1. Chapter 5 Systems of Equations and Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 5.5 Systems of Inequalities

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Graph a linear inequality in two variables. Graph a nonlinear inequality in two variables. Use mathematical models involving linear inequalities. Graph a system of inequalities. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Linear Inequalities in Two Variables and Their Solutions Equations in the form Ax + By = C are straight lines when graphed. If we change the symbol = to we obtain a linear inequality in two variables. A solution of an inequality in two variables, x and y, is an ordered pair of real numbers with the following property: When the x-coordinate is substituted for x and the y-coordinate is substituted for y in the inequality, we obtain a true statement. Each ordered-pair solution is said to satisfy the inequality.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 The Graph of a Linear Inequality in Two Variables The graph of an inequality in two variables is the set of all points whose coordinates satisfy the inequality.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Graphing a Linear Inequality in Two Variables

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Graphing a Linear Inequality in Two Variables Graph: Step 1 Replace the inequality symbol by = and graph the linear equation. We set y = 0 to find the x-intercept. We set x = 0 to find the y-intercept.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Graphing a Linear Inequality in Two Variables (continued) Graph: Step 1 (cont) Replace the inequality symbol by = and graph the linear equation. We have found that the x-intercept is 2 and the y-intercept is – 4. Using the intercepts, we graph the line.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Graphing a Linear Inequality in Two Variables (continued) Graph: Step 2 Choose a test point from one of the half- planes and not from the line. Substitute its coordinates into the inequality. We will test (0, 0). false

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Graphing a Linear Inequality in Two Variables (continued) Graph: Step 3 If a false statement results, shade the half- plane not containing the test point. The inequality is false at (0, 0). We shade the half-plane that does not include (0, 0).

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Graphing a Nonlinear Inequality in Two Variables Graph: Step 1 Replace the inequality symbol by = and graph the nonlinear equation. The graph is a circle of radius 4 with its center at the origin.

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Graphing a Nonlinear Inequality in Two Variables (continued) Graph: Step 2 Choose a test point from one of the half- planes and not from the circle. Substitute its coordinates into the inequality. We will test (0, 0). false

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Graphing a Nonlinear Inequality in Two Variables (continued) Graph: Step 3 If a true statement results, shade the region containing the test point. The false statement tells us that no points inside the circle satisfy the inequality. We shade outside the circle.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Application The latest guidelines, which apply to both men and woman, give healthy weight ranges, rather than specific weights, for your height. If x represents height in inches, and y represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: Show that the point (66, 130) is a solution of this system.

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Application (continued) If x represents height in inches, and y represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: Show that the point (66, 130) is a solution of this system. true

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Application (continued) If x represents height in inches, and y represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: The point (66, 130) satisfies both inequalities. Thus, (66, 130) is a solution for this system. Show that the point (66, 130) is a solution of this system.

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Graphing Systems of Linear Inequalities The solution set of a system of linear inequalities in two variables is the set of all ordered pairs that satisfy each inequality in the system. Thus, to graph a system of inequalities in two variables, begin by graphing each individual inequality in the same rectangular coordinate system. Then find the region, if there is one, that is common to every graph in the system. This region of intersection gives a picture of the system’s solution set.

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Graphing a System of Linear Inequalities Graph the solution set of the system: Step 1 Replace the inequality symbol by = and graph the linear equations. Find the intercepts.

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Graphing a System of Linear Inequalities (continued) Graph the solution set of the system: Step 1 (cont) Replace the inequality symbol by = and graph the linear equations. The intercepts for are (6, 0) and (0, –2). The intercepts for are (–3, 0) and (0, –2).

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Graphing a System of Linear Inequalities (continued) Graph the solution set of the system: Step 2 We will test the half-planes formed by each line. (0,0) is a solution for both inequalities.

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Graphing a System of Linear Inequalities (continued) Graph the solution set of the system: Step 3 We shade the solution region.