of fertilizer and peat moss for his garden. Fertilizer costs ?2 a bag and peat moss costs ?5 a bag. Jake\u2019s van can hold at most 20 bags.<\/p>\r\n<p id=\"fs-id1167833056343\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Can he buy 15 bags of fertilizer and 4 bags of peat moss?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Can he buy 10 bags of fertilizer and 10 bags of peat moss?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829744328\">\r\n<p id=\"fs-id1167829744330\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}f\\ge 0\\hfill \\\\ p\\ge 0\\hfill \\\\ f+p\\le 20\\hfill \\\\ 2f+5p\\le 50\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167833339148\" data-alt=\"The figure shows the graph of the inequalities f plus p less than or equal to twenty and two f and five p less than or equal to fifty. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_235_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities f plus p less than or equal to twenty and two f and five p less than or equal to fifty. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> yes\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> no\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832950886\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167833057432\">\r\n<p id=\"fs-id1167833057434\">Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than ?500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is ?3 and for a package the cost is ?7.<\/p>\r\n<p id=\"fs-id1167836558183\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Can she mail 60 cards and 26 packages?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Can she mail 90 cards and 40 packages?\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829695320\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829695322\">\r\n\r\nJuan is studying for his final exams in chemistry and algebra. he knows he only has 24 hours to study, and it will take him at least three times as long to study for algebra than chemistry.\r\n<p id=\"fs-id1167836788650\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Can he spend 4 hours on chemistry and 20 hours on algebra?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Can he spend 6 hours on chemistry and 18 hours on algebra?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836601793\">\r\n<p id=\"fs-id1167836287747\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}c\\ge 0\\hfill \\\\ a\\ge 0\\hfill \\\\ c+a\\le 24\\hfill \\\\ a\\ge 3c\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167833135357\" data-alt=\"The figure shows the graph of the inequalities c plus a less than or equal to twenty four and a greater than or equal to three times c. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_237_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities c plus a less than or equal to twenty four and a greater than or equal to three times c. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> yes\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> no\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836511611\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836511613\">\r\n<p id=\"fs-id1167836511616\">Jocelyn is pregnant and so she needs to eat at least 500 more calories a day than usual. When buying groceries one day with a budget of ?15 for the extra food, she buys bananas that have 90 calories each and chocolate granola bars that have 150 calories each. The bananas cost ?0.35 each and the granola bars cost ?2.50 each.<\/p>\r\n<p id=\"fs-id1167826171973\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Could she buy 5 bananas and 6 granola bars?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Could she buy 3 bananas and 4 granola bars?\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824731940\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167824731942\">\r\n<p id=\"fs-id1167829909635\">Mark is attempting to build muscle mass and so he needs to eat at least an additional 80 grams of protein a day. A bottle of protein water costs ?3.20 and a protein bar costs ?1.75. The protein water supplies 27 grams of protein and the bar supplies 16 gram. If he has ?10 dollars to spend<\/p>\r\n<p id=\"fs-id1167833076719\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Could he buy 3 bottles of protein water and 1 protein bar?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Could he buy no bottles of protein water and 5 protein bars?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833339515\">\r\n\r\n<span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}w\\ge 0\\hfill \\\\ b\\ge 0\\hfill \\\\ 27w+16b&gt;80\\hfill \\\\ 3.20w+1.75b\\le 10\\hfill \\end{array}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167833386413\" data-alt=\"The figure shows the graph of the inequalities twenty seven times w plus sixteen times b greater than eighty and three point two times w plus one point seven five b less than or equal to ten. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_239_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities twenty seven times w plus sixteen times b greater than eighty and three point two times w plus one point seven five b less than or equal to ten. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> no\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> yes\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836731285\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836731288\">\r\n<p id=\"fs-id1167836731290\">Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least 35 more grams of protein each day and no more than an additional 200 calories daily. An ounce of cheddar cheese has 7 grams of protein and 110 calories. An ounce of parmesan cheese has 11 grams of protein and 22 calories.<\/p>\r\n<p id=\"fs-id1167833025309\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Could she eat 1 ounce of cheddar cheese and 3 ounces of parmesan cheese?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Could she eat 2 ounces of cheddar cheese and 1 ounce of parmesan cheese?\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836553814\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836553817\">\r\n<p id=\"fs-id1167836553819\">Mark is increasing his exercise routine by running and walking at least 4 miles each day. His goal is to burn a minimum of 1500 calories from this exercise. Walking burns 270 calories\/mile and running burns 650 calories.<\/p>\r\n<p id=\"fs-id1167833135425\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Could he meet his goal by walking 3 miles and running 1 mile?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Could he his goal by walking 2 miles and running 2 mile\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836493827\">\r\n<p id=\"fs-id1167836493829\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}w\\ge 0\\hfill \\\\ r\\ge 0\\hfill \\\\ w+r\\ge 4\\hfill \\\\ 270w+650r\\ge 1500\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167836585157\" data-alt=\"The figure shows the graph of the inequalities w plus r greater than or equals to four and two seventy w plus six fifty r greater than or equal to fifteen hundred. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_241_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities w plus r greater than or equals to four and two seventy w plus six fifty r greater than or equal to fifteen hundred. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> no\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> yes\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836615624\">\r\n<h4 data-type=\"title\">Writing Exercises<\/h4>\r\n<div data-type=\"exercise\" id=\"fs-id1167836554970\">\r\n<div data-type=\"problem\" id=\"fs-id1167836554972\">\r\n\r\nGraph the inequality \\(x-y\\ge 3.\\) How do you know which side of the line \\(x-y=3\\) should be shaded?\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836689087\">\r\n<div data-type=\"problem\" id=\"fs-id1167836689089\">\r\n<p id=\"fs-id1167830087738\">Graph the system \\(\\left\\{\\begin{array}{c}x+2y\\le 6\\hfill \\\\ y\\ge -\\frac{1}{2}x-4\\hfill \\end{array}.\\) What does the solution mean?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836502447\">\r\n<p id=\"fs-id1167836502449\">Answers will vary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833024556\">\r\n<h4 data-type=\"title\">Self Check<\/h4>\r\n<p id=\"fs-id1167836547975\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167824584580\" data-alt=\"The figure shows a table with four columns and four rows. The first row is the title row. The four titles are I can\u2026, confidently, with some help and No \u2013 I don\u2019t get it! In the second row of the first column, the text says \u2018determine whether an ordered pair is a solution of a system of linear inequalities.\u2019 In the third row of the first column, the text says \u2018solve applications of systems of inequalities.\u2019\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a table with four columns and four rows. The first row is the title row. The four titles are I can\u2026, confidently, with some help and No \u2013 I don\u2019t get it! In the second row of the first column, the text says \u2018determine whether an ordered pair is a solution of a system of linear inequalities.\u2019 In the third row of the first column, the text says \u2018solve applications of systems of inequalities.\u2019\" \/><\/span>\r\n<p id=\"fs-id1167836296437\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1167836390206\">\r\n<h3 data-type=\"title\">Chapter Review Exercises<\/h3>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836704892\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/a236420d-b79a-43e3-8f7c-1b8e18c5045e\" class=\"target-chapter\">Solve Systems of Linear Equations with Two Variables<\/a><\/h4>\r\n<p id=\"fs-id1167829644764\"><strong data-effect=\"bold\">Determine Whether an Ordered Pair is a Solution of a System of Equations<\/strong>.<\/p>\r\n<p id=\"fs-id1167836321756\">In the following exercises, determine if the following points are solutions to the given system of equations.<\/p>\r\n\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167833412561\">\r\n<p id=\"fs-id1167833412564\">\\(\\left\\{\\begin{array}{c}x+3y=-9\\hfill \\\\ 2x-4y=12\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(\\left(-3,-2\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(0,-3\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829809271\">\r\n<div data-type=\"problem\" id=\"fs-id1167829809273\">\r\n<p id=\"fs-id1167829809275\">\\(\\left\\{\\begin{array}{c}x+y=8\\hfill \\\\ y=x-4\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(\\left(6,2\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(9,-1\\right)\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829634159\">\r\n<p id=\"fs-id1167829580545\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829741600\"><strong data-effect=\"bold\">Solve a System of Linear Equations by Graphing<\/strong><\/p>\r\n<p id=\"fs-id1167836415885\">In the following exercises, solve the following systems of equations by graphing.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836390561\">\r\n<div data-type=\"problem\" id=\"fs-id1167836390563\">\r\n<p id=\"fs-id1167836390565\">\\(\\left\\{\\begin{array}{c}3x+y=6\\hfill \\\\ x+3y=-6\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836690408\">\r\n<div data-type=\"problem\" id=\"fs-id1167836690411\">\r\n<p id=\"fs-id1167836690413\">\\(\\left\\{\\begin{array}{c}x+4y=-1\\hfill \\\\ x=3\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836705411\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167829747204\" data-alt=\"The figure shows the graph of equations x plus four times y equal to minus one and x equal to three. Two intersecting lines are shown.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_322_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of equations x plus four times y equal to minus one and x equal to three. Two intersecting lines are shown.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\left(3,-1\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833030870\">\r\n<div data-type=\"problem\" id=\"fs-id1167833030872\">\r\n<p id=\"fs-id1167833030875\">\\(\\left\\{\\begin{array}{c}2x-y=5\\hfill \\\\ 4x-2y=10\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836689138\">\r\n<div data-type=\"problem\" id=\"fs-id1167836689140\">\r\n<p id=\"fs-id1167836689142\">\\(\\left\\{\\begin{array}{c}\\text{\u2212}x+2y=4\\hfill \\\\ y=\\frac{1}{2}x-3\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836789431\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167836789434\" data-alt=\"The figure shows the graph for the equations minus x plus two times y equal to four and y equal to half x minus three. Two parallel lines are shown.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_324_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph for the equations minus x plus two times y equal to four and y equal to half x minus three. Two parallel lines are shown.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nno solution\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836519173\">In the following exercises, without graphing determine the number of solutions and then classify the system of equations.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167825901073\">\r\n<div data-type=\"problem\" id=\"fs-id1167825901076\">\r\n<p id=\"fs-id1167825901078\">\\(\\left\\{\\begin{array}{c}y=\\frac{2}{5}x+2\\hfill \\\\ -2x+5y=10\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829719704\">\r\n<div data-type=\"problem\" id=\"fs-id1167833202536\">\r\n<p id=\"fs-id1167833202538\">\\(\\left\\{\\begin{array}{c}3x+2y=6\\hfill \\\\ y=-3x+4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829704947\">\r\n<p id=\"fs-id1167829704950\">one solution, consistent system, independent equations<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836283200\">\r\n<div data-type=\"problem\" id=\"fs-id1167836551623\">\r\n<p id=\"fs-id1167836551625\">\\(\\left\\{\\begin{array}{c}5x-4y=0\\hfill \\\\ y=\\frac{5}{4}x-5\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829739814\"><strong data-effect=\"bold\">Solve a System of Equations by Substitution<\/strong><\/p>\r\n<p id=\"fs-id1167829627719\">In the following exercises, solve the systems of equations by substitution.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829627722\">\r\n<div data-type=\"problem\" id=\"fs-id1167829832930\">\r\n<p id=\"fs-id1167829832932\">\\(\\left\\{\\begin{array}{c}3x-2y=2\\hfill \\\\ y=\\frac{1}{2}x+3\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836698903\">\r\n\r\n\\(\\left(4,5\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836544387\">\r\n<div data-type=\"problem\" id=\"fs-id1167836544389\">\r\n<p id=\"fs-id1167821895442\">\\(\\left\\{\\begin{array}{c}x-y=0\\hfill \\\\ 2x+5y=-14\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836386679\">\r\n<div data-type=\"problem\" id=\"fs-id1167836386681\">\r\n<p id=\"fs-id1167836386683\">\\(\\left\\{\\begin{array}{c}y=-2x+7\\hfill \\\\ y=\\frac{2}{3}x-1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836327160\">\r\n<p id=\"fs-id1167836327162\">\\(\\left(3,1\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826046358\">\r\n<div data-type=\"problem\" id=\"fs-id1167826046360\">\r\n<p id=\"fs-id1167836573256\">\\(\\left\\{\\begin{array}{c}y=-5x\\hfill \\\\ 5x+y=6\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836666697\">\r\n<div data-type=\"problem\" id=\"fs-id1167836666699\">\r\n<p id=\"fs-id1167836730049\">\\(\\left\\{\\begin{array}{c}y=-\\frac{1}{3}x+2\\hfill \\\\ x+3y=6\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826077166\">\r\n<p id=\"fs-id1167826077168\">infinitely many solutions<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836287290\"><strong data-effect=\"bold\">Solve a System of Equations by Elimination<\/strong><\/p>\r\n<p id=\"fs-id1167836287296\">In the following exercises, solve the systems of equations by elimination<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167824773957\">\r\n<div data-type=\"problem\" id=\"fs-id1167824773959\">\r\n<p id=\"fs-id1167824773961\">\\(\\left\\{\\begin{array}{c}x+y=12\\hfill \\\\ x-y=-10\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829737940\">\r\n<div data-type=\"problem\" id=\"fs-id1167826046341\">\r\n<p id=\"fs-id1167826046343\">\\(\\left\\{\\begin{array}{c}3x-8y=20\\hfill \\\\ x+3y=1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826132850\">\r\n<p id=\"fs-id1167833052812\">\\(\\left(4,-1\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829958013\">\r\n<div data-type=\"problem\" id=\"fs-id1167829958015\">\r\n<p id=\"fs-id1167829958017\">\\(\\left\\{\\begin{array}{c}9x+4y=2\\hfill \\\\ 5x+3y=5\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829627584\">\r\n<div data-type=\"problem\" id=\"fs-id1167829627586\">\r\n<p id=\"fs-id1167829627589\">\\(\\left\\{\\begin{array}{c}\\frac{1}{3}x-\\frac{1}{2}y=1\\hfill \\\\ \\frac{3}{4}x-y=\\frac{5}{2}\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829644664\">\r\n<p id=\"fs-id1167829644666\">\\(\\left(6,2\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836690368\">\r\n<div data-type=\"problem\" id=\"fs-id1167836690370\">\r\n<p id=\"fs-id1167833310434\">\\(\\left\\{\\begin{array}{c}\\text{\u2212}x+3y=8\\hfill \\\\ 2x-6y=-20\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836626839\"><strong data-effect=\"bold\">Choose the Most Convenient Method to Solve a System of Linear Equations<\/strong><\/p>\r\n<p id=\"fs-id1167836626846\">In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836687102\">\r\n<div data-type=\"problem\" id=\"fs-id1167836687104\">\r\n<p id=\"fs-id1167836687106\">\\(\\left\\{\\begin{array}{c}6x-5y=27\\hfill \\\\ 3x+10y=-24\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833239499\">\r\n<p id=\"fs-id1167833239501\">elimination<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833239507\">\r\n<div data-type=\"problem\" id=\"fs-id1167829619825\">\r\n<p id=\"fs-id1167829619827\">\\(\\left\\{\\begin{array}{c}y=3x-9\\hfill \\\\ 4x-5y=23\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836408465\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/560ed071-7c89-41ad-8282-94b385324dd9\" class=\"target-chapter\">Solve Applications with Systems of Equations<\/a><\/h4>\r\n<p id=\"fs-id1167836712373\"><strong data-effect=\"bold\">Solve Direct Translation Applications<\/strong><\/p>\r\n<p id=\"fs-id1167836712380\">In the following exercises, translate to a system of equations and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836415081\">\r\n<div data-type=\"problem\" id=\"fs-id1167836415083\">\r\n<p id=\"fs-id1167836415085\">Mollie wants to plant 200 bulbs in her garden, all irises and tulips. She wants to plant three times as many tulips as irises. How many irises and how many tulips should she plant?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836415091\">\r\n<p id=\"fs-id1167829747161\">50 irises and 150 tulips<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829747167\">\r\n<div data-type=\"problem\" id=\"fs-id1167829747169\">\r\n<p id=\"fs-id1167829747171\">Ashanti has been offered positions by two phone companies. The first company pays a salary of ?22,000 plus a commission of ?100 for each contract sold. The second pays a salary of ?28,000 plus a commission of ?25 for each contract sold. How many contract would need to be sold to make the total pay the same?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1167836611755\">\r\n<p id=\"fs-id1167836611757\">Leroy spent 20 minutes jogging and 40 minutes cycling and burned 600 calories. The next day, Leroy swapped times, doing 40 minutes of jogging and 20 minutes of cycling and burned the same number of calories. How many calories were burned for each minute of jogging and how many for each minute of cycling?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836611764\">\r\n<p id=\"fs-id1167836714034\">10 calories jogging and 10 calories cycling<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836714039\">\r\n<div data-type=\"problem\" id=\"fs-id1167836714042\">\r\n<p id=\"fs-id1167836714044\">Troy and Lisa were shopping for school supplies. Each purchased different quantities of the same notebook and calculator. Troy bought four notebooks and five calculators for ?116. Lisa bought two notebooks and three calculators for ?68. Find the cost of each notebook and each thumb drive.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836714046\"><strong data-effect=\"bold\">Solve Geometry Applications<\/strong><\/p>\r\n<p id=\"fs-id1167826206187\">In the following exercises, translate to a system of equations and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167826206190\">\r\n<div data-type=\"problem\" id=\"fs-id1167826206192\">\r\n<p id=\"fs-id1167826206194\">The difference of two supplementary angles is 58 degrees. Find the measures of the angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833057704\">\r\n<p id=\"fs-id1167833057706\">119, 61<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833057711\">\r\n<div data-type=\"problem\" id=\"fs-id1167836645569\">\r\n<p id=\"fs-id1167836645571\">Two angles are complementary. The measure of the larger angle is five more than four times the measure of the smaller angle. Find the measures of both angles.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833186148\">\r\n<div data-type=\"problem\" id=\"fs-id1167833186150\">\r\n<p id=\"fs-id1167833186152\">The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167826130274\">\r\n<p id=\"fs-id1167826130276\">\\(35\\text{\u00b0}\\) and \\(55\\text{\u00b0}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829810710\">\r\n<div data-type=\"problem\" id=\"fs-id1167829810712\">\r\n<p id=\"fs-id1167829810714\">Becca is hanging a 28 foot floral garland on the two sides and top of a pergola to prepare for a wedding. The height is four feet less than the width. Find the height and width of the pergola.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836620062\">\r\n<div data-type=\"problem\" id=\"fs-id1167836620064\">\r\n<p id=\"fs-id1167836620066\">The perimeter of a city rectangular park is 1428 feet. The length is 78 feet more than twice the width. Find the length and width of the park.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829790114\">\r\n<p id=\"fs-id1167829790116\">the length is 450 feet, the width is 264 feet<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829790121\"><strong data-effect=\"bold\">Solve Uniform Motion Applications<\/strong><\/p>\r\n<p id=\"fs-id1167825696609\">In the following exercises, translate to a system of equations and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167825696613\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167825696615\">\r\n<p id=\"fs-id1167825696617\">Sheila and Lenore were driving to their grandmother\u2019s house. Lenore left one hour after Sheila. Sheila drove at a rate of 45 mph, and Lenore drove at a rate of 60 mph. How long will it take for Lenore to catch up to Sheila?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833056470\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167833056472\">\r\n<p id=\"fs-id1167833056474\">Bob left home, riding his bike at a rate of 10 miles per hour to go to the lake. Cheryl, his wife, left 45 minutes \\(\\left(\\frac{3}{4}\\) hour) later, driving her car at a rate of 25 miles per hour. How long will it take Cheryl to catch up to Bob?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836684835\">\r\n<p id=\"fs-id1167824811868\">\\(\\frac{1}{2}\\) an hour<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836685478\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836685480\">\r\n<p id=\"fs-id1167836685482\">Marcus can drive his boat 36 miles down the river in three hours but takes four hours to return upstream. Find the rate of the boat in still water and the rate of the current.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836477237\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836477239\">\r\n\r\nA passenger jet can fly 804 miles in 2 hours with a tailwind but only 776 miles in 2 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829696497\">\r\n<p id=\"fs-id1167829696499\">the rate of the jet is 395 mph, the rate of the wind is 7 mph<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829696506\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/4cdf0dd0-1221-47ea-9e33-0562d98f6eec\" class=\"target-chapter\">Solve Mixture Applications with Systems of Equations<\/a><\/h4>\r\n<p id=\"fs-id1167825949254\"><strong data-effect=\"bold\">Solve Mixture Applications with Systems of Equations<\/strong><\/p>\r\n<p id=\"fs-id1167832951099\">For the following exercises, translate to a system of equations and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167832951102\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167832951104\">\r\n\r\nLynn paid a total of ?2,780 for 261 tickets to the theater. Student tickets cost ?10 and adult tickets cost ?15. How many student tickets and how many adult tickets did Lynn buy?\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836444773\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836502937\">\r\n<p id=\"fs-id1167836502939\">Priam has dimes and pennies in a cup holder in his car. The total value of the coins is ?4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836728066\">\r\n<p id=\"fs-id1167836728068\">41 dimes and 11 pennies<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836391929\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836391931\">\r\n<p id=\"fs-id1167836391933\">Yumi wants to make 12 cups of party mix using candies and nuts. Her budget requires the party mix to cost her ?1.29 per cup. The candies are ?2.49 per cup and the nuts are ?0.69 per cup. How many cups of candies and how many cups of nuts should she use?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836349275\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836349277\">\r\n<p id=\"fs-id1167836349279\">A scientist needs 70 liters of a 40% solution of alcohol. He has a 30% and a 60% solution available. How many liters of the 30% and how many liters of the 60% solutions should he mix to make the 40% solution?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167823012042\">\r\n<p id=\"fs-id1167823012045\">\\(46\\frac{2}{3}\\) liters of 30% solution, \\(23\\frac{1}{3}\\) liters of 60% solution<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167832925457\"><strong data-effect=\"bold\">Solve Interest Applications<\/strong><\/p>\r\n<p id=\"fs-id1167832925463\">For the following exercises, translate to a system of equations and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829859217\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829859219\">\r\n<p id=\"fs-id1167829859221\">Jack has ?12,000 to invest and wants to earn 7.5% interest per year. He will put some of the money into a savings account that earns 4% per year and the rest into CD account that earns 9% per year. How much money should he put into each account?