3-3 Solving Systems of Linear Inequalities Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up 1. Graph 2x – y > 4. Determine if the given ordered pair is a solution of the system of equations. 2y – x = –6 2x + y = 2 2. (2, –2) yes x – y = –1 x + 2y = 2 3. (–4, 3) no

Objective Solve systems of linear inequalities.

Vocabulary system of linear inequalities

When a problem uses phrases like “greater than” or “no more than,” you can model the situation using a system of linear inequalities. A system of linear inequalities is a set of two or more linear inequalities with the same variables. The solution to a system of inequalities is often an infinite set of points that can be represented graphically by shading. When you graph multiple inequalities on the same graph, the region where the shadings overlap is the solution region.

Example 1A: Graphing Systems of Inequalities Graph the system of inequalities. y ≥ –x + 2 y < – 3 For y < – 3, graph the dashed boundary line y = – 3, and shade below it. For y ≥ –x + 2, graph the solid boundary line y = –x + 2, and shade above it. The overlapping region is the solution region.

Example 1B: Graphing Systems of Inequalities Graph each system of inequalities. y < –3x + 2 y ≥ –1 For y < –3x + 2, graph the dashed boundary line y = –3x + 2, and shade below it. For y ≥ –1, graph the solid boundary line y = –1, and shade above it.

Example 1B Continued Check Choose a point in the solution region, such as (0, 0), and test it in both inequalities. y < –3x + 2 y ≥ –1 0 < –3(0) + 2 0 ≥ –1 0 < 2  0 ≥ –1  The test point satisfies both inequalities, so the solution region is correct.

Check It Out! Example 2 Leyla is selling hot dogs and spicy sausages at the fair. She has only 40 buns, so she can sell no more than a total of 40 hot dogs and spicy sausages. Each hot dog sells for $2, and each sausage sells for $2.50. Leyla needs at least $90 in sales to meet her goal. Write and graph a system of inequalities that models this situation.

Check It Out! Example 2 Continued Let d represent the number of hot dogs, and let s represent the number of sausages. The total number of buns Leyla has can be modeled by the inequality d + s ≤ 40. The amount of money that Leyla needs to meet her goal can be modeled by 2d + 2.5s ≥ 90. d  0 s  0 The system of inequalities is . d + s ≤ 40 2d + 2.5s ≥ 90

Check It Out! Example 2 Continued Graph the solid boundary line d + s = 40, and shade below it. Graph the solid boundary line 2d + 2.5s ≥ 90, and shade above it. The overlapping region is the solution region.

Check It Out! Example 2 Continued Check Test the point (5, 32) in both inequalities. This point represents selling 5 hot dogs and 32 sausages. 2d + 2.5s ≥ 90 d + s ≤ 40 5 + 32 ≤ 40 2(5) + 2.5(32) ≥ 90 37 ≤ 40  90 ≥ 90 

Systems of inequalities may contain more than two inequalities.

Example 3: Geometry Application Graph the system of inequalities, and classify the figure created by the solution region. x ≥ –2 x ≤ 3 y ≥ –x + 1 y ≤ 4

Example 3 Continued Graph the solid boundary line x = –2 and shade to the right of it. Graph the solid boundary line x = 3, and shade to the left of it. Graph the solid boundary line y = –x + 1, and shade above it. Graph the solid boundary line y = 4, and shade below it. The overlapping region is the solution region.

Lesson Quiz: Part I 1. Graph the system of inequalities and classify the figure created by the solution region. y ≤ x – 2 y ≥ –2x – 2 x ≤ 4 x ≥ 1 trapezoid

Lesson Quiz: Part II 2. The cross-country team is selling water bottles to raise money for the team. The price of the water bottle is $3 for students and $5 for everyone else. The team needs to raise at least $400 and has 100 water bottles. Write and graph a system of inequalities that can be used to determine when the team will meet its goal.

Lesson Quiz: Part II Continued x + y ≤ 100 3x + 5y ≥ 400