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OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Systems of Inequalities Graph a linear inequality in two variables. Graph systems of linear inequalities in two variables. Graph a nonlinear inequality in two variables. Graph systems of nonlinear inequalities in two variables. SECTION 8.4 1 2 3 4

3 © 2010 Pearson Education, Inc. All rights reserved Definitions The statements x + y > 4, 2x + 3y < 7, y ≥ x, and x + y ≤ 9 are examples of linear inequalities in the variables x and y. A solution of an inequality in two variables x and y is an ordered pair (a, b) that results in a true statement when x is replaced by a, and y is replaced by b in the inequality. The set of all solutions of an inequality is called the solution set of the inequality. The graph of an inequality in two variables is the graph of the solution set of the inequality.

4 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Graphing a Linear Inequality Graph the inequality 2x + y > 6. Solution First, graph the equation 2x + y = 6. Notice that if we solve this equation for y, we have y = 6 − 2x.

5 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Graphing a Linear Inequality Solution continued Any point P(a, b) whose y-coordinate, b, is greater than 6 – 2a must be above this line.

6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Graphing a Linear Inequality Solution continued The graph of y > −2x + 6, which is equivalent to 2x + y > 6, consists of all points in the plane that lie above the line 2x + y = 6.

7 © 2010 Pearson Education, Inc. All rights reserved PROCEDURE FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES Step 1Replace the inequality symbol by an equals (=) sign. Step 2Sketch the graph of the corresponding equation in Step 1. Use a dashed line for the boundary if the given inequality sign is, and a solid line if the inequality symbol is ≤ or ≥.

8 © 2010 Pearson Education, Inc. All rights reserved PROCEDURE FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES Step 3The graph in Step 2 divides the plane into two regions. Select a test point in the either region, but not on the graph of the equation in Step 1.

9 © 2010 Pearson Education, Inc. All rights reserved PROCEDURE FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES (ii) If the test point’s coordinates do not satisfy the given inequality, shade the region that does not contain the test point. Step 4 (i) If the coordinates of the test point satisfy the given inequality, then so do all the points in that region. Shade that region. The shaded region (including the boundary if it is solid) is the graph of the given inequality.

10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Graphing Inequalities Sketch the graph of each of the following inequalities. a. x ≥ 2 b. y < 3 c. x + y < 4 Solution a. Step 1 Change the ≥ to =: x = 2 Step 2 Graph x = 2 with a solid line. Step 3 Test (0, 0). 0 ≥ 2 is a false statement.

11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Graphing Inequalities Solution continued Step 3 continued The region not containing (0, 0), together with the vertical line, is the solution set. Step 4 Shade the solution set.

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Graphing Inequalities Solution continued Step 2 Graph y = 3 with a dashed line. Step 3 Test (0, 0). 0 < 3 is a true statement. The region containing (0, 0) is the solution set. Step 4 Shade the solution set. b. Step 1 Change the < to =: y = 3

13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Graphing Inequalities Solution continued Step 2 Graph x + y = 4 with a dashed line. Step 3 Test (0, 0). 0 + 0 < 4 is a true statement. The region containing (0, 0) is the solution set. Step 4 Shade the solution set. c. Step 1 Change the < to = : x + y = 4

14 © 2010 Pearson Education, Inc. All rights reserved SYSTEMS OF LINEAR INEQUALITIES IN TWO VARIABLES An ordered pair (a, b) is a solution of a system of inequalities involving two variables x and y if and only if, when x is replaced by a and y is replaced by b in each inequality of the system, all resulting statements are true. The solution set of a system of inequalities is the intersection of the solution sets of all the inequalities in the system.

15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Graphing a System of Two Inequalities Graph the solution set of the system of Solution Graph each inequality separately. Step 1 2x + 3y = 6 Step 2 Sketch as a dashed line by joining the points (0, 2) and (3, 0). inequalities: Step 3Test (0, 0). 2(0) + 3(0) > 6 is a false statement. Step 4 Shade the solution set.

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Graphing a System of Two Inequalities Solution continued Now graph the second inequality.