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829924270\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829924273\">\r\n<p id=\"fs-id1167829924275\">When she graduates college, Linda will owe ?43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was ?1,585. What is the amount of each loan?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836556586\">\r\n<p id=\"fs-id1167836556588\">?29,000 for the federal loan, ?14,000 for the private loan<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836309828\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/56fc492e-b71a-4a4a-bab4-b95331687fd3\" class=\"target-chapter\">Solve Systems of Equations with Three Variables<\/a><\/h4>\r\n<p id=\"fs-id1167836387009\"><strong data-effect=\"bold\">Solve Systems of Equations with Three Variables<\/strong><\/p>\r\n<p id=\"fs-id1167836387016\">In the following exercises, determine whether the ordered triple is a solution to the system.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836387020\">\r\n<div data-type=\"problem\" id=\"fs-id1167829908520\">\r\n<p id=\"fs-id1167829908522\">\\(\\left\\{\\begin{array}{c}3x-4y-3z=2\\hfill \\\\ 2x-6y+z=3\\hfill \\\\ 2x+3y-2z=3\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(\\left(2,3,-1\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(3,1,3\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836705096\">\r\n<div data-type=\"problem\" id=\"fs-id1167836705098\">\r\n<p id=\"fs-id1167829957958\">\\(\\left\\{\\begin{array}{c}y=\\frac{2}{3}x-2\\hfill \\\\ x+3y-z=15\\hfill \\\\ x-3y+z=-2\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(\\left(-6,5,\\frac{1}{2}\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(5,\\frac{4}{3},-3\\right)\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836620221\">\r\n<p id=\"fs-id1167836620223\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167824738895\"><strong data-effect=\"bold\">Solve a System of Linear Equations with Three Variables<\/strong><\/p>\r\n<p id=\"fs-id1167829683650\">In the following exercises, solve the system of equations.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829683653\">\r\n<div data-type=\"problem\" id=\"fs-id1167829683656\">\r\n<p id=\"fs-id1167829683658\">\\(\\left\\{\\begin{array}{c}3x-5y+4z=5\\hfill \\\\ 5x+2y+z=0\\hfill \\\\ 2x+3y-2z=3\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829690837\">\r\n<div data-type=\"problem\" id=\"fs-id1167829690839\">\r\n<p id=\"fs-id1167829690841\">\\(\\left\\{\\begin{array}{c}x+\\frac{5}{2}y+z=-2\\hfill \\\\ 2x+2y+\\frac{1}{2}z=-4\\hfill \\\\ \\frac{1}{3}x-y-z=1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167824763564\">\r\n<p id=\"fs-id1167824763566\">\\(\\left(-3,2,-4\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836322233\">\r\n<div data-type=\"problem\" id=\"fs-id1167836322235\">\r\n<p id=\"fs-id1167836322237\">\\(\\left\\{\\begin{array}{c}5x+3y=-6\\hfill \\\\ 2y+3z=-1\\hfill \\\\ 7x+z=1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833240289\">\r\n<div data-type=\"problem\" id=\"fs-id1167829748636\">\r\n<p id=\"fs-id1167829748638\">\\(\\left\\{\\begin{array}{c}2x+3y+z=12\\hfill \\\\ x+y+z=9\\hfill \\\\ 3x+4y+2z=20\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832940462\">\r\n<p id=\"fs-id1167832940465\">no solution<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832940470\">\r\n<div data-type=\"problem\" id=\"fs-id1167836792711\">\r\n<p id=\"fs-id1167836792713\">\\(\\left\\{\\begin{array}{c}\\text{\u2212}x-3y+2z=14\\hfill \\\\ \\text{\u2212}x+2y-3z=-4\\hfill \\\\ 3x+y-2z=6\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836387828\"><strong data-effect=\"bold\">Solve Applications using Systems of Linear Equations with Three Variables<\/strong><\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836387834\">\r\n<div data-type=\"problem\" id=\"fs-id1167836387836\">\r\n<p id=\"fs-id1167829597480\">After attending a major league baseball game, the patrons often purchase souvenirs. If a family purchases 4 t-shirts, a cap and 1 stuffed animal their total is ?135. A couple buys 2 t-shirts, a cap and 3 stuffed animals for their nieces and spends ?115. Another couple buys 2 t-shirts, a cap and 1 stuffed animal and their total is ?85. What is the cost of each item?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829597487\">\r\n<p id=\"fs-id1167829597489\">25, 20, 15<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829689026\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/6a4b38a5-e5d8-4f37-ac52-172097013b24\" class=\"target-chapter\">Solve Systems of Equations Using Matrices<\/a><\/h4>\r\n<p id=\"fs-id1167826188078\"><strong data-effect=\"bold\">Write the Augmented Matrix for a System of Equations.<\/strong><\/p>\r\n<p id=\"fs-id1167826188084\">Write each system of linear equations as an augmented matrix.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167826188087\">\r\n<div data-type=\"problem\" id=\"fs-id1167825824095\">\r\n<p id=\"fs-id1167825824097\">\\(\\left\\{\\begin{array}{c}3x-y=-1\\hfill \\\\ -2x+2y=5\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833086715\">\r\n<div data-type=\"problem\" id=\"fs-id1167833086717\">\r\n<p id=\"fs-id1167833086720\">\\(\\left\\{\\begin{array}{c}4x+3y=-2\\hfill \\\\ x-2y-3z=7\\hfill \\\\ 2x-y+2z=-6\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829696207\">\r\n<p id=\"fs-id1167829696209\">\\(\\left[\\begin{array}{cccccccccc}\\hfill 4&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -2\\\\ \\hfill 1&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 7\\\\ \\hfill 2&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -6\\end{array}\\right]\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829893644\">Write the system of equations that that corresponds to the augmented matrix.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829893647\">\r\n<div data-type=\"problem\" id=\"fs-id1167829893649\">\r\n<p id=\"fs-id1167836613622\">\\(\\left[\\begin{array}{cccc}2\\hfill &amp; &amp; &amp; \\hfill -4\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -3\\end{array}\\phantom{\\rule{1em}{0ex}}|\\phantom{\\rule{1em}{0ex}}\\begin{array}{c}\\hfill -2\\\\ \\hfill -1\\end{array}\\right]\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832977348\">\r\n<div data-type=\"problem\" id=\"fs-id1167832977350\">\r\n<p id=\"fs-id1167832977352\">\\(\\left[\\begin{array}{ccccccc}\\hfill 1&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -3\\\\ \\hfill 1&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 0\\\\ \\hfill 0&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 2\\end{array}\\phantom{\\rule{1em}{0ex}}|\\phantom{\\rule{1em}{0ex}}\\begin{array}{c}\\hfill -1\\\\ \\hfill -2\\\\ \\hfill 3\\end{array}\\right]\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167825703256\">\r\n<p id=\"fs-id1167833380503\">\\(\\left\\{\\begin{array}{c}x-3z=-1\\hfill \\\\ x-2y=-27\\hfill \\\\ \\text{\u2212}y+2z=3\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167833186073\">In the following exercises, perform the indicated operations on the augmented matrices.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167833186076\">\r\n<div data-type=\"problem\" id=\"fs-id1167833186078\">\r\n<p id=\"fs-id1167833186080\">\\(\\left[\\begin{array}{cccc}4\\hfill &amp; &amp; &amp; \\hfill -6\\\\ 3\\hfill &amp; &amp; &amp; \\hfill 2\\end{array}\\phantom{\\rule{1em}{0ex}}|\\phantom{\\rule{1em}{0ex}}\\begin{array}{c}\\hfill -3\\\\ \\hfill 1\\end{array}\\right]\\)<\/p>\r\n<p id=\"fs-id1167824780927\"><span class=\"token\">\u24d0<\/span> Interchange rows 2 and 1.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Multiply row 1 by 4.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Multiply row 2 by 3 and add to row 1.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836573286\">\r\n<div data-type=\"problem\" id=\"fs-id1167833142230\">\r\n<p id=\"fs-id1167833142232\">\\(\\left[\\begin{array}{ccccccc}1\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -2\\\\ 2\\hfill &amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -1\\\\ 4\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -3\\end{array}\\phantom{\\rule{1em}{0ex}}|\\phantom{\\rule{1em}{0ex}}\\begin{array}{c}\\hfill 4\\\\ \\hfill -3\\\\ \\hfill -1\\end{array}\\right]\\)<\/p>\r\n<p id=\"fs-id1167836752836\"><span class=\"token\">\u24d0<\/span> Interchange rows 2 and 3.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Multiply row 1 by 2.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Multiply row 3 by \\(-2\\) and add to row 2.\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167824774461\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(\\left[\\begin{array}{cccccccccc}1\\hfill &amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 4\\\\ 4\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -1\\\\ 2\\hfill &amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -3\\end{array}\\right]\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left[\\begin{array}{cccccccccc}2\\hfill &amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill 8\\\\ 4\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -1\\\\ 2\\hfill &amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -3\\end{array}\\right]\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(\\left[\\begin{array}{cccccccccc}2\\hfill &amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill 8\\\\ 4\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -1\\\\ 0\\hfill &amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 5\\end{array}\\right]\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836791631\"><strong data-effect=\"bold\">Solve Systems of Equations Using Matrices<\/strong><\/p>\r\n<p id=\"fs-id1167836791637\">In the following exercises, solve each system of equations using a matrix.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829594309\">\r\n<div data-type=\"problem\" id=\"fs-id1167829594311\">\r\n<p id=\"fs-id1167829594313\">\\(\\left\\{\\begin{array}{c}4x+y=6\\hfill \\\\ x-y=4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836293784\">\r\n<div data-type=\"problem\" id=\"fs-id1167836293786\">\r\n<p id=\"fs-id1167836293788\">\\(\\left\\{\\begin{array}{c}2x-y+3z=-3\\hfill \\\\ \\text{\u2212}x+2y-z=10\\hfill \\\\ x+y+z=5\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836292856\">\r\n<p id=\"fs-id1167836292859\">\\(\\left(-2,5,-2\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836341414\">\r\n<div data-type=\"problem\" id=\"fs-id1167836432792\">\r\n<p id=\"fs-id1167836432794\">\\(\\left\\{\\begin{array}{c}2y+3z=-1\\hfill \\\\ 5x+3y=-6\\hfill \\\\ 7x+z=1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833310524\">\r\n<div data-type=\"problem\" id=\"fs-id1167833310526\">\r\n<p id=\"fs-id1167833310529\">\\(\\left\\{\\begin{array}{c}x+2y-3z=-1\\hfill \\\\ x-3y+z=1\\hfill \\\\ 2x-y-2z=2\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833274705\">\r\n<p id=\"fs-id1167833274707\">no solution<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829715988\">\r\n<div data-type=\"problem\" id=\"fs-id1167829715990\">\r\n<p id=\"fs-id1167829715993\">\\(\\left\\{\\begin{array}{c}x+y-3z=-1\\hfill \\\\ y-z=0\\hfill \\\\ \\text{\u2212}x+2y=1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829785269\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/0ab06d7a-f85b-4c4d-b56b-aa826ea612a4\" class=\"target-chapter\">Solve Systems of Equations Using Determinants<\/a><\/h4>\r\n<p id=\"fs-id1167830092974\"><strong data-effect=\"bold\">Evaluate the Determinant of a 2 \u00d7 2 Matrix<\/strong><\/p>\r\n<p id=\"fs-id1167830092980\">In the following exercise, evaluate the determinate of the square matrix.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167830092984\">\r\n<div data-type=\"problem\" id=\"fs-id1167830092986\">\r\n\r\n\\(\\left[\\begin{array}{cccc}8\\hfill &amp; &amp; &amp; \\hfill -4\\\\ 5\\hfill &amp; &amp; &amp; \\hfill -3\\end{array}\\right]\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836685512\">\r\n<p id=\"fs-id1167836685514\">\\(-4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836753517\"><strong data-effect=\"bold\">Evaluate the Determinant of a 3 \u00d7 3 Matrix<\/strong><\/p>\r\n<p id=\"fs-id1167836753524\">In the following exercise, find and then evaluate the indicated minors.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836753527\">\r\n<div data-type=\"problem\" id=\"fs-id1167836753529\">\r\n<p id=\"fs-id1167829942346\">\\(|\\begin{array}{ccccccc}\\hfill -1&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 2\\\\ \\hfill 4&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -1\\\\ \\hfill -2&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -3\\end{array}|;\\) Find the minor <span class=\"token\">\u24d0<\/span> \\({a}_{1}\\) <span class=\"token\">\u24d1<\/span> \\({b}_{1}\\) <span class=\"token\">\u24d2<\/span> \\({c}_{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167825024028\">In the following exercise, evaluate each determinant by expanding by minors along the first row.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836616233\">\r\n<div data-type=\"problem\" id=\"fs-id1167836616235\">\r\n<p id=\"fs-id1167836616237\">\\(|\\begin{array}{ccccccc}\\hfill -2&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -4\\\\ \\hfill 5&amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill 7\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 0\\end{array}|\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167824732987\">\r\n<p id=\"fs-id1167824732989\">21<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829859252\">In the following exercise, evaluate each determinant by expanding by minors.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829859255\">\r\n<div data-type=\"problem\" id=\"fs-id1167829859257\">\r\n<p id=\"fs-id1167829859259\">\\(|\\begin{array}{ccccccc}\\hfill 3&amp; &amp; &amp; \\hfill 5&amp; &amp; &amp; \\hfill 4\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill 0\\\\ \\hfill -2&amp; &amp; &amp; \\hfill 6&amp; &amp; &amp; \\hfill 1\\end{array}|\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167826030458\"><strong data-effect=\"bold\">Use Cramer\u2019s Rule to Solve Systems of Equations<\/strong><\/p>\r\n<p id=\"fs-id1167826030465\">In the following exercises, solve each system of equations using Cramer\u2019s rule<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167824710198\">\r\n<div data-type=\"problem\" id=\"fs-id1167824710200\">\r\n<p id=\"fs-id1167824710202\">\\(\\left\\{\\begin{array}{c}x-3y=-9\\hfill \\\\ 2x+5y=4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836574872\">\r\n<p id=\"fs-id1167836574874\">\\(\\left(-3,2\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826206229\">\r\n<div data-type=\"problem\" id=\"fs-id1167836611683\">\r\n<p id=\"fs-id1167836611685\">\\(\\left\\{\\begin{array}{c}4x-3y+z=7\\hfill \\\\ 2x-5y-4z=3\\hfill \\\\ 3x-2y-2z=-7\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829807833\">\r\n<div data-type=\"problem\" id=\"fs-id1167829807835\">\r\n<p id=\"fs-id1167836611137\">\\(\\left\\{\\begin{array}{c}2x+5y=4\\hfill \\\\ 3y-z=3\\hfill \\\\ 4x+3z=-3\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836787589\">\r\n<p id=\"fs-id1167836787592\">\\(\\left(-3,2,3\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829894130\">\r\n<div data-type=\"problem\" id=\"fs-id1167829894132\">\r\n<p id=\"fs-id1167829894134\">\\(\\left\\{\\begin{array}{c}x+y-3z=-1\\hfill \\\\ y-z=0\\hfill \\\\ \\text{\u2212}x+2y=1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829737696\">\r\n<div data-type=\"problem\" id=\"fs-id1167829737698\">\r\n<p id=\"fs-id1167829737700\">\\(\\left\\{\\begin{array}{c}3x+4y-3z=-2\\hfill \\\\ 2x+3y-z=-12\\hfill \\\\ x+y-2z=6\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836689272\">\r\n<p id=\"fs-id1167836689274\">inconsistent<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836363671\"><strong data-effect=\"bold\">Solve Applications Using Determinants<\/strong><\/p>\r\n<p id=\"fs-id1167836363677\">In the following exercises, determine whether the given points are collinear.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836549078\">\r\n<div data-type=\"problem\" id=\"fs-id1167836549080\">\r\n<p id=\"fs-id1167836549082\">\\(\\left(0,2\\right),\\)\\(\\left(-1,-1\\right),\\) and \\(\\left(-2,4\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167826169690\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/ccd859e9-a2f9-4004-abe9-5823a43ab70f\" class=\"target-chapter\">Graphing Systems of Linear Inequalities<\/a><\/h4>\r\n<strong data-effect=\"bold\">Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities<\/strong>\r\n<p id=\"fs-id1167836331129\">In the following exercises, determine whether each ordered pair is a solution to the system.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836712344\">\r\n<div data-type=\"problem\" id=\"fs-id1167836712346\">\r\n<p id=\"fs-id1167836712348\">\\(\\left\\{\\begin{array}{c}4x+y&gt;6\\hfill \\\\ 3x-y\\le 12\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167824734409\"><span class=\"token\">\u24d0<\/span>\\(\\left(2,-1\\right)\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(3,-2\\right)\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833257706\">\r\n<p id=\"fs-id1167833257708\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836320828\">\r\n<div data-type=\"problem\" id=\"fs-id1167836755237\">\r\n<p id=\"fs-id1167836755239\">\\(\\left\\{\\begin{array}{c}y&gt;\\frac{1}{3}x+2\\hfill \\\\ x-\\frac{1}{4}y\\le 10\\hfill \\end{array}\\)<\/p>\r\n<p id=\"fs-id1167836730240\"><span class=\"token\">\u24d0<\/span>\\(\\left(6,5\\right)\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(15,8\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167833347098\"><strong data-effect=\"bold\">Solve a System of Linear Inequalities by Graphing<\/strong><\/p>\r\n<p id=\"fs-id1167833347105\">In the following exercises, solve each system by graphing.<\/p>\r\n\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167826172302\">\\(\\left\\{\\begin{array}{c}y&lt;3x+1\\hfill \\\\ y\\ge -x-2\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829716918\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167829716921\" data-alt=\"The figure shows the graph of inequalities y less than three times x plus one and y greater than or equal to minus x minus two. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_202_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of inequalities y less than three times x plus one and y greater than or equal to minus x minus two. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167824578795\">\r\n<div data-type=\"problem\" id=\"fs-id1167824578797\">\r\n<p id=\"fs-id1167824578799\">\\(\\left\\{\\begin{array}{c}x-y&gt;-1\\hfill \\\\ y&lt;\\frac{1}{3}x-2\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826171494\">\r\n<div data-type=\"problem\" id=\"fs-id1167826171496\">\r\n<p id=\"fs-id1167829808730\">\\(\\left\\{\\begin{array}{c}2x-3y&lt;6\\hfill \\\\ 3x+4y\\ge 12\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829628080\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167829908492\" data-alt=\"The figure shows the graph of inequalities two times x minus three times y less six and three times x plus four times y greater than or equal to twelve. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_204_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of inequalities two times x minus three times y less six and three times x plus four times y greater than or equal to twelve. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nThe solution is the grey region.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836626881\">\r\n<div data-type=\"problem\" id=\"fs-id1167836626883\">\r\n<p id=\"fs-id1167836626885\">\\(\\left\\{\\begin{array}{c}y\\le -\\frac{3}{4}x+1\\hfill \\\\ x\\ge -5\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833129094\">\r\n<div data-type=\"problem\" id=\"fs-id1167833129096\">\r\n<p id=\"fs-id1167833129098\">\\(\\left\\{\\begin{array}{c}x+3y&lt;5\\hfill \\\\ y\\ge -\\frac{1}{3}x+6\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836798046\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167832966114\" data-alt=\"The figure shows the graph of inequalities x plus three times y less than five and y greater than or equal to minus one third x plus six. Two parallel lines, one in red and the other in blue, are shown. An area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_206_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of inequalities x plus three times y less than five and y greater than or equal to minus one third x plus six. Two parallel lines, one in red and the other in blue, are shown. An area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\nNo solution.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836692578\">\r\n<div data-type=\"problem\" id=\"fs-id1167836692580\">\r\n<p id=\"fs-id1167836692582\">\\(\\left\\{\\begin{array}{c}y\\ge 2x-5\\hfill \\\\ -6x+3y&gt;-4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836731539\"><strong data-effect=\"bold\">Solve Applications of Systems of Inequalities<\/strong><\/p>\r\n<p id=\"fs-id1167836731546\">In the following exercises, translate to a system of inequalities and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836529685\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167836529687\">\r\n<p id=\"fs-id1167836529689\">Roxana makes bracelets and necklaces and sells them at the farmers\u2019 market. She sells the bracelets for ?12 each and the necklaces for ?18 each. At the market next weekend she will have room to display no more than 40 pieces, and she needs to sell at least ?500 worth in order to earn a profit.<\/p>\r\n<p id=\"fs-id1167836529692\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Should she display 26 bracelets and 14 necklaces?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Should she display 39 bracelets and 1 necklace?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836616137\">\r\n<p id=\"fs-id1167836616139\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}b\\ge 0\\hfill \\\\ n\\ge 0\\hfill \\\\ b+n\\le 40\\hfill \\\\ 12b+18n\\ge 500\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167826131486\" data-alt=\"The figure shows the graph of b plus n equal to forty and twelve b plus eighteen n equal to five hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_315_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of b plus n equal to forty and twelve b plus eighteen n equal to five hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> yes\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> no\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829752821\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1167829752823\">\r\n<p id=\"fs-id1167829752825\">Annie has a budget of ?600 to purchase paperback books and hardcover books for her classroom. She wants the number of hardcover to be at least 5 more than three times the number of paperback books. Paperback books cost ?4 each and hardcover books cost ?15 each.<\/p>\r\n<p id=\"fs-id1167832989101\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Can she buy 8 paperback books and 40 hardcover books?\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Can she buy 10 paperback books and 37 hardcover books?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1167833346751\">\r\n<h3 data-type=\"title\">Chapter Practice Test<\/h3>\r\n<p id=\"fs-id1167829906586\">In the following exercises, solve the following systems by graphing.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829906589\">\r\n<div data-type=\"problem\" id=\"fs-id1167829906591\">\r\n<p id=\"fs-id1167829906593\">\\(\\left\\{\\begin{array}{c}x-y=5\\hfill \\\\ x+2y=-4\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836663833\">\r\n\r\n<span data-type=\"media\" id=\"fs-id1167833076666\" data-alt=\"The figure shows the graph of inequalities h equal to three p plus five and four times p plus fifteen times h equal to six hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_317_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of inequalities h equal to three p plus five and four times p plus fifteen times h equal to six hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\left(2,-3\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829877873\">\r\n<div data-type=\"problem\" id=\"fs-id1167829877876\">\r\n<p id=\"fs-id1167833158737\">\\(\\left\\{\\begin{array}{c}x-y&gt;-2\\hfill \\\\ y\\le 3x+1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167833339402\">In the following exercises, solve each system of equations. Use either substitution or elimination.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829908677\">\r\n<div data-type=\"problem\" id=\"fs-id1167829908679\">\r\n<p id=\"fs-id1167829908681\">\\(\\left\\{\\begin{array}{c}x+4y=6\\hfill \\\\ -2x+y=-3\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836705196\">\r\n<p id=\"fs-id1167836705198\">\\(\\left(2,1\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829624883\">\r\n<div data-type=\"problem\" id=\"fs-id1167829624885\">\r\n<p id=\"fs-id1167829624887\">\\(\\left\\{\\begin{array}{c}-3x+4y=25\\hfill \\\\ x-5y=-23\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833058733\">\r\n<div data-type=\"problem\" id=\"fs-id1167829808045\">\r\n<p id=\"fs-id1167829808047\">\\(\\left\\{\\begin{array}{c}x+y-z=-1\\hfill \\\\ 2x-y+2z=8\\hfill \\\\ -3x+2y+z=-9\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829879454\">\r\n<p id=\"fs-id1167833326439\">\\(\\left(2,-2,1\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167826206143\">Solve the system of equations using a matrix.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836525066\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1167836525070\">\\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836485678\">\r\n<div data-type=\"problem\" id=\"fs-id1167836485680\">\r\n<p id=\"fs-id1167836485682\">\\(\\left\\{\\begin{array}{c}-3x+y+z=-4\\hfill \\\\ \\text{\u2212}x+2y-2z=1\\hfill \\\\ 2x-y-z=-1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833141974\">\r\n<p id=\"fs-id1167833141977\">\\(\\left(5,7,4\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836738051\">Solve using Cramer\u2019s rule.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836738055\">\r\n<div data-type=\"problem\" id=\"fs-id1167836738057\">\r\n<p id=\"fs-id1167829850382\">\\(\\left\\{\\begin{array}{c}3x+y=-3\\hfill \\\\ 2x+3y=6\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836619203\">\r\n<div data-type=\"problem\" id=\"fs-id1167836619205\">\r\n<p id=\"fs-id1167836540111\">Evaluate the determinant by expanding by minors:<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(|\\begin{array}{ccccccc}\\hfill 3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -2\\\\ \\hfill 2&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 4\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -3\\end{array}|.