© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 7 Algebra: Graphs, Functions, and Linear Systems

© 2010 Pearson Prentice Hall. All rights reserved Linear Inequalities in Two Variables

© 2010 Pearson Prentice Hall. All rights reserved. Objectives 1.Graph a linear inequality in two variables. 2.Use mathematical models involving linear inequalities. 3.Graph a system of linear inequalities. 3

© 2010 Pearson Prentice Hall. All rights reserved. Linear Inequalities in Two Variables and Their Solutions If we change the symbol = in the equation Ax + By = C to >, <, ≥, or ≤, we obtain a linear inequality in two variables. For example, x + y < 2 and 3x – 5y ≥ 15 are linear inequalities in two variables. A solution of an inequality in two variables, x and y, is an ordered pair of real numbers such that when the x- coordinate is substituted for x and the y-coordinate is substituted for y in the inequality and we obtain a true statement. 4

© 2010 Pearson Prentice Hall. All rights reserved. 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

© 2010 Pearson Prentice Hall. All rights reserved. Graph: 3x – 5y ≥ 15. Solution: Step 1 We need to graph 3x – 5y = 15. We can use intercepts to graph this line. We set y = 0 to We set x = 0 to find the x-intercept. find the y-intercept. 3x – 5y = 15 3x – 5y = 15 3x – 5 · 0 = 15 3 · 0 – 5y = 15 3x = 15 −5y = 15 x = 5 y = −3 Example 1: Graphing a Linear Inequality in Two Variables 6

© 2010 Pearson Prentice Hall. All rights reserved. The x-intercept is 5, so the line passes through (5,0). The y-intercept is −3, so the line passes through (0,−3). Step 2 We choose (0,0) as a test point. 3x – 5y ≥ 15 3 · 0 – 5 · 0 ≥ 15 0 – 0 ≥ 15 0 ≥ 15 NOT TRUE! Example 1 continued 7

© 2010 Pearson Prentice Hall. All rights reserved. Step 3 Since the statement is false, we shade the half- plane that does not include the test point (0,0). Thus, the graph with the shading is the solution to the given inequality. Example 1 continued 8

© 2010 Pearson Prentice Hall. All rights reserved. Graph: Solution: Step 1 We need to graph Since the inequality > is given, we use a dashed line. Example 2: The Graph of a Linear Inequality in Two Variables 9

© 2010 Pearson Prentice Hall. All rights reserved. Step 2 We choose a test point not on the line, (1,1), which lies in the half-plane above the line. TRUE! Step 3 Since the statement is true, then we shade the half-plane that includes the test point (1,1). Example 2 continued 10

© 2010 Pearson Prentice Hall. All rights reserved. Graphing Linear Inequalities without Using Test Points For the vertical line x = a: If x > a, shade the half- plane to the right of x = a. If x < a, shade the half- plane to the left of x = a. For the horizontal line y = b: If y > b, shade the half-plane above y = b. If y < b, shade the half-plane below y = b. 11

© 2010 Pearson Prentice Hall. All rights reserved. Graph each inequality in a rectangular coordinate system:a. y ≤ −3b. x > 2 Solution: a. y ≤ −3b. x > 2 Example 3: Graphing Inequalities Without Using Test Points 12

© 2010 Pearson Prentice Hall. All rights reserved. Modeling with Systems of Linear Inequalities Just as two or more linear equations make up a system of linear equations, two or more linear inequalities make up a system of linear inequalities. A solution of a system of linear inequalities in two variables is an ordered pair that satisfies each inequalities in the system. 13

© 2010 Pearson Prentice Hall. All rights reserved. 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. 14

© 2010 Pearson Prentice Hall. All rights reserved. Example 5: Graphing a System of Linear Inequalities Graph the solution set of the system: x – y < 1 2x + 3y ≥ 12. Solution: Replacing each inequality symbol with an equal sign indicates that we need to graph x – y = 1 and 2x + 3y = 12. We can use intercepts to graph these lines. 15

© 2010 Pearson Prentice Hall. All rights reserved. Example 5 continued 16

© 2010 Pearson Prentice Hall. All rights reserved. Now we are ready to graph the solution set of the system of linear inequalities. Example 5 continued 17