17 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Graphing a System of Two Inequalities Solution continued Step 2 Sketch as a solid line by joining the points (0, 0) and (1, 1). Step 3Test (1, 0). 0 – 1 ≤ 6 is a true statement. Step 4 Shade the solution set. Step 1 y – x = 0

18 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Graphing a System of Two Inequalities Solution continued The graph of the solution set of inequalities (1) and (2) is the region where the shading overlaps.

19 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a System of Three Linear Inequalities Sketch the graph and label the vertices of the solution set of the system of linear inequalities. Solution On the same coordinate plane, sketch the graphs of the three linear equations that correspond to the three inequalities. All of the equations are graphed as solid lines.

20 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a System of Three Linear Inequalities Solution continued Use (0, 0) as the test point for each inequality. (i) The region on or below 3x + 2y = 11 is in the solution set of inequality (1).

21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a System of Three Linear Inequalities Solution continued (ii) The region on or above x – y = 2 is in the solution set of inequality (2). (iii) The region on or below 7x – 2y = –1 belongs to the solution set of inequality (3).

22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a System of Three Linear Inequalities Solution continued The solution set is the shaded region, including the sides of the triangle. The vertices are obtained by solving each pair of equations in the system.

23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a System of Three Linear Inequalities Solution continued a. To find the vertex (3, 1) solve the system Solve equation (2) for x and substitute x = 2 + y in equation (1). Back-substitute y = 1 in equation (2) to find the vertex (3, 1).

24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a System of Three Linear Inequalities Solution continued b. Solve the following system of equations by the substitution method to find the vertex (–1, –3): c. Solve the following system of equations by the elimination method to find the vertex (1, 4):

25 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Graph a nonlinear inequality in two variables. Step 1 Replace the inequality symbol with an equal (=) sign. Step 2 Sketch the graph of the corresponding equation in Step 1. Use a dashed curve if the given inequality sign is and a solid curve if the inequality symbol is ≤ or ≥. 25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Graphing a Nonlinear Inequality in Two Variables EXAMPLE Graph y > x 2 – 2. 1. y = x 2 – 2 2.

26 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Graph a nonlinear inequality in two variables. Step 3 The graph in Step 2 divides the plane into two regions. Select a test point in either region, but not on the graph. 26 © 2010 Pearson Education, Inc. All rights reserved Graphing a Nonlinear Inequality in Two Variables EXAMPLE Graph y > x 2 – 2. 3. EXAMPLE 6

27 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Graph a nonlinear inequality in two variables. Step 4 (i) If the coordinates of the test point satisfy the inequality, then so do all of the points in that region. Shade that region. (ii) If the test point’s coordinates do not satisfy the inequality, shade the region that does not contain the test point. 27 © 2010 Pearson Education, Inc. All rights reserved Graphing a Nonlinear Inequality in Two Variables EXAMPLE Graph y > x 2 – 2. 4. 0 > 0 2 – 2 = –2, so the test point (0, 0) satisfies the inequality. EXAMPLE 6

28 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Solving a Nonlinear System of Inequalities Graph the solution set of the following system of Solution Graph each inequality separately in the same coordinate plane. Since (0, 0) is not a solution of any the corresponding equations, use (0, 0) as a test point for each inequality. inequalities:

29 © 2010 Pearson Education, Inc. All rights reserved Solving a Nonlinear System of Inequalities Solution continued Step 2 Sketch as a solid curve with vertex (0, 4). Step 3Test (0, 0). 0 ≤ 4 – 0 is a true statement. Step 4 Shade the region. Step 1 y = 4 – x 2 EXAMPLE 7

30 © 2010 Pearson Education, Inc. All rights reserved Solving a Nonlinear System of Inequalities Solution continued Step 2 Sketch as a solid line through (0, –3) & (2, 0). Step 3Test (0, 0). 0 ≥ 0 – 3 is a true statement. Step 4 Shade the region. Step 1 EXAMPLE 7

31 © 2010 Pearson Education, Inc. All rights reserved Solving a Nonlinear System of Inequalities Solution continued Step 3Test (0, 0). 0 ≥ 0 – 3 is a true statement. Step 4 Shade the region. Step 1 y = –6x – 3 Step 2 Sketch as a solid line through (0, –3) EXAMPLE 7

32 © 2010 Pearson Education, Inc. All rights reserved Solving a Nonlinear System of Inequalities Solution continued The region common to all three graphs is the graph of the solution set of the given system of inequalities. EXAMPLE 7