\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836549180\">\r\n<p id=\"fs-id1167836549182\">\\(9\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829746255\">In the following exercises, translate to a system of equations and solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829746258\">\r\n<div data-type=\"problem\" id=\"fs-id1167829746261\">\r\n<p id=\"fs-id1167836530781\">Greg is paddling his canoe upstream, against the current, to a fishing spot 10 miles away. If he paddles upstream for 2.5 hours and his return trip takes 1.25 hours, find the speed of the current and his paddling speed in still water.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167826204892\">\r\n<div data-type=\"problem\" id=\"fs-id1167826204895\">\r\n<p id=\"fs-id1167833169125\">A pharmacist needs 20 liters of a 2% saline solution. He has a 1% and a 5% solution available. How many liters of the 1% and how many liters of the 5% solutions should she mix to make the 2% solution?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833169131\">\r\n<p id=\"fs-id1167833169133\">15 liters of 1% solution, 5 liters of 5% solution<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833339844\">\r\n<div data-type=\"problem\" id=\"fs-id1167833339846\">\r\n<p id=\"fs-id1167833339848\">Arnold invested ?64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received ?4,500 in interest in one year?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829696408\">\r\n<div data-type=\"problem\" id=\"fs-id1167829696410\">\r\n<p id=\"fs-id1167829696413\">The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of ?65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of ?140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 can of popcorn for a total sales of ?250. What is the cost of each item?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829719019\">\r\n<p id=\"fs-id1167829719021\">The candy cost ?20; the cookies cost ?5; and the popcorn cost ?10.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829696415\">\r\n<div data-type=\"problem\" id=\"fs-id1167829696417\">\r\n<p id=\"fs-id1167836514875\">The manufacturer of a granola bar spends ?1.20 to make each bar and sells them for ?2. The manufacturer also has fixed costs each month of ?8,000.<\/p>\r\n<span class=\"token\">\u24d0<\/span> Find the cost function <em data-effect=\"italics\">C<\/em> when <em data-effect=\"italics\">x<\/em> granola bars are manufactured\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Find the revenue function <em data-effect=\"italics\">R<\/em> when <em data-effect=\"italics\">x<\/em> granola bars are sold.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Show the break-even point by graphing both the Revenue and Cost functions on the same grid.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> Find the break-even point. Interpret what the break-even point means.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829720516\">\r\n<div data-type=\"problem\" id=\"fs-id1167829720518\">\r\n<p id=\"fs-id1167829720520\">Translate to a system of inequalities and solve.<\/p>\r\n<p id=\"fs-id1167829720524\">Andi wants to spend no more than ?50 on Halloween treats. She wants to buy candy bars that cost ?1 each and lollipops that cost ?0.50 each, and she wants the number of lollipops to be at least three times the number of candy bars.<\/p>\r\n<p id=\"fs-id1167836729309\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Graph the system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> Can she buy 20 candy bars and 40 lollipops?\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836399851\">\r\n<p id=\"fs-id1167836399853\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}C\\ge 0\\hfill \\\\ L\\ge 0\\hfill \\\\ C+0.5L\\le 50\\hfill \\\\ L\\ge 3C\\hfill \\end{array}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829627822\" data-alt=\"The figure shows the graph of two equations. Two intersecting lines, one in red and the other in blue, are shown. The red line passes through origin. An area is shown in grey.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_319_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of two equations. Two intersecting lines, one in red and the other in blue, are shown. The red line passes through origin. An area is shown in grey.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> no\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> yes\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"glossary\" class=\"textbox shaded\">\r\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\r\n<dl id=\"fs-id1167824740928\">\r\n \t<dt>system of linear inequalities<\/dt>\r\n \t<dd id=\"fs-id1167824740934\">Two or more linear inequalities grouped together form a system of linear inequalities.<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Determine whether an ordered pair is a solution of a system of linear inequalities<\/li>\n<li>Solve a system of linear inequalities by graphing<\/li>\n<li>Solve applications of systems of inequalities<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829586823\" class=\"be-prepared\">\n<p id=\"fs-id1167836535609\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167829905390\" type=\"1\">\n<li>Solve the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0bbcacebb4e493a8583e39734c5dae8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;&#60;&#53;&#97;&#43;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"104\" style=\"vertical-align: -2px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835324646\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Determine whether the ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f8eee42c12e59b5fdd0c4faa2504837_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/> is a solution to the system <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e318fca1ca20ca28c799a5d0a826873_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#50;&#120;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/f15d09b6-aae4-4ad3-a9e6-c9d1ac2436e3#fs-id1167835253977\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836521500\">\n<h3 data-type=\"title\">Determine whether an ordered pair is a solution of a system of linear inequalities<\/h3>\n<p id=\"fs-id1167836550209\">The definition of a <span data-type=\"term\">system of linear inequalities<\/span> is very similar to the definition of a system of linear equations.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836791176\">\n<div data-type=\"title\">System of Linear Inequalities<\/div>\n<p id=\"fs-id1167833329231\">Two or more linear inequalities grouped together form a system of linear inequalities.<\/p>\n<\/div>\n<p>A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836441020\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7ccc7d8506205cc0bbc13b242bfb269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#52;&#121;&#92;&#103;&#101;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#60;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1167829597513\">To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> that make both inequalities true.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829717259\">\n<div data-type=\"title\">Solutions of a System of Linear Inequalities<\/div>\n<p id=\"fs-id1167826170280\">Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.<\/p>\n<p>The solution of a system of linear inequalities is shown as a shaded region in the <em data-effect=\"italics\">x, y<\/em> coordinate system that includes all the points whose ordered pairs make the inequalities true.<\/p>\n<\/div>\n<p>To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829650722\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167826132435\">\n<div data-type=\"problem\" id=\"fs-id1167829717048\">\n<p id=\"fs-id1167836549293\">Determine whether the ordered pair is a solution to the system <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-374c61b35b62d587a710ed3e0b6e1ade_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#52;&#121;&#92;&#103;&#101;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#60;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167836663929\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ab12b849de44ae6a90ccee83f8e1018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4307e3633578c7e56f8f767895b20497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833338505\">\n<p id=\"fs-id1167836737853\"><span class=\"token\">\u24d0<\/span> Is the ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ab12b849de44ae6a90ccee83f8e1018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> a solution?<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832935757\" data-alt=\"We substitute x equal to negative 2 and y equal to 4 into both inequalities. First inequality is x plus 4 times y greater than or equal to 10. So negative 2 plus 4 open parentheses 4 close parenthesis is greater than or equal to 10 or not. 14 is greater than or equal to 10 is true. Second inequality, 3 times x minus 2 times y is less than 12. Three open parentheses negative 2 close parentheses minus two open parentheses 4 close parentheses is less than 12 or not. Negative 14 is less than 12 is true.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"We substitute x equal to negative 2 and y equal to 4 into both inequalities. First inequality is x plus 4 times y greater than or equal to 10. So negative 2 plus 4 open parentheses 4 close parenthesis is greater than or equal to 10 or not. 14 is greater than or equal to 10 is true. Second inequality, 3 times x minus 2 times y is less than 12. Three open parentheses negative 2 close parentheses minus two open parentheses 4 close parentheses is less than 12 or not. Negative 14 is less than 12 is true.\" \/><\/span><\/p>\n<p id=\"fs-id1167836688862\">The ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ab12b849de44ae6a90ccee83f8e1018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> made both inequalities true. Therefore <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ab12b849de44ae6a90ccee83f8e1018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> is a solution to this system.<\/p>\n<p id=\"fs-id1167833018011\"><span class=\"token\">\u24d1<\/span> Is the ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4307e3633578c7e56f8f767895b20497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> a solution?<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836621274\" data-alt=\"We substitute x equal to three and y equal to one into both inequalities. First inequality is x plus four times y greater than or equal to ten. So three plus four open parentheses one close parenthesis is greater than or equal to ten or not. Seven greater than or equal to ten is false. Second inequality, three times x minus two times y is less than twelve. Three open parentheses three close parentheses minus two open parentheses one close parentheses is less than twelve or not. Seven less than 12 holds true.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_002_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"We substitute x equal to three and y equal to one into both inequalities. First inequality is x plus four times y greater than or equal to ten. So three plus four open parentheses one close parenthesis is greater than or equal to ten or not. Seven greater than or equal to ten is false. Second inequality, three times x minus two times y is less than twelve. Three open parentheses three close parentheses minus two open parentheses one close parentheses is less than twelve or not. Seven less than 12 holds true.\" \/><\/span><\/p>\n<p id=\"fs-id1167833138779\">The ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4307e3633578c7e56f8f767895b20497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> made one inequality true, but the other one false. Therefore <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4307e3633578c7e56f8f767895b20497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is not a solution to this system.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836515280\">\n<div data-type=\"problem\" id=\"fs-id1167836544143\">\n<p>Determine whether the ordered pair is a solution to the system: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2bd4d67134771bc6b835924f40efb12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#53;&#121;&#62;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#62;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"137\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167833060161\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a00a53e3c5401663e307951ae0d74c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750280\">\n<p><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829580320\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832981652\">\n<div data-type=\"problem\" id=\"fs-id1167833364848\">\n<p>Determine whether the ordered pair is a solution to the system: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-706224e73359bdaf1d22450b640a4ba6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#62;&#52;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#121;&#60;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"123\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167836554187\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e86393e45c0f6cf9bb7fcf130d3db9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6de38faced02869069c306c45a22c5e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836534637\">\n<p id=\"fs-id1167833023095\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836568053\">\n<h3 data-type=\"title\">Solve a System of Linear Inequalities by Graphing<\/h3>\n<p id=\"fs-id1167829906617\">The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836792628\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a System of Linear Inequalities by Graphing<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833053710\">\n<div data-type=\"problem\" id=\"fs-id1167829719558\">\n<p id=\"fs-id1167833138397\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12e86e57426578442cfc10f599e7f0a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#50;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836558351\"><span data-type=\"media\" id=\"fs-id1167833328924\" data-alt=\"Step 1. Graph the first inequality. We graph y less than 2x minus 1. Graph the boundary line y equal to 2x minus 1. It is a solid line because the inequality sign is less than. Shade in the side of the boundary line where the inequality is true. We choose 0, 0 as a test point. It is a solution to the equation, so we shade in above the boundary line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_003a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1. Graph the first inequality. We graph y less than 2x minus 1. Graph the boundary line y equal to 2x minus 1. It is a solid line because the inequality sign is less than. Shade in the side of the boundary line where the inequality is true. We choose 0, 0 as a test point. It is a solution to the equation, so we shade in above the boundary line.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829745782\" data-alt=\"Step 2. On the same grid, graph the second inequality y less than x plus 1. Graph the boundary line y equal to x plus 1. It is a dashed line because the inequality sign is less than. Shade in the side of that boundary line where the inequality is true. Again, we use 0, 0 as a test point. It is a solution so we shade below the line y equals x plus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_003b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2. On the same grid, graph the second inequality y less than x plus 1. Graph the boundary line y equal to x plus 1. It is a dashed line because the inequality sign is less than. Shade in the side of that boundary line where the inequality is true. Again, we use 0, 0 as a test point. It is a solution so we shade below the line y equals x plus 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829930090\" data-alt=\"Step 3. The solution is the region where the shading overlaps. The point where the boundary lines intersect is not a solution because it is not a solution to y less than x plus 1. The solution is all points in the area bounded by the lines on the bottom left.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_003c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3. The solution is the region where the shading overlaps. The point where the boundary lines intersect is not a solution because it is not a solution to y less than x plus 1. The solution is all points in the area bounded by the lines on the bottom left.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167833186666\" data-alt=\"Step 4. Check by choosing a test point. We use minus 1, minus 1. Substituting in the inequality y less than 2x minus 1, we get minus 1 less than minus 3 which is true. Hence, it is a solution. Similarly, it is also true for the other inequality. The region containing minus 1, minus 1 is the solution to this system.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_003d_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. Check by choosing a test point. We use minus 1, minus 1. Substituting in the inequality y less than 2x minus 1, we get minus 1 less than minus 3 which is true. Hence, it is a solution. Similarly, it is also true for the other inequality. The region containing minus 1, minus 1 is the solution to this system.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833022166\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836299481\">\n<p id=\"fs-id1167836609707\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95e704817c8c48e83fb1433c37aabc60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#60;&#51;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#62;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836543963\">\n<p><span data-type=\"media\" id=\"fs-id1167836600273\" data-alt=\"The figure shows a graph plotted for the inequalities y less than three times x plus two and y greater than minus x minus one. Two lines intersect each other on the graph. An area to the right of both the lines is colored in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_301_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph plotted for the inequalities y less than three times x plus two and y greater than minus x minus one. Two lines intersect each other on the graph. An area to the right of both the lines is colored in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836523171\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836667050\">\n<div data-type=\"problem\" id=\"fs-id1167825913897\">\n<p id=\"fs-id1167833382178\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc0f760701241b677c4ba5f137cb1288_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#51;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><span data-type=\"media\" id=\"fs-id1167833207697\" data-alt=\"The figure shows the graph plotted for the inequalities y less than minus half of x plus three and y less than three times x minus four. Two intersecting lines are shown on the graph. The area bound by the two lines to the bottom right is shown in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_302_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph plotted for the inequalities y less than minus half of x plus three and y less than three times x minus four. Two intersecting lines are shown on the graph. The area bound by the two lines to the bottom right is shown in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"howto\">\n<div data-type=\"title\">Solve a system of linear inequalities by graphing.<\/div>\n<ol id=\"fs-id1167836481249\" class=\"stepwise\" type=\"1\">\n<li>Graph the first inequality.\n<ul data-bullet-style=\"bullet\">\n<li>Graph the boundary line.<\/li>\n<li>Shade in the side of the boundary line where the inequality is true.<\/li>\n<\/ul>\n<\/li>\n<li>On the same grid, graph the second inequality.\n<ul id=\"fs-id1167829811762\" data-bullet-style=\"bullet\">\n<li>Graph the boundary line.<\/li>\n<li>Shade in the side of that boundary line where the inequality is true.<\/li>\n<\/ul>\n<\/li>\n<li>The solution is the region where the shading overlaps.<\/li>\n<li>Check by choosing a test point.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836300564\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836418956\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167824740437\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0e0fcdea15fc4bce442dab3fab50ae6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#62;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167829908693\" class=\"unnumbered unstyled\" summary=\"The figure shows the graph of the inequality x minus y greater than three. It is plotted by graphing x minus y equal to three and testing a point. The intercepts are x equal to three and y equal to minus three and the boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality false so shade the side that does not contain this point in red.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a7e11f41d4b5b66c7b0d70d6dbc2f74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#62;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph <em data-effect=\"italics\">x<\/em> \u2212 <em data-effect=\"italics\">y<\/em> &gt; 3, by graphing <em data-effect=\"italics\">x<\/em> \u2212 <em data-effect=\"italics\">y<\/em> = 3<\/p>\n<div data-type=\"newline\"><\/div>\n<p>and testing a point.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>The intercepts are <em data-effect=\"italics\">x<\/em> = 3 and <em data-effect=\"italics\">y<\/em> = \u22123 and the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>boundary line will be dashed.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Test (0, 0) which makes the inequality false so shade<\/p>\n<div data-type=\"newline\"><\/div>\n<p>(red) the side that does not contain (0, 0).<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836531569\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_004a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-id1167836329255\" class=\"unnumbered unstyled\" summary=\"The figure shows the making of graph of y less than minus one fifth x plus four by graphing y equal to minus one fifth x plus four using the slope m equal to minus one by five and y intercept b equal to four. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality true so the side that contains this point is shaded blue.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-977233750791ab2a40a83a40211f8c02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-167ff240d536e8f45621849d73479973_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>using the slope <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0e35a4e0a70aabb2f4ed03eca173840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/> and <em data-effect=\"italics\">y<\/em>-intercept <em data-effect=\"italics\">b<\/em> = 4.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The boundary line will be dashed<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Test (0, 0) which makes the inequality true, so<\/p>\n<div data-type=\"newline\"><\/div>\n<p>shade (blue) the side that contains (0, 0).<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Choose a test point in the solution and verify that it is a solution to both inequalties.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836531590\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_005a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836515907\">The point of intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice\u2014which appears as the darkest shaded region.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836532492\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829688088\">\n<div data-type=\"problem\" id=\"fs-id1167836615921\">\n<p id=\"fs-id1167833379505\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9625c246932541cb6157d110276b413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#92;&#108;&#101;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836547471\">\n<p><span data-type=\"media\" id=\"fs-id1167836399021\" data-alt=\"The figure shows the graph for the inequalities x plus y less than or equal to two and y greater than or equal to two by three of x minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_303_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph for the inequalities x plus y less than or equal to two and y greater than or equal to two by three of x minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829715839\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826130667\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836613306\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb41c6b82330ac611ecc729558f232a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#50;&#121;&#92;&#108;&#101;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#62;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"123\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836390184\">\n<p><span data-type=\"media\" id=\"fs-id1167836540392\" data-alt=\"The figure shows graph for the inequalities three times x minus two times y less than or equal to six and y greater than or equal to minus one by four of x plus five. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_304_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows graph for the inequalities three times x minus two times y less than or equal to six and y greater than or equal to minus one by four of x plus five. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836355790\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836800431\">\n<p id=\"fs-id1167836506374\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98282ba6f639281a106e5e28ab22779b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#50;&#121;&#60;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#62;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836322831\">\n<table id=\"fs-id1167836507012\" class=\"unnumbered unstyled\" summary=\"The figure shows the making of graph of x minus two times y less than five by graphing x minus two times y equal to five and testing a point The intercepts are x equal to five and y equal to minus two point five. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality true so the side that contains this point is shaded red.