Copyright © Cengage Learning. All rights reserved. Systems of Linear Inequalities SECTION 2.5

2 Learning Objectives 1 Graph linear inequalities given in slope-intercept or standard form 2 Determine the corner points of a solution region of a system of linear inequalities 3 Explain the practical meaning of solutions of linear inequalities in real-world contexts

3 Linear Inequalities

4.

5 Graphing Linear Inequalities

6 The graph of a linear inequality is a region bordered by a line called a boundary line. The solution region of a linear inequality is the set of all points (including the boundary line) that satisfy the inequality.

7 Graphing Linear Inequalities The linear inequality graphing process is summarized as follows.

8 Example 1 – Graphing the Solution Region of a Linear Inequality Graph the solution region of the linear inequality

9 Example 1 – Graphing the Solution Region of a Linear Inequality Graph the solution region of the linear inequality. Solution: We can easily find the x-intercept of the boundary line by dividing the constant term by the coefficient on the x-term. The point (2, 0) is the x-intercept.

10 Example 1 – Solution Graph the solution region of the linear inequality Solution: We find the y-intercept of the boundary line by dividing the constant term by the coefficient on the y-term. The point (0, 4) is the y-intercept. cont’d Figure 2.25

11 Example 1 – Solution As shown in Figure 2.25, we graph the x- and y-intercepts and then draw the line through the intercepts. cont’d Figure 2.25

12 Example 1 – Solution Next we pick the point (0, 0) to plug in to the inequality to see which side to shade, The statement is true, so all points on the same side of the line as (0, 0) are in the solution region. Figure 2.26 shows the shaded solution region. cont’d Figure 2.26

13 Graphing Linear Inequalities If you choose to convert lines from standard to slope-intercept form before graphing, the following properties will help you to quickly identify the solution region without having to check a point.

14 Graphing Systems of Linear Inequalities

15 Example 2 – Graphing the Solution Region of a System of Linear Inequalities Graph the solution region of the following system of linear inequalities.

16 Example 2 – Solution Figure 2.27 cont’d

17 Example 2 – Solution The next two inequalities, x  0 and y  0, limit the solution region to positive values of x and y. The line x = 0 is the vertical axis. The line y = 0 is the horizontal axis. cont’d Figure 2.28

18 Example 2 – Solution Therefore, as shown in Figure 2.28, the solution region of the system of inequalities is the triangular region to the right of the line x = 0, above the line y = 0, and below the line. cont’d Figure 2.28

19 Graphing Systems of Linear Inequalities If it is possible to draw a circle around the solution region, the solution region is bounded. If no circle will enclose the entire solution region, the solution region is unbounded. The solution region in Example 2 is bounded.

20 Example 3 – Unbounded Solution Note, if we cannot draw a circle around the solution, then it is unbounded, and has infinitely many solutions. Figure 2.29

21 Example 4 If the lines have the same slope, they are parallel and will never intersect. Also, if the shaded region never overlaps the system has no solution. That is, no ordered pair exists that satisfies both inequalities. cont’d Figure 2.30

22 Corner Points

23 Corner Points To find the coordinates of a corner point, we solve the system of equations formed by the two intersecting boundary lines that form the corner.

24 Example 6 – Finding the Corner Points of a Solution Region A student earns $15.00 per hour designing web pages and $9.00 per hour supervising a campus tutoring center. She has at most 30 hours per week to work and needs to earn at least $320. (Let w represent the hours designing web pages and t represent the hours tutoring.) a.) Set up the system of linear inequalities that can be used to solve the problem. b.) Graph and shade the system. c.) Is the system bounded or unbounded?

25 t + w < 30 9t + 15w > 310 w > 0 t > 0 Example 6 – Finding the Corner Points of a Solution Region

26 What are the corner points? Example 6 – Finding the Corner Points of a Solution Region