Solving problems with systems of inequalities

<h1>Text Preview</h1> <p>Let’s review systems of three inequalities.</p> <p>You are given a graph of a system of three inequalities.</p> <p>Three points are plotted on the graph. </p> <p>Only one point is a solution. Click on the point that is a solution to the system.</p> <p>Now, let’s move on with today’s lesson.</p> <p>Here is a system of three inequalities: <EQUATION> and <EQUATION> and <EQUATION>.</p> <p>Let’s find its solution.</p> <p>We graph the first inequality.</p> <p>Next, we graph the second inequality.</p> <p>Finally, we graph the third inequality.</p> <p>The solution region has a triangular shape.</p> <p>Let’s label the vertices of this triangle as A, B and C.</p> <p>Now, we wish to find the coordinates of Vertex A. </p> <p>Notice that this point is the intersection of two lines.</p> <p>Vertex A is the solution to a system of equations.</p> <p>These equations describe the two lines.</p> <p>What is this system of equations? Click the "Submit" button after selecting your answer.</p> <p>The two lines are boundaries of the first two inequalities.</p> <p>We can write a related equation for each of these: <font size="+2" face="Times New Roman">2<i>y</i> – <i>x</i> = 1</font> and <font size="+2" face="Times New Roman">3<i>y</i> + <i>x</i> = 9</font>.</p> <p>This is our system of equations.</p> <p>The solution is the ordered pair <font size="+2" face="Times New Roman">(3,2)</font>. These are the coordinates of Vertex A.</p> <p>What are the coordinates of Vertex B? Click the "Submit" button after selecting your answer.</p> <p>We write a system of equations, <font size="+2" face="Times New Roman">2<i>y</i> – <i>x</i> = 1</font> and <font size="+2" face="Times New Roman"><i>y</i> + 2<i>x</i> = 3</font>, that are related to the first and third inequalities.</p> <p>The solution gives the coordinates of Vertex B, which are <font size="+2" face="Times New Roman">(1,1)</font>.</p> <p>Finally, we write a system of equations, <font size="+2" face="Times New Roman">3<i>y</i> + <i>x</i> = 9</font> and <font size="+2" face="Times New Roman"><i>y</i> + 2<i>x</i> = 3</font>, that are related to the second and third equations.</p> <p>Its solution gives Vertex <font size="+2" face="Times New Roman">C</font>, which has the coordinates <font size="+2" face="Times New Roman">(0,3)</font>.</p> <p>Now, try some examples on your own. </p> <p>You are given a system of three inequalities.</p> <p>You also are given the graph of the system.</p> <p>The solution region is a triangle. </p> <p>Find the coordinates of the three vertices. </p> <p>Enter the values into the text boxes. Click the "Submit" button to see the answer.</p> <p>Let’s look at a problem involving a system of inequalities.</p> <p>Carlos is moving math club materials on a cart.</p> <p>There are red and blue boxes.</p> <p>A red box weighs <font size="+2" face="Times New Roman"><i>x</i></font> pounds. </p> <p>A blue box weighs <font size="+2" face="Times New Roman"><i>y</i></font> pounds.</p> <p>The cart has a maximum load limit. So, only certain combinations of red and blue boxes can be loaded on the cart. </p> <p>The possible weights of the boxes are given by a system of inequalities. The graph of that system is shown here.</p> <p>The solution is the shaded region.</p> <p>What is the maximum possible weight of a red box? Click the "Submit" button after selecting your answer.</p> <p>The maximum possible weight of a red box is the maximum value of <font size="+2" face="Times New Roman"><i>x</i></font> in the solution region.</p> <p>This occurs at the solution vertex.</p> <p>We can read the vertex coordinates directly from the graph. They are <font size="+2" face="Times New Roman">(30,10)</font>.</p> <p>We find that the maximum possible weight of a red box is <font size="+2" face="Times New Roman">30</font> pounds.</p> <p id="TextCopyright">Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education</p>