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0238379d0a5f72acebc8b023d70713c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#50;&#121;&#60;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#62;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9ffca6abb59390931d9843fc39a1d4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#60;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f592894dfc669be1583facabdf7c38b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>and testing a point. The intercepts are<\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">x<\/em> = 5 and <em data-effect=\"italics\">y<\/em> = \u22122.5 and the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>boundary line will be dashed.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Test (0, 0) which makes the inequality true, so shade<\/p>\n<div data-type=\"newline\"><\/div>\n<p>(red) the side that contains (0, 0).<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836662872\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_006a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-id1167836330926\" class=\"unnumbered unstyled can-break\" summary=\"The figure shows the making of graph of y greater than minus four by graphing y equal to minus four and recognizing that it is a horizontal line through y equal to minus four. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality true so the side that contains this point is shaded blue.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9d8c1e359f9cca37b1bc79745c8147c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#45;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53923be0c534e9cf06b453317eed3f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>recognizing that it is a horizontal line<\/p>\n<div data-type=\"newline\"><\/div>\n<p>through <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8af9b026845f3fbfb2c441fcaba8be36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/> The boundary line will<\/p>\n<div data-type=\"newline\"><\/div>\n<p>be dashed.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Test (0, 0) which makes the inequality<\/p>\n<div data-type=\"newline\"><\/div>\n<p>true so shade (blue) the side that contains (0, 0).<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829713444\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_007a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836521734\">The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is in the solution and we have already found it to be a solution of each inequality. The point of intersection of the two lines is not included as both boundary lines were dashed.<\/p>\n<p id=\"fs-id1167836319295\">The solution is the area shaded twice\u2014which appears as the darkest shaded region.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836527766\">\n<div data-type=\"problem\" id=\"fs-id1167836508780\">\n<p id=\"fs-id1167833086952\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66b060c260e0aa2924088d90ac6c7d30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#51;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><span data-type=\"media\" id=\"fs-id1167836571625\" data-alt=\"The figure shows graph for the inequalities y greater than or equal to three times x minus two and y less than minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_305_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows graph for the inequalities y greater than or equal to three times x minus two and y less than minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829746816\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833378448\">\n<div data-type=\"problem\" id=\"fs-id1167832939943\">\n<p id=\"fs-id1167833397126\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bae26ec4d94b6adbe1d91cb948bf26c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#62;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#92;&#103;&#101;&#32;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833369871\">\n<p><span data-type=\"media\" id=\"fs-id1167836510670\" data-alt=\"The figure shows graph for the inequalities x greater than or equal to minus four and x minus two times y greater than minus four. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_306_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows graph for the inequalities x greater than or equal to minus four and x minus two times y greater than minus four. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836549970\">Systems of linear inequalities where the boundary lines are parallel might have no solution. We\u2019ll see this in the next example.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836423663\">\n<div data-type=\"problem\" id=\"fs-id1167829783800\">\n<p id=\"fs-id1167829609167\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9495880739a7847bad35d87e7b82b3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#43;&#51;&#121;&#92;&#103;&#101;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836379111\">\n<table id=\"fs-id1167836367953\" class=\"unnumbered unstyled can-break\" summary=\"The figure shows the making of two graphs. First the graph of four times x plus three times y greater than or equal to twelve is made by graphing four times x plus three times y equal to twelve. The intercepts are x equal to three and y equal to four. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality false so the side that contains this point is shaded blue. Next the graph of y less than minus four by three of x plus one is made by graphing y equal to minus four by three of x plus one using the slope m equal to minus four by three and y intercept b equal to one. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality true so the side that contains this point is shaded red.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff42697ceaa606e69a472609a1507cf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#43;&#51;&#121;&#92;&#103;&#101;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22bf38a8d5d9b793a5b2db24fe8f4c0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#51;&#121;&#92;&#103;&#101;&#32;&#49;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -4px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78fad9dbf93f5bbe1e94c2ac7a1f17ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#51;&#121;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>and testing a point. The intercepts are <em data-effect=\"italics\">x<\/em> = 3<\/p>\n<div data-type=\"newline\"><\/div>\n<p>and <em data-effect=\"italics\">y<\/em> = 4 and the boundary line will be solid.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Test (0, 0) which makes the inequality false, so<\/p>\n<div data-type=\"newline\"><\/div>\n<p>shade (red) the side that does not contain (0, 0).<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826206421\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_008a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99a9f07887b88c4b3874909153d098e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08a48b76f35b590c7fe96f2bd1fee4d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>using the slope <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c21ea751b47a79f244d99ee48ee336b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/> and <em data-effect=\"italics\">y<\/em>-intercept<\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> = 1. The boundary line will be dashed.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Test (0, 0) which makes the inequality true, so<\/p>\n<div data-type=\"newline\"><\/div>\n<p>shade (blue) the side that contains (0, 0).<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832981583\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_008b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167833407721\">There is no point in both shaded regions, so the system has no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836378100\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836418859\">\n<div data-type=\"problem\" id=\"fs-id1167836630482\">\n<p id=\"fs-id1167836353188\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-385b76ca430a2abea3f20c18acebc12d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#50;&#121;&#92;&#103;&#101;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836518216\">\n<p><span data-type=\"media\" id=\"fs-id1167829688838\" data-alt=\"The graph of three times x minus two times y greater than or equal to twelve and y greater than or equal to three by two of x plus one is shown. Two intersecting lines are shown. The inequalities do not have a solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_307_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of three times x minus two times y greater than or equal to twelve and y greater than or equal to three by two of x plus one is shown. Two intersecting lines are shown. The inequalities do not have a solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>No solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836352956\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836306660\">\n<div data-type=\"problem\" id=\"fs-id1167836699421\">\n<p id=\"fs-id1167836378871\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5365fe48564820ccfd08edb838271f03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#51;&#121;&#62;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><span data-type=\"media\" id=\"fs-id1167836602905\" data-alt=\"The graph of x plus three times y greater than eight and y less than minus one by three of x minus two is shown. Two intersecting lines are shown. The inequalities do not have a solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_308_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of x plus three times y greater than eight and y less than minus one by three of x minus two is shown. Two intersecting lines are shown. The inequalities do not have a solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>No solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168756714670\">Some systems of linear inequalities where the boundary lines are parallel will have a solution. We\u2019ll see this in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836596568\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829785685\">\n<div data-type=\"problem\" id=\"fs-id1167829746913\">\n<p id=\"fs-id1167836378180\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ffd4aadb494d80c9cc9f521b2f0a2ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#60;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829830048\">\n<table id=\"fs-id1167836390685\" class=\"unnumbered unstyled can-break\" summary=\"The figure shows the making of two graphs. First the graph of y greater than half of x minus four is made by graphing y equal to half of x minus four using the slope m equal to half and y intercept b equal to minus 4. The boundary line is dashed. Test with point with coordinates zero and zero which makes the inequality so the side that contains this point is shaded red. Next the graph of x minus two times y less than minus four is made by graphing x minus two times y equal to minus four. The intercepts are x equal to minus four and y equal to two. The boundary line is dashed.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8b89a458bca86d76ebf19d04c76bc30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#60;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a0719c7367f94c72eac3da953c3689b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -6px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52fb6a706fbcd64f2a7398346ce16a3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>using the slope <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4e7c1468d71a3131e728a60e2da9237_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/> and the intercept<\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">b<\/em> = \u22124. The boundary line will be dashed.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Test (0, 0) which makes the inequality true, so<\/p>\n<div data-type=\"newline\"><\/div>\n<p>shade (red) the side that contains (0, 0).<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836440079\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-864074ac6c923db53d28749a677e43cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#60;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -4px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46b8bb6abd0b3a1f2b13e61ff685657e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>and testing a point. The intercepts are<\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">x<\/em> = \u22124 and <em data-effect=\"italics\">y<\/em> = 2 and the boundary line will be dashed.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Choose a test point in the solution and verify<\/p>\n<div data-type=\"newline\"><\/div>\n<p>that it is a solution to both inequalties.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Test (0, 0) which makes the inequality false, so<\/p>\n<div data-type=\"newline\"><\/div>\n<p>shade (blue) the side that does not contain (0, 0).<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824734677\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167833346510\">No point on the boundary lines is included in the solution as both lines are dashed.<\/p>\n<p id=\"fs-id1167829749840\">The solution is the region that is shaded twice which is also the solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7e38958b22d5edccbe19dbbb589be42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#60;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836537024\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836297393\">\n<div data-type=\"problem\" id=\"fs-id1167833048343\">\n<p>Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a71033e056c4103b98e91b047927620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#51;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#43;&#121;&#92;&#103;&#101;&#32;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"142\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836789202\">\n<p><span data-type=\"media\" id=\"fs-id1167836341685\" data-alt=\"The figure shows the graph of the inequalities y greater than or equal to three times x plus one and minus three times x plus y greater than or equal to minus four. Two parallel lines are shown and the region to the left of both is colored in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_309_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y greater than or equal to three times x plus one and minus three times x plus y greater than or equal to minus four. Two parallel lines are shown and the region to the left of both is colored in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829744169\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836611939\">\n<div data-type=\"problem\" id=\"fs-id1167836523596\">\n<p>Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80d4f99ae73f1ee3b6848aab4b9faa0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#52;&#121;&#92;&#108;&#101;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><span data-type=\"media\" id=\"fs-id1167833058054\" data-alt=\"The figure shows the graph of the inequalities y less than or equal to minus one fourth of x plus 2 and x plus four times y less than or equal to four. Two parallel lines are shown and the region to the bottom of both is colored in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_310_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y less than or equal to minus one fourth of x plus 2 and x plus four times y less than or equal to four. Two parallel lines are shown and the region to the bottom of both is colored in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829744348\">\n<h3 data-type=\"title\">Solve Applications of Systems of Inequalities<\/h3>\n<p id=\"fs-id1167829843274\">The first thing we\u2019ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so we add inequalities to the system as additional requirements.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836556132\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829620350\">\n<div data-type=\"problem\" id=\"fs-id1167836510446\">\n<p id=\"fs-id1167829695726\">Christy sells her photographs at a booth at a street fair. At the start of the day, she wants to have at least 25 photos to display at her booth. Each small photo she displays costs her ?4 and each large photo costs her ?10. She doesn\u2019t want to spend more than ?200 on photos to display.<\/p>\n<p id=\"fs-id1167829694291\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Could she display 10 small and 20 large photos?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Could she display 20 large and 10 small photos?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836731946\">\n<p id=\"fs-id1167836390111\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0810833e1e68e798f68b2f8fb4c130fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#115;&#109;&#97;&#108;&#108;&#32;&#112;&#104;&#111;&#116;&#111;&#115;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#108;&#97;&#114;&#103;&#101;&#32;&#112;&#104;&#111;&#116;&#111;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"293\" style=\"vertical-align: -15px;\" \/><\/p>\n<p id=\"fs-id1167836649828\">To find the system of equations translate the information.<\/p>\n<p id=\"fs-id1167836693211\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-158afe26cc2bd0e0fbc5429f048957fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#104;&#101;&#32;&#119;&#97;&#110;&#116;&#115;&#32;&#116;&#111;&#32;&#104;&#97;&#118;&#101;&#32;&#97;&#116;&#32;&#108;&#101;&#97;&#115;&#116;&#32;&#50;&#53;&#32;&#112;&#104;&#111;&#116;&#111;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#115;&#109;&#97;&#108;&#108;&#32;&#112;&#108;&#117;&#115;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#108;&#97;&#114;&#103;&#101;&#32;&#115;&#104;&#111;&#117;&#108;&#100;&#32;&#98;&#101;&#32;&#97;&#116;&#32;&#108;&#101;&#97;&#115;&#116;&#32;&#50;&#53;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#121;&#92;&#103;&#101;&#32;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#52;&#32;&#102;&#111;&#114;&#32;&#101;&#97;&#99;&#104;&#32;&#115;&#109;&#97;&#108;&#108;&#32;&#97;&#110;&#100;&#32;&#63;&#49;&#48;&#32;&#102;&#111;&#114;&#32;&#101;&#97;&#99;&#104;&#32;&#108;&#97;&#114;&#103;&#101;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#110;&#111;&#32;&#109;&#111;&#114;&#101;&#32;&#116;&#104;&#97;&#110;&#32;&#63;&#50;&#48;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#120;&#43;&#49;&#48;&#121;&#92;&#108;&#101;&#32;&#50;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#115;&#109;&#97;&#108;&#108;&#32;&#112;&#104;&#111;&#116;&#111;&#115;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#103;&#114;&#101;&#97;&#116;&#101;&#114;&#32;&#116;&#104;&#97;&#110;&#32;&#111;&#114;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#32;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#108;&#97;&#114;&#103;&#101;&#32;&#112;&#104;&#111;&#116;&#111;&#115;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#103;&#114;&#101;&#97;&#116;&#101;&#114;&#32;&#116;&#104;&#97;&#110;&#32;&#111;&#114;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#32;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"324\" width=\"529\" style=\"vertical-align: -190px;\" \/><\/p>\n<p id=\"fs-id1167829645186\">We have our system of equations.<\/p>\n<div data-type=\"equation\" id=\"fs-id1169146740783\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8dc52fd7d4d48b932ca2662af0b01e67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#92;&#103;&#101;&#32;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#49;&#48;&#121;&#92;&#108;&#101;&#32;&#50;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"139\" style=\"vertical-align: -38px;\" \/><\/div>\n<p id=\"fs-id1167829693372\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46bb299f47d94e1927057d14ab78801f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-407245eaac2a01ec21d965527d39162c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> (both are greater than or equal to) all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.<\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167833245815\" class=\"unnumbered unstyled can-break\" summary=\"The figure shows the making of two graphs. First the inequality x plus y greater than or equal to twenty five is made by graphing x plus y equal to twenty five as a solid line. The point with coordinates zero and zero does not make the inequality true so the side that does not include this point is shaded red. Next the inequality four times x plus ten times y greater than or equal to two hundred is made by graphing four times x plus ten times y equal to two hundred. The point with coordinates zero and zero does make the inequality true so the side which contains this point is shaded blue.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6c2a2d1a1ddd3a016f1d8e2df233090_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#121;&#92;&#103;&#101;&#32;&#50;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/> graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc16ce691347f0958ad8738f7901f946_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#121;&#61;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/> as a solid line.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Choose (0, 0) as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point (0, 0).<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>To graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4feaf298f31bcb619b0a76b15b27f219_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#49;&#48;&#121;&#92;&#108;&#101;&#32;&#50;&#48;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"122\" style=\"vertical-align: -4px;\" \/> graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11afbcc85dfdcdae5b6a65d4928d3756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#49;&#48;&#121;&#61;&#50;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"118\" style=\"vertical-align: -4px;\" \/> as a solid line.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Choose (0, 0) as a test point. Since it does make the inequality true, shade (blue) the side that include the point (0, 0).<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829614356\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_010a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836434591\">The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly-shaded section are included in the solution as are the points on the <em data-effect=\"italics\">x<\/em>-axis from (25, 0) to (55, 0).<\/p>\n<p><span class=\"token\">\u24d2<\/span> To determine if 10 small and 20 large photos would work, we look at the graph to see if the point (10, 20) is in the solution region. We could also test the point to see if it is a solution of both equations.<\/p>\n<p id=\"fs-id1167836378351\">It is not, Christy would not display 10 small and 20 large photos.<\/p>\n<p id=\"fs-id1167833224670\"><span class=\"token\">\u24d3<\/span> To determine if 20 small and 10 large photos would work, we look at the graph to see if the point (20, 10) is in the solution region. We could also test the point to see if it is a solution of both equations.<\/p>\n<p id=\"fs-id1167833050449\">It is, so Christy could choose to display 20 small and 10 large photos.<\/p>\n<p id=\"fs-id1167836537096\">Notice that we could also test the possible solutions by substituting the values into each inequality.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836706462\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167822971721\">\n<div data-type=\"problem\" id=\"fs-id1167836688596\">\n<p id=\"fs-id1167829718513\">A trailer can carry a maximum weight of 160 pounds and a maximum volume of 15 cubic feet. A microwave oven weighs 30 pounds and has 2 cubic feet of volume, while a printer weighs 20 pounds and has 3 cubic feet of space.<\/p>\n<p id=\"fs-id1167836327777\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Could 4 microwaves and 2 printers be carried on this trailer?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Could 7 microwaves and 3 printers be carried on this trailer?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829685153\">\n<p id=\"fs-id1167836287444\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae1bfb9bbe073c4c85c69814913697cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#48;&#109;&#43;&#50;&#48;&#112;&#92;&#108;&#101;&#32;&#49;&#54;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#109;&#43;&#51;&#112;&#92;&#108;&#101;&#32;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"152\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836300551\" data-alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_311_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> yes<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> no<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829745732\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833316799\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836312119\">Mary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least 5 more than the number of pencils. The pencils cost ?2 and the answer sheets cost ?1. Mary\u2019s budget for these supplies allows for a maximum cost of ?400.<\/p>\n<p id=\"fs-id1167836392171\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Could Mary purchase 100 pencils and 100 answer sheets?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Could Mary purchase 150 pencils and 150 answer sheets?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826102750\">\n<p id=\"fs-id1167836689366\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64e05514c6708d7fcbffef2b417b8771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#97;&#92;&#103;&#101;&#32;&#112;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#43;&#50;&#112;&#92;&#108;&#101;&#32;&#52;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836610639\" data-alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_312_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> no<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> no<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168756762423\">When we use variables other than <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> to define an unknown quantity, we must change the names of the axes of the graph as well.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836507645\">\n<div data-type=\"problem\" id=\"fs-id1167836628288\">\n<p id=\"fs-id1167836295269\">Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers and cookies, and he doesn\u2019t want to spend more than ?5. At the hamburger restaurant near his college, each hamburger has 240 calories and costs ?1.40. Each cookie has 160 calories and costs ?0.50.<\/p>\n<p id=\"fs-id1167833396849\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Could he eat 3 hamburgers and 1 cookie?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Could he eat 2 hamburgers and 4 cookies?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836518697\">\n<p id=\"fs-id1167836329245\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c769933ceb49909ea925c02692f5ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#104;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#104;&#97;&#109;&#98;&#117;&#114;&#103;&#101;&#114;&#115;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#99;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#99;&#111;&#111;&#107;&#105;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"286\" style=\"vertical-align: -12px;\" \/><\/p>\n<p id=\"fs-id1167833021820\">To find the system of equations translate the information.<\/p>\n<p id=\"fs-id1167836539916\">The calories from hamburgers at 240 calories each, plus the calories from cookies at 160 calories each must be more that 800.<\/p>\n<p id=\"fs-id1167836621351\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-706ef77abe1a67548dcc7955eefca130_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#52;&#48;&#104;&#43;&#49;&#54;&#48;&#99;&#92;&#103;&#101;&#32;&#56;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#97;&#109;&#111;&#117;&#110;&#116;&#32;&#115;&#112;&#101;&#110;&#116;&#32;&#111;&#110;&#32;&#104;&#97;&#109;&#98;&#117;&#114;&#103;&#101;&#114;&#115;&#32;&#97;&#116;&#32;&#63;&#49;&#46;&#52;&#48;&#32;&#101;&#97;&#99;&#104;&#44;&#32;&#112;&#108;&#117;&#115;&#32;&#116;&#104;&#101;&#32;&#97;&#109;&#111;&#117;&#110;&#116;&#32;&#115;&#112;&#101;&#110;&#116;&#32;&#111;&#110;&#32;&#99;&#111;&#111;&#107;&#105;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#32;&#63;&#48;&#46;&#53;&#48;&#32;&#101;&#97;&#99;&#104;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#110;&#111;&#32;&#109;&#111;&#114;&#101;&#32;&#116;&#104;&#97;&#110;&#32;&#63;&#53;&#46;&#48;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#46;&#52;&#48;&#104;&#43;&#48;&#46;&#53;&#48;&#99;&#92;&#108;&#101;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#104;&#97;&#109;&#98;&#117;&#114;&#103;&#101;&#114;&#115;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#103;&#114;&#101;&#97;&#116;&#101;&#114;&#32;&#116;&#104;&#97;&#110;&#32;&#111;&#114;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#32;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#104;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#99;&#111;&#111;&#107;&#105;&#101;&#115;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#103;&#114;&#101;&#97;&#116;&#101;&#114;&#32;&#116;&#104;&#97;&#110;&#32;&#111;&#114;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#32;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#99;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"302\" width=\"642\" style=\"vertical-align: -146px;\" \/><\/p>\n<p id=\"fs-id1171792683972\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c5ae6014c921bea8cdbfa8b63113851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#104;&#97;&#118;&#101;&#32;&#111;&#117;&#114;&#32;&#115;&#121;&#115;&#116;&#101;&#109;&#32;&#111;&#102;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#52;&#48;&#104;&#43;&#49;&#54;&#48;&#99;&#92;&#103;&#101;&#32;&#56;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#49;&#46;&#52;&#48;&#104;&#43;&#48;&#46;&#53;&#48;&#99;&#92;&#108;&#101;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#104;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#99;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"515\" style=\"vertical-align: -38px;\" \/><\/p>\n<p id=\"fs-id1167833310128\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0fd8a6336f593c4bc5dc5a64b5c2912_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#62;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1076b6618f7ebf8e374b3eba79f41fbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#62;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"54\" style=\"vertical-align: 0px;\" \/> (both are greater than or equal to) all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.<\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167833136536\" class=\"unnumbered unstyled can-break\" summary=\"The graph of inequality two forty times h plus one sixty times c greater than or equal to eight hundred is shown. It is made by graphing two forty h plus one sixty c equal to eight hundred as a solid line. The point with coordinates zero and zero does not make the inequality true so the side which does not include this point is shaded red.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-998c15fe71c069a00e05711d16a69b81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#48;&#104;&#43;&#49;&#54;&#48;&#99;&#92;&#103;&#101;&#32;&#56;&#48;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"147\" style=\"vertical-align: -4px;\" \/> graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de00912023c4953f1c00dafec004952d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#48;&#104;&#43;&#49;&#54;&#48;&#99;&#61;&#56;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"143\" style=\"vertical-align: -2px;\" \/> as a solid line.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Choose (0, 0) as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point (0, 0).<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833136523\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_011a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167833239251\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef8decd4436333396ac12d03bf755753_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#52;&#48;&#104;&#43;&#48;&#46;&#53;&#48;&#99;&#92;&#108;&#101;&#32;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"138\" style=\"vertical-align: -3px;\" \/> The boundary line is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a51eba4a9ca13ba051d466668daae1e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#52;&#48;&#104;&#43;&#48;&#46;&#53;&#48;&#99;&#61;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"138\" style=\"vertical-align: -2px;\" \/> We test (0, 0) and it makes the inequality true. We shade the side of the line that includes (0, 0).<\/p>\n<p id=\"fs-id1167836512397\">The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly shaded section are included in the solution as are the points on the <em data-effect=\"italics\">x<\/em>-axis from (5, 0) to (10, 0).<\/p>\n<p id=\"fs-id1167833135690\"><span class=\"token\">\u24d2<\/span> To determine if 3 hamburgers and 2 cookies would meet Omar\u2019s criteria, we see if the point (3, 2) is in the solution region. It is, so Omar might choose to eat 3 hamburgers and 2 cookies.<\/p>\n<p id=\"fs-id1167829942426\"><span class=\"token\">\u24d3<\/span> To determine if 2 hamburgers and 4 cookies would meet Omar\u2019s criteria, we see if the point (2, 4) is in the solution region. It is, Omar might choose to eat 2 hamburgers and 4 cookies.<\/p>\n<p id=\"fs-id1167836559981\">We could also test the possible solutions by substituting the values into each inequality.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829619410\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829718986\">\n<div data-type=\"problem\" id=\"fs-id1167836706434\">\n<p id=\"fs-id1167836391752\">Tension needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only ?25 to spend on the extra food he needs and will spend it on ?0.75 donuts which have 360 calories each and ?2 energy drinks which have 110 calories.<\/p>\n<p id=\"fs-id1167836516260\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities that models this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Can he buy 8 donuts and 4 energy drinks and satisfy his caloric needs?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Can he buy 1 donut and 3 energy drinks and satisfy his caloric needs?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836493823\">\n<p id=\"fs-id1167836705878\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ef5c013f8b6a656f1767fd284946ab8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#48;&#46;&#55;&#53;&#100;&#43;&#50;&#101;&#92;&#108;&#101;&#32;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#54;&#48;&#100;&#43;&#49;&#49;&#48;&#101;&#92;&#103;&#101;&#32;&#49;&#48;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"171\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829579782\" data-alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_313_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> yes<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> no<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833338266\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833407364\">\n<div data-type=\"problem\" id=\"fs-id1167829619467\">\n<p id=\"fs-id1167829924727\">Philip\u2019s doctor tells him he should add at least 1,000 more calories per day to his usual diet. Philip wants to buy protein bars that cost ?1.80 each and have 140 calories and juice that costs ?1.25 per bottle and have 125 calories. He doesn\u2019t want to spend more than ?12.<\/p>\n<p id=\"fs-id1167833086963\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities that models this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Can he buy 3 protein bars and 5 bottles of juice?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Can he buy 5 protein bars and 3 bottles of juice?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836282522\">\n<p id=\"fs-id1167836386502\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13ae2dbe9c7b9e6910e338c1d9730e0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#49;&#52;&#48;&#112;&#43;&#49;&#50;&#53;&#106;&#92;&#103;&#101;&#32;&#49;&#48;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#49;&#46;&#56;&#48;&#112;&#43;&#49;&#46;&#50;&#53;&#106;&#92;&#108;&#101;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"171\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836522128\" data-alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_314_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> yes<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> no<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"media-2\">\n<p id=\"fs-id1167832980753\">Access these online resources for additional instruction and practice with solving systems of linear inequalities by graphing.<\/p>\n<ul id=\"fs-id1167836690512\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37sysineqgraph\">Solving Systems of Linear Inequalities by Graphing<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37sysineqgraph2\">Systems of Linear Inequalities<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836622961\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836610518\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Solutions of a System of Linear Inequalities:<\/strong> Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the <em data-effect=\"italics\">x, y<\/em> coordinate system that includes all the points whose ordered pairs make the inequalities true.<\/li>\n<li><strong data-effect=\"bold\">How to solve a system of linear inequalities by graphing.<\/strong>\n<ol id=\"fs-id1167833310565\" class=\"stepwise\" type=\"1\">\n<li>Graph the first inequality.\n<div data-type=\"newline\"><\/div>\n<p>Graph the boundary line.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Shade in the side of the boundary line where the inequality is true.<\/li>\n<li>On the same grid, graph the second inequality.\n<div data-type=\"newline\"><\/div>\n<p>Graph the boundary line.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Shade in the side of that boundary line where the inequality is true.<\/li>\n<li>The solution is the region where the shading overlaps.<\/li>\n<li>Check by choosing a test point.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836551834\">\n<h3 data-type=\"title\">Section Exercises<\/h3>\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829907198\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p><strong data-effect=\"bold\">Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities<\/strong><\/p>\n<p id=\"fs-id1167836521359\">In the following exercises, determine whether each ordered pair is a solution to the system.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836608480\">\n<div data-type=\"problem\" id=\"fs-id1167836701180\">\n<p id=\"fs-id1167836697073\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4db0c9b1b94c9b8445bb448c3cf856a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#62;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#92;&#108;&#101;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167829624252\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f054d79e93ab302f68adcad7d4c3ec21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f63e6ea0c8474555192b305e3472481d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836333949\">\n<div data-type=\"problem\" id=\"fs-id1167836349238\">\n<p id=\"fs-id1167836334178\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8148640e6997c811a49b407ab3cb3a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#45;&#121;&#60;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#50;&#121;&#62;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"139\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167836294884\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d89831e962b1131c0edfd43df352881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2347704000ad2e9ae878a8427611b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825703167\">\n<p><span class=\"token\">\u24d0<\/span> false <span class=\"token\">\u24d1<\/span> true<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833017814\">\n<div data-type=\"problem\" id=\"fs-id1167836334160\">\n<p id=\"fs-id1167833412555\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad6a9436a447b306704d8e106c739e26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#92;&#108;&#101;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"104\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167836743453\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-caaaf19d132bc66bf2d0f1bfca39c6b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#54;&#44;&#32;&#8722;&#52;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e45c05c27110103f1a1b55d37f16a5e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#51;&#44;&#32;&#48;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836688087\">\n<div data-type=\"problem\" id=\"fs-id1167836574907\">\n<p id=\"fs-id1167836619659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e5cd97cb71c5d0673d0d5454f522c10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#50;&#121;&#60;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"112\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167829749517\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ef6c187a0599fedf1caa75800c24233_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8f40a634f7934f02c606fb27e274d8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#56;&#44;&#32;&#51;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836514155\">\n<p id=\"fs-id1167833310849\"><span class=\"token\">\u24d0<\/span> false <span class=\"token\">\u24d1<\/span> true<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829713662\">\n<div data-type=\"problem\" id=\"fs-id1167836288276\">\n<p id=\"fs-id1167836557972\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-649d19f2669635af3cd312c6de473640_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#55;&#120;&#43;&#50;&#121;&#62;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#45;&#121;&#92;&#108;&#101;&#32;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167829905595\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0bae1365d9dfb6731eecda331429b5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#50;&#44;&#32;&#51;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dea82112c101c70aff4cbf51c648e977_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#55;&#44;&#32;&#8722;&#49;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836408854\">\n<div data-type=\"problem\" id=\"fs-id1167833224335\">\n<p id=\"fs-id1167833138577\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b01edb3e7d5cf659c5499e798b810c9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#54;&#120;&#45;&#53;&#121;&#60;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#55;&#121;&#62;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"139\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167836738279\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-486025cc72c03895eb87780e179ddb47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#49;&#44;&#32;&#8722;&#51;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e1d5cdfff5f95646f6e786b78296c9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#8722;&#52;&#44;&#32;&#52;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829691536\">\n<p id=\"fs-id1167836538843\"><span class=\"token\">\u24d0<\/span> false <span class=\"token\">\u24d1<\/span> true<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833071549\"><strong data-effect=\"bold\">Solve a System of Linear Inequalities by Graphing<\/strong><\/p>\n<p id=\"fs-id1167836622335\">In the following exercises, solve each system by graphing.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836508300\">\n<p id=\"fs-id1167836408694\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37d6d59a5df9b7844bec633dd95e5767_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#108;&#101;&#32;&#51;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#62;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836536081\">\n<div data-type=\"problem\" id=\"fs-id1167830123178\">\n<p id=\"fs-id1167836694111\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed5dc17d2a4e5fd07f45d52d5fff5f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#60;&#45;&#50;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#45;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826132859\">\n<p><span data-type=\"media\" id=\"fs-id1167824773950\" data-alt=\"The figure shows the graph of inequalities y less than minus two times x plus two and y greater than or equal to minus x minus one. Two intersecting lines are shown, one in red and the other in blue. The area bound by the two lines is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_203_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of inequalities y less than minus two times x plus two and y greater than or equal to minus x minus one. Two intersecting lines are shown, one in red and the other in blue. The area bound by the two lines is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833048440\">\n<div data-type=\"problem\" id=\"fs-id1167833128530\">\n<p id=\"fs-id1167833056596\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ac0e3186f612203ef3229505f40a6f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#60;&#50;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836607397\">\n<div data-type=\"problem\" id=\"fs-id1167836692537\">\n<p id=\"fs-id1167829908011\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d3b37edbe09edac8095923179e193dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#62;&#50;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836538932\">\n<p><span data-type=\"media\" id=\"fs-id1167836509816\" data-alt=\"The figure shows the graph of the inequalities y greater than or equal to minus two by three x plus two and y greater than two times x minus three. Two intersecting lines, one in red and the other in blue, are shown. The region bound by them is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_205_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y greater than or equal to minus two by three x plus two and y greater than two times x minus three. Two intersecting lines, one in red and the other in blue, are shown. The region bound by them is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833055904\">\n<div data-type=\"problem\" id=\"fs-id1167833350378\">\n<p id=\"fs-id1167836512297\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc7f10bf7537e171837e5cb33f535772_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#62;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836727811\">\n<div data-type=\"problem\" id=\"fs-id1167829743662\">\n<p id=\"fs-id1167836629307\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09ca49c3d774a3c07bd05d508be4014f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#121;&#60;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836629128\">\n<p><span data-type=\"media\" id=\"fs-id1167836685960\" data-alt=\"The figure shows the graph of the inequalities x minus two times y less than four and y less than x minus two. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_207_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities x minus two times y less than four and y less than x minus two. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836705535\">\n<div data-type=\"problem\" id=\"fs-id1167829755866\">\n<p id=\"fs-id1167829755868\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d9fd8527dd44f23c9d52f43a7754afe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#121;&#92;&#103;&#101;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836448216\">\n<div data-type=\"problem\" id=\"fs-id1167836448218\">\n<p id=\"fs-id1167836322981\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1db35d3741ed0b350f3c96338de6ba16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#52;&#121;&#92;&#103;&#101;&#32;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836598601\">\n<p><span data-type=\"media\" id=\"fs-id1167833191635\" data-alt=\"The figure shows the graph of the inequalities two times x plus four times y greater than or equal to eight and y less than or equal to minus three fourth of x. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_209_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities two times x plus four times y greater than or equal to eight and y less than or equal to minus three fourth of x. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836391716\">\n<p id=\"fs-id1167836391718\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47e960df8c74191ede88b27a3e741143_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#53;&#121;&#60;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#52;&#121;&#92;&#103;&#101;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833271854\">\n<div data-type=\"problem\" id=\"fs-id1167836535416\">\n<p id=\"fs-id1167836535418\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8851c0cbfb4eceb5accb78f3a70de7d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#50;&#121;&#92;&#108;&#101;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#52;&#120;&#45;&#50;&#121;&#62;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836615038\">\n<p><span data-type=\"media\" id=\"fs-id1167824764655\" data-alt=\"The figure shows the graph of the inequalities three times x minus two times y less than or equal to six and minus four times x minus two times y greater than eight. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_211_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities three times x minus two times y less than or equal to six and minus four times x minus two times y greater than eight. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824773992\">\n<div data-type=\"problem\" id=\"fs-id1167833031765\">\n<p id=\"fs-id1167833031767\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67b317b724f448b49a717b922719a3b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#50;&#121;&#62;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#51;&#121;&#92;&#103;&#101;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836508671\">\n<div data-type=\"problem\" id=\"fs-id1167833387018\">\n<p id=\"fs-id1167833387020\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3d1eddbf10a71e06b224a1cdbd19ba3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#62;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#92;&#103;&#101;&#32;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833056781\">\n<p><span data-type=\"media\" id=\"fs-id1167836614346\" data-alt=\"The figure shows the graph of the inequalities two times x plus y greater than minus six and minus x plus two times y greater than or equal to minus four. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_213_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities two times x plus y greater than minus six and minus x plus two times y greater than or equal to minus four. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836717278\">\n<div data-type=\"problem\" id=\"fs-id1167836717280\">\n<p id=\"fs-id1167833350465\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8482f4bfc902bb065c3b610da8cde58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#50;&#121;&#60;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#108;&#101;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833053554\">\n<div data-type=\"problem\" id=\"fs-id1167833053556\">\n<p id=\"fs-id1167836477283\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ace5e5c0da46023857a88de8690e7f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#121;&#62;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#108;&#101;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833018419\">\n<p><span data-type=\"media\" id=\"fs-id1167833018422\" data-alt=\"The figure shows the graph of the inequalities x minus three times y greater than four and y less than or equal to minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_215_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities x minus three times y greater than four and y less than or equal to minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836484677\">\n<div data-type=\"problem\" id=\"fs-id1167836484679\">\n<p id=\"fs-id1167833057051\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a360d3c3c09797ea6614b0db16e5542_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#92;&#108;&#101;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836596407\">\n<div data-type=\"problem\" id=\"fs-id1167836596409\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34f8e7259ad00649f4e42639d99dbad2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#92;&#103;&#101;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836606749\">\n<p><span data-type=\"media\" id=\"fs-id1167836318353\" data-alt=\"The figure shows the graph of the inequality y less than or equal to minus two by three times x plus five and x greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_217_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequality y less than or equal to minus two by three times x plus five and x greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836663874\">\n<div data-type=\"problem\" id=\"fs-id1167836663876\">\n<p id=\"fs-id1167836663878\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f63ab4c635e95aa90bf7865225bf6012_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"103\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836363505\">\n<div data-type=\"problem\" id=\"fs-id1167836363507\">\n<p id=\"fs-id1167836664850\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90f78875a378e3a4040dd5ec8e167346_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836506503\">\n<p><span data-type=\"media\" id=\"fs-id1167836607100\" data-alt=\"The figure shows the graph of the inequalities y less than or equal to minus half x plus three and y less than one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_219_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y less than or equal to minus half x plus three and y less than one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836506581\">\n<div data-type=\"problem\" id=\"fs-id1167836506583\">\n<p id=\"fs-id1167829877906\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-736e3c16a5aa9022b03c41895b263c23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#52;&#121;&#60;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#60;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833060370\">\n<div data-type=\"problem\" id=\"fs-id1167833060372\">\n<p id=\"fs-id1167833025692\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ef0a61c98b88025ba89e793622a267a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#43;&#53;&#121;&#62;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#62;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836729905\">\n<p><span data-type=\"media\" id=\"fs-id1167836729908\" data-alt=\"The figure shows the graph of the inequalities minus three times x plus five times y greater than ten and x greater than minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_221_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities minus three times x plus five times y greater than ten and x greater than minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836792647\">\n<div data-type=\"problem\" id=\"fs-id1167836620418\">\n<p id=\"fs-id1167836620420\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4718a8b8d6746229bfbc0991ec716d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#92;&#103;&#101;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#108;&#101;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"62\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829890995\">\n<div data-type=\"problem\" id=\"fs-id1167829890997\">\n<p id=\"fs-id1167836621261\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15ef4ba24459c57fb1cc632e33bd950e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#92;&#108;&#101;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"75\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829742976\">\n<p><span data-type=\"media\" id=\"fs-id1167833364682\" data-alt=\"The figure shows the graph of the inequalities x less than or equal to minus one and y greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_223_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities x less than or equal to minus one and y greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836554483\">\n<div data-type=\"problem\" id=\"fs-id1167836554485\">\n<p id=\"fs-id1167836375864\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7df75f1affd3064f1ad13a652bb10a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#52;&#121;&#62;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833058265\">\n<div data-type=\"problem\" id=\"fs-id1167836535017\">\n<p id=\"fs-id1167836535019\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10e412744f70d85de664e351af694009_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#121;&#92;&#103;&#101;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"103\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829745862\">\n<p><span data-type=\"media\" id=\"fs-id1167836531493\" data-alt=\"The figure shows the graph of the inequalities x minus three times y greater than or equal to six and y greater than one third of x plus one. Two non intersecting lines, one in blue and the other in red, are shown.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_225_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities x minus three times y greater than or equal to six and y greater than one third of x plus one. Two non intersecting lines, one in blue and the other in red, are shown.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>No solution.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833250778\">\n<div data-type=\"problem\" id=\"fs-id1167833250780\">\n<p id=\"fs-id1167833382481\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f11c676c87d1356e866818bf07e4174e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#50;&#120;&#43;&#54;&#121;&#60;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#121;&#62;&#50;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836697594\">\n<div data-type=\"problem\" id=\"fs-id1167836697596\">\n<p id=\"fs-id1167836697598\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a07d33be573076198a63cce39de14e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#43;&#54;&#121;&#62;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#121;&#92;&#108;&#101;&#32;&#50;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"133\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833197311\">\n<p><span data-type=\"media\" id=\"fs-id1167829754453\" data-alt=\"The figure shows the graph of the inequalities minus three times x plus six times y greater than twelve and four times y less than or equal to two times x minus four. Two non intersecting lines, one in blue and the other in red, are shown.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_227_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities minus three times x plus six times y greater than twelve and four times y less than or equal to two times x minus four. Two non intersecting lines, one in blue and the other in red, are shown.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>No solution.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833056019\">\n<p id=\"fs-id1167833056021\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4430bdf11370415434fa6af721007153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#45;&#51;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#121;&#62;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836608976\">\n<div data-type=\"problem\" id=\"fs-id1167833186286\">\n<p id=\"fs-id1167833186288\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-866b2a826f061ad6edb5b858ab8e7394_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#52;&#121;&#92;&#103;&#101;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829586638\">\n<p><span data-type=\"media\" id=\"fs-id1167836547001\" data-alt=\"The figure shows the graph of the inequalities y greater than or equal to minus half x minus one and minus two times x plus four times y greater than or equal to four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_229_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y greater than or equal to minus half x minus one and minus two times x plus four times y greater than or equal to four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833346852\">\n<div data-type=\"problem\" id=\"fs-id1167826182813\">\n<p id=\"fs-id1167826182815\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f70c02d3b1f54e5e5119ab80cfcb331_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#52;&#121;&#60;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833066673\">\n<div data-type=\"problem\" id=\"fs-id1167825913939\">\n<p id=\"fs-id1167825913941\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d72f4b0a648f56f0aab7b19f8f7056d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#51;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#43;&#121;&#62;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836440537\">\n<p><span data-type=\"media\" id=\"fs-id1167833272751\" data-alt=\"The figure shows the graph of the inequalities y greater than or equal to three times x minus one and minus three times x plus y greater than minus four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_231_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y greater than or equal to three times x minus one and minus three times x plus y greater than minus four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836700468\">\n<div data-type=\"problem\" id=\"fs-id1167836356885\">\n<p id=\"fs-id1167836356887\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7300a29d3cb36b49e8f041210f68e407_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#121;&#62;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#54;&#121;&#62;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833008893\">\n<div data-type=\"problem\" id=\"fs-id1167833008895\">\n<p id=\"fs-id1167836701006\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24ca8fd86cb891210b53553e190fb400_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#43;&#52;&#121;&#60;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836523933\">\n<p><span data-type=\"media\" id=\"fs-id1167836663644\" data-alt=\"The figure shows the graph of the inequalities y less than three by fourth x minus two and minus three x plus four y less than seven. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_233_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities y less than three by fourth x minus two and minus three x plus four y less than seven. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836596247\"><strong data-effect=\"bold\">Solve Applications of Systems of Inequalities<\/strong><\/p>\n<p id=\"fs-id1167836609964\">In the following exercises, translate to a system of inequalities and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836609967\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836611857\">\n<p id=\"fs-id1167836611859\">Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for ?15 and the landscapes for ?10. She needs to sell at least ?800 worth of drawings in order to earn a profit.<\/p>\n<p id=\"fs-id1167833326352\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Will she make a profit if she sells 20 portraits and 35 landscapes?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Will she make a profit if she sells 50 portraits and 20 landscapes?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836523534\">\n<p id=\"fs-id1167836409011\">Jake does not want to spend more than ?50 on bags of fertilizer and peat moss for his garden. Fertilizer costs ?2 a bag and peat moss costs ?5 a bag. Jake\u2019s van can hold at most 20 bags.<\/p>\n<p id=\"fs-id1167833056343\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Can he buy 15 bags of fertilizer and 4 bags of peat moss?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Can he buy 10 bags of fertilizer and 10 bags of peat moss?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829744328\">\n<p id=\"fs-id1167829744330\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-836ef5ada90d4c71e5705f3461092381_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#102;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#112;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#102;&#43;&#112;&#92;&#108;&#101;&#32;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#102;&#43;&#53;&#112;&#92;&#108;&#101;&#32;&#53;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"121\" style=\"vertical-align: -38px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167833339148\" data-alt=\"The figure shows the graph of the inequalities f plus p less than or equal to twenty and two f and five p less than or equal to fifty. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_235_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities f plus p less than or equal to twenty and two f and five p less than or equal to fifty. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> yes<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832950886\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833057432\">\n<p id=\"fs-id1167833057434\">Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than ?500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is ?3 and for a package the cost is ?7.<\/p>\n<p id=\"fs-id1167836558183\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Can she mail 60 cards and 26 packages?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Can she mail 90 cards and 40 packages?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829695320\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829695322\">\n<p>Juan is studying for his final exams in chemistry and algebra. he knows he only has 24 hours to study, and it will take him at least three times as long to study for algebra than chemistry.<\/p>\n<p id=\"fs-id1167836788650\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Can he spend 4 hours on chemistry and 20 hours on algebra?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Can he spend 6 hours on chemistry and 18 hours on algebra?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836601793\">\n<p id=\"fs-id1167836287747\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-401ea3a44b3cfacf90784f8e77af4a26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#99;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#99;&#43;&#97;&#92;&#108;&#101;&#32;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#92;&#103;&#101;&#32;&#51;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"101\" style=\"vertical-align: -38px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167833135357\" data-alt=\"The figure shows the graph of the inequalities c plus a less than or equal to twenty four and a greater than or equal to three times c. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_237_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities c plus a less than or equal to twenty four and a greater than or equal to three times c. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> yes<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836511611\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836511613\">\n<p id=\"fs-id1167836511616\">Jocelyn is pregnant and so she needs to eat at least 500 more calories a day than usual. When buying groceries one day with a budget of ?15 for the extra food, she buys bananas that have 90 calories each and chocolate granola bars that have 150 calories each. The bananas cost ?0.35 each and the granola bars cost ?2.50 each.<\/p>\n<p id=\"fs-id1167826171973\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Could she buy 5 bananas and 6 granola bars?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Could she buy 3 bananas and 4 granola bars?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824731940\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167824731942\">\n<p id=\"fs-id1167829909635\">Mark is attempting to build muscle mass and so he needs to eat at least an additional 80 grams of protein a day. A bottle of protein water costs ?3.20 and a protein bar costs ?1.75. The protein water supplies 27 grams of protein and the bar supplies 16 gram. If he has ?10 dollars to spend<\/p>\n<p id=\"fs-id1167833076719\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Could he buy 3 bottles of protein water and 1 protein bar?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Could he buy no bottles of protein water and 5 protein bars?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833339515\">\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10ed947ce316dea2a85dedea4dc51f4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#119;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#98;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#55;&#119;&#43;&#49;&#54;&#98;&#62;&#56;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#46;&#50;&#48;&#119;&#43;&#49;&#46;&#55;&#53;&#98;&#92;&#108;&#101;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"168\" style=\"vertical-align: -38px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167833386413\" data-alt=\"The figure shows the graph of the inequalities twenty seven times w plus sixteen times b greater than eighty and three point two times w plus one point seven five b less than or equal to ten. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_239_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities twenty seven times w plus sixteen times b greater than eighty and three point two times w plus one point seven five b less than or equal to ten. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> no<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> yes<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836731285\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836731288\">\n<p id=\"fs-id1167836731290\">Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least 35 more grams of protein each day and no more than an additional 200 calories daily. An ounce of cheddar cheese has 7 grams of protein and 110 calories. An ounce of parmesan cheese has 11 grams of protein and 22 calories.<\/p>\n<p id=\"fs-id1167833025309\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Could she eat 1 ounce of cheddar cheese and 3 ounces of parmesan cheese?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Could she eat 2 ounces of cheddar cheese and 1 ounce of parmesan cheese?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836553814\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836553817\">\n<p id=\"fs-id1167836553819\">Mark is increasing his exercise routine by running and walking at least 4 miles each day. His goal is to burn a minimum of 1500 calories from this exercise. Walking burns 270 calories\/mile and running burns 650 calories.<\/p>\n<p id=\"fs-id1167833135425\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Could he meet his goal by walking 3 miles and running 1 mile?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Could he his goal by walking 2 miles and running 2 mile<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836493827\">\n<p id=\"fs-id1167836493829\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3d903b0af08159bf9422c95037cba49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#119;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#114;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#119;&#43;&#114;&#92;&#103;&#101;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#55;&#48;&#119;&#43;&#54;&#53;&#48;&#114;&#92;&#103;&#101;&#32;&#49;&#53;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"177\" style=\"vertical-align: -38px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836585157\" data-alt=\"The figure shows the graph of the inequalities w plus r greater than or equals to four and two seventy w plus six fifty r greater than or equal to fifteen hundred. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_241_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of the inequalities w plus r greater than or equals to four and two seventy w plus six fifty r greater than or equal to fifteen hundred. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> no<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836615624\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167836554970\">\n<div data-type=\"problem\" id=\"fs-id1167836554972\">\n<p>Graph the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8db8d4a4fda1a581634d774a78e016de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#121;&#92;&#103;&#101;&#32;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/> How do you know which side of the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b709ed39dd85d3c913cd6508754675ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#121;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\" \/> should be shaded?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836689087\">\n<div data-type=\"problem\" id=\"fs-id1167836689089\">\n<p id=\"fs-id1167830087738\">Graph the system <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63fac594a8b95d52bedabe3a713066ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#121;&#92;&#108;&#101;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"130\" style=\"vertical-align: -17px;\" \/> What does the solution mean?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836502447\">\n<p id=\"fs-id1167836502449\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833024556\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167836547975\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167824584580\" data-alt=\"The figure shows a table with four columns and four rows. The first row is the title row. The four titles are I can\u2026, confidently, with some help and No \u2013 I don\u2019t get it! In the second row of the first column, the text says \u2018determine whether an ordered pair is a solution of a system of linear inequalities.\u2019 In the third row of the first column, the text says \u2018solve applications of systems of inequalities.\u2019\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a table with four columns and four rows. The first row is the title row. The four titles are I can\u2026, confidently, with some help and No \u2013 I don\u2019t get it! In the second row of the first column, the text says \u2018determine whether an ordered pair is a solution of a system of linear inequalities.\u2019 In the third row of the first column, the text says \u2018solve applications of systems of inequalities.\u2019\" \/><\/span><\/p>\n<p id=\"fs-id1167836296437\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p>\n<\/div>\n<\/div>\n<div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1167836390206\">\n<h3 data-type=\"title\">Chapter Review Exercises<\/h3>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836704892\">\n<h4 data-type=\"title\"><a href=\"\/contents\/a236420d-b79a-43e3-8f7c-1b8e18c5045e\" class=\"target-chapter\">Solve Systems of Linear Equations with Two Variables<\/a><\/h4>\n<p id=\"fs-id1167829644764\"><strong data-effect=\"bold\">Determine Whether an Ordered Pair is a Solution of a System of Equations<\/strong>.<\/p>\n<p id=\"fs-id1167836321756\">In the following exercises, determine if the following points are solutions to the given system of equations.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833412561\">\n<p id=\"fs-id1167833412564\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d07536f66af9fc8ac17178a125906ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#51;&#121;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#52;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a17ffffadfb8456567f4803e943d15a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-017f20ea0c6fda3470cedb20ea0b5537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829809271\">\n<div data-type=\"problem\" id=\"fs-id1167829809273\">\n<p id=\"fs-id1167829809275\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9d6395a8debcb2709bcc28ef4c6b0db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"93\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccc35b92a6567ced556cb46473589564_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ef2cb721f4e314e492442d7e4973839_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829634159\">\n<p id=\"fs-id1167829580545\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829741600\"><strong data-effect=\"bold\">Solve a System of Linear Equations by Graphing<\/strong><\/p>\n<p id=\"fs-id1167836415885\">In the following exercises, solve the following systems of equations by graphing.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836390561\">\n<div data-type=\"problem\" id=\"fs-id1167836390563\">\n<p id=\"fs-id1167836390565\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85e175b8b7c696dcfcf96718a696d93e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836690408\">\n<div data-type=\"problem\" id=\"fs-id1167836690411\">\n<p id=\"fs-id1167836690413\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91de93bcefc0b67579689a0236c49650_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#52;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836705411\">\n<p><span data-type=\"media\" id=\"fs-id1167829747204\" data-alt=\"The figure shows the graph of equations x plus four times y equal to minus one and x equal to three. Two intersecting lines are shown.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_322_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of equations x plus four times y equal to minus one and x equal to three. Two intersecting lines are shown.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833030870\">\n<div data-type=\"problem\" id=\"fs-id1167833030872\">\n<p id=\"fs-id1167833030875\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e726188de4fddc3e83953c444f89d5ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#50;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836689138\">\n<div data-type=\"problem\" id=\"fs-id1167836689140\">\n<p id=\"fs-id1167836689142\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00045687fabb321ce3dfcafbfd057c28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"104\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836789431\">\n<p><span data-type=\"media\" id=\"fs-id1167836789434\" data-alt=\"The figure shows the graph for the equations minus x plus two times y equal to four and y equal to half x minus three. Two parallel lines are shown.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_324_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph for the equations minus x plus two times y equal to four and y equal to half x minus three. Two parallel lines are shown.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>no solution<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836519173\">In the following exercises, without graphing determine the number of solutions and then classify the system of equations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167825901073\">\n<div data-type=\"problem\" id=\"fs-id1167825901076\">\n<p id=\"fs-id1167825901078\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f97e051b46dd1ae62116d0378d35dadd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#53;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829719704\">\n<div data-type=\"problem\" id=\"fs-id1167833202536\">\n<p id=\"fs-id1167833202538\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-467c2091fe41199dc0216dc605676cba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#50;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#51;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829704947\">\n<p id=\"fs-id1167829704950\">one solution, consistent system, independent equations<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836283200\">\n<div data-type=\"problem\" id=\"fs-id1167836551623\">\n<p id=\"fs-id1167836551625\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b58394e3091d64e96ca7d4f6a5b1be41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#53;&#120;&#45;&#52;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#120;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829739814\"><strong data-effect=\"bold\">Solve a System of Equations by Substitution<\/strong><\/p>\n<p id=\"fs-id1167829627719\">In the following exercises, solve the systems of equations by substitution.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829627722\">\n<div data-type=\"problem\" id=\"fs-id1167829832930\">\n<p id=\"fs-id1167829832932\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f73ef1fdd54fbea1b34c54cc58b2e88a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#50;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836698903\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b385c1ebb27da6a9f60a8f21a49f0483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836544387\">\n<div data-type=\"problem\" id=\"fs-id1167836544389\">\n<p id=\"fs-id1167821895442\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-793b0bcabc60ccc14ebffe3f1d1185c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#61;&#45;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836386679\">\n<div data-type=\"problem\" id=\"fs-id1167836386681\">\n<p id=\"fs-id1167836386683\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b847bf394ac5169fcd0f0a4a121b2a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#50;&#120;&#43;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836327160\">\n<p id=\"fs-id1167836327162\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4307e3633578c7e56f8f767895b20497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826046358\">\n<div data-type=\"problem\" id=\"fs-id1167826046360\">\n<p id=\"fs-id1167836573256\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62c56d99523de3cde0cf96af9ade35ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#53;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836666697\">\n<div data-type=\"problem\" id=\"fs-id1167836666699\">\n<p id=\"fs-id1167836730049\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49fd56eead91d05d6a3cfedf21a292b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826077166\">\n<p id=\"fs-id1167826077168\">infinitely many solutions<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836287290\"><strong data-effect=\"bold\">Solve a System of Equations by Elimination<\/strong><\/p>\n<p id=\"fs-id1167836287296\">In the following exercises, solve the systems of equations by elimination<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167824773957\">\n<div data-type=\"problem\" id=\"fs-id1167824773959\">\n<p id=\"fs-id1167824773961\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2c8581042c20606af4c9dd3559c0cc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#61;&#45;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829737940\">\n<div data-type=\"problem\" id=\"fs-id1167826046341\">\n<p id=\"fs-id1167826046343\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4389cbc003571a8875cd65af795b7b44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#56;&#121;&#61;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826132850\">\n<p id=\"fs-id1167833052812\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6de38faced02869069c306c45a22c5e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829958013\">\n<div data-type=\"problem\" id=\"fs-id1167829958015\">\n<p id=\"fs-id1167829958017\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96a58dc5af8d883b1f68dce0a019b234_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#57;&#120;&#43;&#52;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#51;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829627584\">\n<div data-type=\"problem\" id=\"fs-id1167829627586\">\n<p id=\"fs-id1167829627589\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c07f8d0af2b6176c1a17c1d0fcaf678b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829644664\">\n<p id=\"fs-id1167829644666\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccc35b92a6567ced556cb46473589564_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836690368\">\n<div data-type=\"problem\" id=\"fs-id1167836690370\">\n<p id=\"fs-id1167833310434\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c84e67ebbbde6a28e2c3e05dd088533_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#51;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#54;&#121;&#61;&#45;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836626839\"><strong data-effect=\"bold\">Choose the Most Convenient Method to Solve a System of Linear Equations<\/strong><\/p>\n<p id=\"fs-id1167836626846\">In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836687102\">\n<div data-type=\"problem\" id=\"fs-id1167836687104\">\n<p id=\"fs-id1167836687106\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b33549695a975874db45fe3113324bea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#54;&#120;&#45;&#53;&#121;&#61;&#50;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#49;&#48;&#121;&#61;&#45;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"143\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833239499\">\n<p id=\"fs-id1167833239501\">elimination<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833239507\">\n<div data-type=\"problem\" id=\"fs-id1167829619825\">\n<p id=\"fs-id1167829619827\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c587730ef61797c922ccfcae5d84f88b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#51;&#120;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#53;&#121;&#61;&#50;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836408465\">\n<h4 data-type=\"title\"><a href=\"\/contents\/560ed071-7c89-41ad-8282-94b385324dd9\" class=\"target-chapter\">Solve Applications with Systems of Equations<\/a><\/h4>\n<p id=\"fs-id1167836712373\"><strong data-effect=\"bold\">Solve Direct Translation Applications<\/strong><\/p>\n<p id=\"fs-id1167836712380\">In the following exercises, translate to a system of equations and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836415081\">\n<div data-type=\"problem\" id=\"fs-id1167836415083\">\n<p id=\"fs-id1167836415085\">Mollie wants to plant 200 bulbs in her garden, all irises and tulips. She wants to plant three times as many tulips as irises. How many irises and how many tulips should she plant?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836415091\">\n<p id=\"fs-id1167829747161\">50 irises and 150 tulips<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829747167\">\n<div data-type=\"problem\" id=\"fs-id1167829747169\">\n<p id=\"fs-id1167829747171\">Ashanti has been offered positions by two phone companies. The first company pays a salary of ?22,000 plus a commission of ?100 for each contract sold. The second pays a salary of ?28,000 plus a commission of ?25 for each contract sold. How many contract would need to be sold to make the total pay the same?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836611755\">\n<p id=\"fs-id1167836611757\">Leroy spent 20 minutes jogging and 40 minutes cycling and burned 600 calories. The next day, Leroy swapped times, doing 40 minutes of jogging and 20 minutes of cycling and burned the same number of calories. How many calories were burned for each minute of jogging and how many for each minute of cycling?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836611764\">\n<p id=\"fs-id1167836714034\">10 calories jogging and 10 calories cycling<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836714039\">\n<div data-type=\"problem\" id=\"fs-id1167836714042\">\n<p id=\"fs-id1167836714044\">Troy and Lisa were shopping for school supplies. Each purchased different quantities of the same notebook and calculator. Troy bought four notebooks and five calculators for ?116. Lisa bought two notebooks and three calculators for ?68. Find the cost of each notebook and each thumb drive.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836714046\"><strong data-effect=\"bold\">Solve Geometry Applications<\/strong><\/p>\n<p id=\"fs-id1167826206187\">In the following exercises, translate to a system of equations and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826206190\">\n<div data-type=\"problem\" id=\"fs-id1167826206192\">\n<p id=\"fs-id1167826206194\">The difference of two supplementary angles is 58 degrees. Find the measures of the angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833057704\">\n<p id=\"fs-id1167833057706\">119, 61<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833057711\">\n<div data-type=\"problem\" id=\"fs-id1167836645569\">\n<p id=\"fs-id1167836645571\">Two angles are complementary. The measure of the larger angle is five more than four times the measure of the smaller angle. Find the measures of both angles.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833186148\">\n<div data-type=\"problem\" id=\"fs-id1167833186150\">\n<p id=\"fs-id1167833186152\">The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826130274\">\n<p id=\"fs-id1167826130276\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6eedb3526de357fd1a26da5d7ce355d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ad9be44031b1a74ffc05457eb10b6fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829810710\">\n<div data-type=\"problem\" id=\"fs-id1167829810712\">\n<p id=\"fs-id1167829810714\">Becca is hanging a 28 foot floral garland on the two sides and top of a pergola to prepare for a wedding. The height is four feet less than the width. Find the height and width of the pergola.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836620062\">\n<div data-type=\"problem\" id=\"fs-id1167836620064\">\n<p id=\"fs-id1167836620066\">The perimeter of a city rectangular park is 1428 feet. The length is 78 feet more than twice the width. Find the length and width of the park.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829790114\">\n<p id=\"fs-id1167829790116\">the length is 450 feet, the width is 264 feet<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829790121\"><strong data-effect=\"bold\">Solve Uniform Motion Applications<\/strong><\/p>\n<p id=\"fs-id1167825696609\">In the following exercises, translate to a system of equations and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167825696613\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167825696615\">\n<p id=\"fs-id1167825696617\">Sheila and Lenore were driving to their grandmother\u2019s house. Lenore left one hour after Sheila. Sheila drove at a rate of 45 mph, and Lenore drove at a rate of 60 mph. How long will it take for Lenore to catch up to Sheila?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833056470\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833056472\">\n<p id=\"fs-id1167833056474\">Bob left home, riding his bike at a rate of 10 miles per hour to go to the lake. Cheryl, his wife, left 45 minutes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd2b5c456310ac5bea78c6d2acf6f4f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -7px;\" \/> hour) later, driving her car at a rate of 25 miles per hour. How long will it take Cheryl to catch up to Bob?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836684835\">\n<p id=\"fs-id1167824811868\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> an hour<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836685478\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836685480\">\n<p id=\"fs-id1167836685482\">Marcus can drive his boat 36 miles down the river in three hours but takes four hours to return upstream. Find the rate of the boat in still water and the rate of the current.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836477237\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836477239\">\n<p>A passenger jet can fly 804 miles in 2 hours with a tailwind but only 776 miles in 2 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829696497\">\n<p id=\"fs-id1167829696499\">the rate of the jet is 395 mph, the rate of the wind is 7 mph<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829696506\">\n<h4 data-type=\"title\"><a href=\"\/contents\/4cdf0dd0-1221-47ea-9e33-0562d98f6eec\" class=\"target-chapter\">Solve Mixture Applications with Systems of Equations<\/a><\/h4>\n<p id=\"fs-id1167825949254\"><strong data-effect=\"bold\">Solve Mixture Applications with Systems of Equations<\/strong><\/p>\n<p id=\"fs-id1167832951099\">For the following exercises, translate to a system of equations and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832951102\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832951104\">\n<p>Lynn paid a total of ?2,780 for 261 tickets to the theater. Student tickets cost ?10 and adult tickets cost ?15. How many student tickets and how many adult tickets did Lynn buy?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836444773\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836502937\">\n<p id=\"fs-id1167836502939\">Priam has dimes and pennies in a cup holder in his car. The total value of the coins is ?4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836728066\">\n<p id=\"fs-id1167836728068\">41 dimes and 11 pennies<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836391929\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836391931\">\n<p id=\"fs-id1167836391933\">Yumi wants to make 12 cups of party mix using candies and nuts. Her budget requires the party mix to cost her ?1.29 per cup. The candies are ?2.49 per cup and the nuts are ?0.69 per cup. How many cups of candies and how many cups of nuts should she use?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836349275\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836349277\">\n<p id=\"fs-id1167836349279\">A scientist needs 70 liters of a 40% solution of alcohol. He has a 30% and a 60% solution available. How many liters of the 30% and how many liters of the 60% solutions should he mix to make the 40% solution?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167823012042\">\n<p id=\"fs-id1167823012045\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2109e856dbdc851a05334be2edecec70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#54;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"27\" style=\"vertical-align: -6px;\" \/> liters of 30% solution, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9227fb8dc7ab13b623fa2f2dbb249e71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"27\" style=\"vertical-align: -6px;\" \/> liters of 60% solution<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832925457\"><strong data-effect=\"bold\">Solve Interest Applications<\/strong><\/p>\n<p id=\"fs-id1167832925463\">For the following exercises, translate to a system of equations and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829859217\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829859219\">\n<p id=\"fs-id1167829859221\">Jack has ?12,000 to invest and wants to earn 7.5% interest per year. He will put some of the money into a savings account that earns 4% per year and the rest into CD account that earns 9% per year. How much money should he put into each account?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829924270\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829924273\">\n<p id=\"fs-id1167829924275\">When she graduates college, Linda will owe ?43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was ?1,585. What is the amount of each loan?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836556586\">\n<p id=\"fs-id1167836556588\">?29,000 for the federal loan, ?14,000 for the private loan<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836309828\">\n<h4 data-type=\"title\"><a href=\"\/contents\/56fc492e-b71a-4a4a-bab4-b95331687fd3\" class=\"target-chapter\">Solve Systems of Equations with Three Variables<\/a><\/h4>\n<p id=\"fs-id1167836387009\"><strong data-effect=\"bold\">Solve Systems of Equations with Three Variables<\/strong><\/p>\n<p id=\"fs-id1167836387016\">In the following exercises, determine whether the ordered triple is a solution to the system.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836387020\">\n<div data-type=\"problem\" id=\"fs-id1167829908520\">\n<p id=\"fs-id1167829908522\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4511ce6e446223778f1e4c7fc488793_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#52;&#121;&#45;&#51;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#54;&#121;&#43;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"152\" style=\"vertical-align: -28px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3192a88b114ea0b2799caf22405de3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed349cfc50391f6bcebabc6c89669d23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836705096\">\n<div data-type=\"problem\" id=\"fs-id1167836705098\">\n<p id=\"fs-id1167829957958\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4a245364abc258de5ea12d92a6e3973_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#45;&#122;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"147\" style=\"vertical-align: -28px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-797ad99adb3ef2fbd7a2283fd3745ab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#53;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"70\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce7aa838e0429abb5822644b14b709b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"71\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836620221\">\n<p id=\"fs-id1167836620223\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167824738895\"><strong data-effect=\"bold\">Solve a System of Linear Equations with Three Variables<\/strong><\/p>\n<p id=\"fs-id1167829683650\">In the following exercises, solve the system of equations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829683653\">\n<div data-type=\"problem\" id=\"fs-id1167829683656\">\n<p id=\"fs-id1167829683658\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba9dc063ea452d762aa7f3850fb9f1c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#53;&#121;&#43;&#52;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#50;&#121;&#43;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"152\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829690837\">\n<div data-type=\"problem\" id=\"fs-id1167829690839\">\n<p id=\"fs-id1167829690841\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2cd34715cfe2a0f5128d79d39239d34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#121;&#43;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#50;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#122;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#121;&#45;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"168\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824763564\">\n<p id=\"fs-id1167824763566\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0efca593ca814981c8f4606d8b5fde1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836322233\">\n<div data-type=\"problem\" id=\"fs-id1167836322235\">\n<p id=\"fs-id1167836322237\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24d9703f59dfdee781586414803a458d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#53;&#120;&#43;&#51;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#121;&#43;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"126\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833240289\">\n<div data-type=\"problem\" id=\"fs-id1167829748636\">\n<p id=\"fs-id1167829748638\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aea6601b106cf1465bade3cf7be311ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#51;&#121;&#43;&#122;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#43;&#122;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#52;&#121;&#43;&#50;&#122;&#61;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"161\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832940462\">\n<p id=\"fs-id1167832940465\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832940470\">\n<div data-type=\"problem\" id=\"fs-id1167836792711\">\n<p id=\"fs-id1167836792713\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-859ab04b7559fd6f0e2de3f8354e3d4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#45;&#51;&#121;&#43;&#50;&#122;&#61;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#45;&#51;&#122;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#121;&#45;&#50;&#122;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"157\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836387828\"><strong data-effect=\"bold\">Solve Applications using Systems of Linear Equations with Three Variables<\/strong><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836387834\">\n<div data-type=\"problem\" id=\"fs-id1167836387836\">\n<p id=\"fs-id1167829597480\">After attending a major league baseball game, the patrons often purchase souvenirs. If a family purchases 4 t-shirts, a cap and 1 stuffed animal their total is ?135. A couple buys 2 t-shirts, a cap and 3 stuffed animals for their nieces and spends ?115. Another couple buys 2 t-shirts, a cap and 1 stuffed animal and their total is ?85. What is the cost of each item?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829597487\">\n<p id=\"fs-id1167829597489\">25, 20, 15<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829689026\">\n<h4 data-type=\"title\"><a href=\"\/contents\/6a4b38a5-e5d8-4f37-ac52-172097013b24\" class=\"target-chapter\">Solve Systems of Equations Using Matrices<\/a><\/h4>\n<p id=\"fs-id1167826188078\"><strong data-effect=\"bold\">Write the Augmented Matrix for a System of Equations.<\/strong><\/p>\n<p id=\"fs-id1167826188084\">Write each system of linear equations as an augmented matrix.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826188087\">\n<div data-type=\"problem\" id=\"fs-id1167825824095\">\n<p id=\"fs-id1167825824097\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4775405c4ad30a631b3c8c0a7bc570a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#50;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"124\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833086715\">\n<div data-type=\"problem\" id=\"fs-id1167833086717\">\n<p id=\"fs-id1167833086720\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18645470d80dab2f4e574fa4faf71b11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#43;&#51;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#45;&#51;&#122;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#43;&#50;&#122;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"157\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829696207\">\n<p id=\"fs-id1167829696209\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-847541d47daab64b78b2ef7a2a122111_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"253\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829893644\">Write the system of equations that that corresponds to the augmented matrix.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829893647\">\n<div data-type=\"problem\" id=\"fs-id1167829893649\">\n<p id=\"fs-id1167836613622\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38a503f43e2f73635ac58480c3636d52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"186\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832977348\">\n<div data-type=\"problem\" id=\"fs-id1167832977350\">\n<p id=\"fs-id1167832977352\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a5c89085f0c00c986105ad74e95119b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"261\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825703256\">\n<p id=\"fs-id1167833380503\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-534ca7ba53dc993af663fea121afd654_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#45;&#50;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#121;&#43;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"126\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833186073\">In the following exercises, perform the indicated operations on the augmented matrices.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833186076\">\n<div data-type=\"problem\" id=\"fs-id1167833186078\">\n<p id=\"fs-id1167833186080\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ffbd45de2677c2ac226d34e67ee7e89c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"186\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167824780927\"><span class=\"token\">\u24d0<\/span> Interchange rows 2 and 1.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Multiply row 1 by 4.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Multiply row 2 by 3 and add to row 1.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836573286\">\n<div data-type=\"problem\" id=\"fs-id1167833142230\">\n<p id=\"fs-id1167833142232\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88051c98b86b211e7bb10bc3b30971d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"261\" style=\"vertical-align: -28px;\" \/><\/p>\n<p id=\"fs-id1167836752836\"><span class=\"token\">\u24d0<\/span> Interchange rows 2 and 3.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Multiply row 1 by 2.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Multiply row 3 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> and add to row 2.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824774461\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ec62e3a43eb35b94109238378c3802d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"253\" style=\"vertical-align: -28px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0d29d845afef1fdb502c58112db6d68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"253\" style=\"vertical-align: -28px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef11c41011943a402cbeef8fad8bc7fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#92;&#92;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"253\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836791631\"><strong data-effect=\"bold\">Solve Systems of Equations Using Matrices<\/strong><\/p>\n<p id=\"fs-id1167836791637\">In the following exercises, solve each system of equations using a matrix.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829594309\">\n<div data-type=\"problem\" id=\"fs-id1167829594311\">\n<p id=\"fs-id1167829594313\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8426b5499aa3b47db2fb6d6163973257_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#43;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836293784\">\n<div data-type=\"problem\" id=\"fs-id1167836293786\">\n<p id=\"fs-id1167836293788\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c49251fc919acd821214c6a2b66c351_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#121;&#43;&#51;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#45;&#122;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#43;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"157\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836292856\">\n<p id=\"fs-id1167836292859\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90eae3c25410039b1747284ff02a8fde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#53;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836341414\">\n<div data-type=\"problem\" id=\"fs-id1167836432792\">\n<p id=\"fs-id1167836432794\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac1fd9dbb34b3d6893bb17cc3705dcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#121;&#43;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#51;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"126\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833310524\">\n<div data-type=\"problem\" id=\"fs-id1167833310526\">\n<p id=\"fs-id1167833310529\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0268b788043e6da4709aed42f991d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#121;&#45;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#45;&#50;&#122;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"156\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833274705\">\n<p id=\"fs-id1167833274707\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829715988\">\n<div data-type=\"problem\" id=\"fs-id1167829715990\">\n<p id=\"fs-id1167829715993\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25c76eb10dc5ed1d13d391f44dbfe3aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#45;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#45;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"147\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829785269\">\n<h4 data-type=\"title\"><a href=\"\/contents\/0ab06d7a-f85b-4c4d-b56b-aa826ea612a4\" class=\"target-chapter\">Solve Systems of Equations Using Determinants<\/a><\/h4>\n<p id=\"fs-id1167830092974\"><strong data-effect=\"bold\">Evaluate the Determinant of a 2 \u00d7 2 Matrix<\/strong><\/p>\n<p id=\"fs-id1167830092980\">In the following exercise, evaluate the determinate of the square matrix.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830092984\">\n<div data-type=\"problem\" id=\"fs-id1167830092986\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e07f514e1555bb44a89b619ae7c5b74e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"107\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836685512\">\n<p id=\"fs-id1167836685514\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836753517\"><strong data-effect=\"bold\">Evaluate the Determinant of a 3 \u00d7 3 Matrix<\/strong><\/p>\n<p id=\"fs-id1167836753524\">In the following exercise, find and then evaluate the indicated minors.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836753527\">\n<div data-type=\"problem\" id=\"fs-id1167836753529\">\n<p id=\"fs-id1167829942346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4e56e8e5b3c8c468acbd8de110aec2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"193\" style=\"vertical-align: -22px;\" \/> Find the minor <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d475a40b209f4b36b08b4dd3abcd0f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"14\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4da37acd0e5d0b3aeee14efbcfc1202_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167825024028\">In the following exercise, evaluate each determinant by expanding by minors along the first row.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836616233\">\n<div data-type=\"problem\" id=\"fs-id1167836616235\">\n<p id=\"fs-id1167836616237\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27b3aa18e4fe943dd9dab313f3cf97b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"187\" style=\"vertical-align: -23px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824732987\">\n<p id=\"fs-id1167824732989\">21<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829859252\">In the following exercise, evaluate each determinant by expanding by minors.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829859255\">\n<div data-type=\"problem\" id=\"fs-id1167829859257\">\n<p id=\"fs-id1167829859259\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e06677eb9a3bcec5fd9e26aee9a908f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"160\" style=\"vertical-align: -23px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826030458\"><strong data-effect=\"bold\">Use Cramer\u2019s Rule to Solve Systems of Equations<\/strong><\/p>\n<p id=\"fs-id1167826030465\">In the following exercises, solve each system of equations using Cramer\u2019s rule<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167824710198\">\n<div data-type=\"problem\" id=\"fs-id1167824710200\">\n<p id=\"fs-id1167824710202\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49ff151ef7419336d89331d2bd5ec391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#121;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836574872\">\n<p id=\"fs-id1167836574874\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f307aa6b65fb69739e6c2b7458409de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826206229\">\n<div data-type=\"problem\" id=\"fs-id1167836611683\">\n<p id=\"fs-id1167836611685\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46af7d0154576c52b63bbcb285e6c8e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#53;&#121;&#45;&#52;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#45;&#50;&#122;&#61;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"166\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829807833\">\n<div data-type=\"problem\" id=\"fs-id1167829807835\">\n<p id=\"fs-id1167836611137\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd958c5a425944a914f86a59bb431a44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#53;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#121;&#45;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#51;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"126\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836787589\">\n<p id=\"fs-id1167836787592\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cccb83f742cb3d351a80d5b7c423823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829894130\">\n<div data-type=\"problem\" id=\"fs-id1167829894132\">\n<p id=\"fs-id1167829894134\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25c76eb10dc5ed1d13d391f44dbfe3aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#45;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#45;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"147\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829737696\">\n<div data-type=\"problem\" id=\"fs-id1167829737698\">\n<p id=\"fs-id1167829737700\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f45a19c3eea57ce3b02e4bc58e777033_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#52;&#121;&#45;&#51;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#122;&#61;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#45;&#50;&#122;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"165\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836689272\">\n<p id=\"fs-id1167836689274\">inconsistent<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836363671\"><strong data-effect=\"bold\">Solve Applications Using Determinants<\/strong><\/p>\n<p id=\"fs-id1167836363677\">In the following exercises, determine whether the given points are collinear.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836549078\">\n<div data-type=\"problem\" id=\"fs-id1167836549080\">\n<p id=\"fs-id1167836549082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43116796bd5b25cb2c6f27ca89f6fd40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-383ffb8dfe258fa508a14d15ba003120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ab12b849de44ae6a90ccee83f8e1018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167826169690\">\n<h4 data-type=\"title\"><a href=\"\/contents\/ccd859e9-a2f9-4004-abe9-5823a43ab70f\" class=\"target-chapter\">Graphing Systems of Linear Inequalities<\/a><\/h4>\n<p><strong data-effect=\"bold\">Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities<\/strong><\/p>\n<p id=\"fs-id1167836331129\">In the following exercises, determine whether each ordered pair is a solution to the system.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836712344\">\n<div data-type=\"problem\" id=\"fs-id1167836712346\">\n<p id=\"fs-id1167836712348\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45e1d804bbd20fc9b141442dfa2452ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#43;&#121;&#62;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#121;&#92;&#108;&#101;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167824734409\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25c4864d7eaa7ae5b2fe81ae29cf46af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dbaec542907415eac32615dfae0ae911_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833257706\">\n<p id=\"fs-id1167833257708\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836320828\">\n<div data-type=\"problem\" id=\"fs-id1167836755237\">\n<p id=\"fs-id1167836755239\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-937a5dc0e584b2010c206492ae5afab6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#121;&#92;&#108;&#101;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"113\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167836730240\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71459a30eb1bcf4f7794fb38957a6457_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4342a67bb7c6d83046df964bd6a27a39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833347098\"><strong data-effect=\"bold\">Solve a System of Linear Inequalities by Graphing<\/strong><\/p>\n<p id=\"fs-id1167833347105\">In the following exercises, solve each system by graphing.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167826172302\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-926a9c0dc81cc576a34918c43264e408_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#60;&#51;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#45;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"106\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829716918\">\n<p><span data-type=\"media\" id=\"fs-id1167829716921\" data-alt=\"The figure shows the graph of inequalities y less than three times x plus one and y greater than or equal to minus x minus two. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_202_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of inequalities y less than three times x plus one and y greater than or equal to minus x minus two. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824578795\">\n<div data-type=\"problem\" id=\"fs-id1167824578797\">\n<p id=\"fs-id1167824578799\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc293d3c1840c19fd52eacb9beecf36b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#62;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"106\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826171494\">\n<div data-type=\"problem\" id=\"fs-id1167826171496\">\n<p id=\"fs-id1167829808730\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3eeeea4dc46b3b9688fd3628c30bdc2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#51;&#121;&#60;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#52;&#121;&#92;&#103;&#101;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829628080\">\n<p><span data-type=\"media\" id=\"fs-id1167829908492\" data-alt=\"The figure shows the graph of inequalities two times x minus three times y less six and three times x plus four times y greater than or equal to twelve. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_204_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of inequalities two times x minus three times y less six and three times x plus four times y greater than or equal to twelve. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The solution is the grey region.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836626881\">\n<div data-type=\"problem\" id=\"fs-id1167836626883\">\n<p id=\"fs-id1167836626885\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b44cb5d033949af223afc31cb9fa0060_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#92;&#103;&#101;&#32;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833129094\">\n<div data-type=\"problem\" id=\"fs-id1167833129096\">\n<p id=\"fs-id1167833129098\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-633225a268a9fe7ae6db037cdaaaf7c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#51;&#121;&#60;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836798046\">\n<p><span data-type=\"media\" id=\"fs-id1167832966114\" data-alt=\"The figure shows the graph of inequalities x plus three times y less than five and y greater than or equal to minus one third x plus six. Two parallel lines, one in red and the other in blue, are shown. An area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_206_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of inequalities x plus three times y less than five and y greater than or equal to minus one third x plus six. Two parallel lines, one in red and the other in blue, are shown. An area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>No solution.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836692578\">\n<div data-type=\"problem\" id=\"fs-id1167836692580\">\n<p id=\"fs-id1167836692582\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ee7ee9203a7518d684157c79a103609_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#50;&#120;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#54;&#120;&#43;&#51;&#121;&#62;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"139\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836731539\"><strong data-effect=\"bold\">Solve Applications of Systems of Inequalities<\/strong><\/p>\n<p id=\"fs-id1167836731546\">In the following exercises, translate to a system of inequalities and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836529685\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836529687\">\n<p id=\"fs-id1167836529689\">Roxana makes bracelets and necklaces and sells them at the farmers\u2019 market. She sells the bracelets for ?12 each and the necklaces for ?18 each. At the market next weekend she will have room to display no more than 40 pieces, and she needs to sell at least ?500 worth in order to earn a profit.<\/p>\n<p id=\"fs-id1167836529692\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Should she display 26 bracelets and 14 necklaces?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Should she display 39 bracelets and 1 necklace?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836616137\">\n<p id=\"fs-id1167836616139\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d2c6f4bca5bc01bf6c208c5759110f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#98;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#110;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#98;&#43;&#110;&#92;&#108;&#101;&#32;&#52;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#49;&#50;&#98;&#43;&#49;&#56;&#110;&#92;&#103;&#101;&#32;&#53;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"147\" style=\"vertical-align: -38px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167826131486\" data-alt=\"The figure shows the graph of b plus n equal to forty and twelve b plus eighteen n equal to five hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_315_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of b plus n equal to forty and twelve b plus eighteen n equal to five hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> yes<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752821\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829752823\">\n<p id=\"fs-id1167829752825\">Annie has a budget of ?600 to purchase paperback books and hardcover books for her classroom. She wants the number of hardcover to be at least 5 more than three times the number of paperback books. Paperback books cost ?4 each and hardcover books cost ?15 each.<\/p>\n<p id=\"fs-id1167832989101\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Can she buy 8 paperback books and 40 hardcover books?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Can she buy 10 paperback books and 37 hardcover books?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1167833346751\">\n<h3 data-type=\"title\">Chapter Practice Test<\/h3>\n<p id=\"fs-id1167829906586\">In the following exercises, solve the following systems by graphing.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829906589\">\n<div data-type=\"problem\" id=\"fs-id1167829906591\">\n<p id=\"fs-id1167829906593\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-166c187e295708e0e2f733856132dbae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#50;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836663833\">\n<p><span data-type=\"media\" id=\"fs-id1167833076666\" data-alt=\"The figure shows the graph of inequalities h equal to three p plus five and four times p plus fifteen times h equal to six hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_317_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of inequalities h equal to three p plus five and four times p plus fifteen times h equal to six hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1564d1d2328bb6bd9e7b30e6d573d2fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829877873\">\n<div data-type=\"problem\" id=\"fs-id1167829877876\">\n<p id=\"fs-id1167833158737\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ba2baad77902dcfc1443f99496d1988_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#62;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#108;&#101;&#32;&#51;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"106\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833339402\">In the following exercises, solve each system of equations. Use either substitution or elimination.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829908677\">\n<div data-type=\"problem\" id=\"fs-id1167829908679\">\n<p id=\"fs-id1167829908681\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca06fee572b400f78d32a5fe483c58a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#52;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836705196\">\n<p id=\"fs-id1167836705198\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb160c5e6177bdd7a1d220c410258e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624883\">\n<div data-type=\"problem\" id=\"fs-id1167829624885\">\n<p id=\"fs-id1167829624887\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d351356a9efac6cabb3776dbdb7c5c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#43;&#52;&#121;&#61;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#53;&#121;&#61;&#45;&#50;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"133\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833058733\">\n<div data-type=\"problem\" id=\"fs-id1167829808045\">\n<p id=\"fs-id1167829808047\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8dbad08cb55a7c9f44faaa654af2b1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#45;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#43;&#50;&#122;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#43;&#50;&#121;&#43;&#122;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"171\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829879454\">\n<p id=\"fs-id1167833326439\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec1978b6de7b1a19abde1b097b9a8063_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826206143\">Solve the system of equations using a matrix.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836525066\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836525070\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67d56114ed51dc4537e8df2392156c29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836485678\">\n<div data-type=\"problem\" id=\"fs-id1167836485680\">\n<p id=\"fs-id1167836485682\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5b832ea9b1009f88916b2f58db208ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#43;&#121;&#43;&#122;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#45;&#50;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#45;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"162\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833141974\">\n<p id=\"fs-id1167833141977\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7307d8b16707f014143c59fffe4de9a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#55;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836738051\">Solve using Cramer\u2019s rule.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836738055\">\n<div data-type=\"problem\" id=\"fs-id1167836738057\">\n<p id=\"fs-id1167829850382\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45f258bd81326e2e2f3b3c56094f8285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836619203\">\n<div data-type=\"problem\" id=\"fs-id1167836619205\">\n<p id=\"fs-id1167836540111\">Evaluate the determinant by expanding by minors:<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ce5ac3855ad1ebb0aff9647135efd64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"193\" style=\"vertical-align: -23px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836549180\">\n<p id=\"fs-id1167836549182\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e99b7c7ddd421682faf67e5b5f865882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829746255\">In the following exercises, translate to a system of equations and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829746258\">\n<div data-type=\"problem\" id=\"fs-id1167829746261\">\n<p id=\"fs-id1167836530781\">Greg is paddling his canoe upstream, against the current, to a fishing spot 10 miles away. If he paddles upstream for 2.5 hours and his return trip takes 1.25 hours, find the speed of the current and his paddling speed in still water.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826204892\">\n<div data-type=\"problem\" id=\"fs-id1167826204895\">\n<p id=\"fs-id1167833169125\">A pharmacist needs 20 liters of a 2% saline solution. He has a 1% and a 5% solution available. How many liters of the 1% and how many liters of the 5% solutions should she mix to make the 2% solution?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833169131\">\n<p id=\"fs-id1167833169133\">15 liters of 1% solution, 5 liters of 5% solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833339844\">\n<div data-type=\"problem\" id=\"fs-id1167833339846\">\n<p id=\"fs-id1167833339848\">Arnold invested ?64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received ?4,500 in interest in one year?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829696408\">\n<div data-type=\"problem\" id=\"fs-id1167829696410\">\n<p id=\"fs-id1167829696413\">The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of ?65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of ?140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 can of popcorn for a total sales of ?250. What is the cost of each item?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829719019\">\n<p id=\"fs-id1167829719021\">The candy cost ?20; the cookies cost ?5; and the popcorn cost ?10.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829696415\">\n<div data-type=\"problem\" id=\"fs-id1167829696417\">\n<p id=\"fs-id1167836514875\">The manufacturer of a granola bar spends ?1.20 to make each bar and sells them for ?2. The manufacturer also has fixed costs each month of ?8,000.<\/p>\n<p><span class=\"token\">\u24d0<\/span> Find the cost function <em data-effect=\"italics\">C<\/em> when <em data-effect=\"italics\">x<\/em> granola bars are manufactured<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Find the revenue function <em data-effect=\"italics\">R<\/em> when <em data-effect=\"italics\">x<\/em> granola bars are sold.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Show the break-even point by graphing both the Revenue and Cost functions on the same grid.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Find the break-even point. Interpret what the break-even point means.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829720516\">\n<div data-type=\"problem\" id=\"fs-id1167829720518\">\n<p id=\"fs-id1167829720520\">Translate to a system of inequalities and solve.<\/p>\n<p id=\"fs-id1167829720524\">Andi wants to spend no more than ?50 on Halloween treats. She wants to buy candy bars that cost ?1 each and lollipops that cost ?0.50 each, and she wants the number of lollipops to be at least three times the number of candy bars.<\/p>\n<p id=\"fs-id1167836729309\"><span class=\"token\">\u24d0<\/span> Write a system of inequalities to model this situation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Graph the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Can she buy 20 candy bars and 40 lollipops?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836399851\">\n<p id=\"fs-id1167836399853\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9e88f6450d015aaa97395aaecea35e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#67;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#76;&#92;&#103;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#67;&#43;&#48;&#46;&#53;&#76;&#92;&#108;&#101;&#32;&#53;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#76;&#92;&#103;&#101;&#32;&#51;&#67;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"133\" style=\"vertical-align: -38px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829627822\" data-alt=\"The figure shows the graph of two equations. Two intersecting lines, one in red and the other in blue, are shown. The red line passes through origin. An area is shown in grey.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_07_319_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of two equations. Two intersecting lines, one in red and the other in blue, are shown. The red line passes through origin. An area is shown in grey.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> no<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167824740928\">\n<dt>system of linear inequalities<\/dt>\n<dd id=\"fs-id1167824740934\">Two or more linear inequalities grouped together form a system of linear inequalities.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4822","chapter","type-chapter","status-publish","hentry"],"part":2305,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4822","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4822\/revisions"}],"predecessor-version":[{"id":5082,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4822\/revisions\/5082"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/2305"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4822\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=4822"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=4822"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=4822"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=4822"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}