Algebra 2 Systems of Inequalities Lesson 3-3

Goals Goal To solve a linear systems of linear inequalities. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary None

Essential Question Big Idea: Solving Equations and Inequalities How do you know which region on a graph is the solution of a system of linear inequalities?

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. Definition

An ordered pair satisfies a system of linear inequalities if it makes each inequality in the system a true statement. Solution to a System of Linear Inequalities Example: Determine if the ordered pair (3, – 2) is a solution to the system of linear inequalities. True (3, – 2) is a solution to the system.

Graph the solution of the system: The boundary line is 3x + y = 9. The boundary line is 2x + 5y = 10 Solution 3x + y < 9 2x + 5y  10 The solution will be the set of all points that satisfy both of the inequalities in the system. y x 4 4 Solution to a System of Linear Inequalities on a Graph

Graph the solution of the system. 3x + y  9 3x + 5y  15 x  0 y  0 These two inequalities show that the graph must lie in Quadrant I. y x 4 4 3x + y = 9 3x + 5y = 15 Solution to a System of Linear Inequalities on a Graph

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:

If you are unsure which direction to shade, use the origin as a test point. Helpful Hint

Graph each system of inequalities. y ≥ –1 y < –3x + 2 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:

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 The test point satisfies both inequalities, so the solution region is correct. Example: continued

Example: No solution Graph both inequalities. Answer: No solution. The graphs do not overlap, so the solutions have no points in common and there is no solution to the system. Solve the system of inequalities by graphing.

Graph the system of inequalities. 2x + y > 1.5 x – 3y < 6 For x – 3y < 6, graph the dashed boundary line y = – 2, and shade above it. For 2x + y > 1.5, graph the dashed boundary line y = –2x + 1.5, and shade above it. The overlapping region is the solution region. Your Turn:

Graph each system of inequalities. y ≤ 4 2x + y < 1 For 2x + y < 1, graph the dashed boundary line y = –3x +2, and shade below it. For y ≤ 4, graph the solid boundary line y = 4, and shade below it. The overlapping region is the solution region. Your Turn:

Check Choose a point in the solution region, such as (0, 0), and test it in both directions. The test point satisfies both inequalities, so the solution region is correct. y ≤ 4 2x + y < 1 0 ≤ 4 2(0) + 0 < 1 0 < 1 Your Turn: continued

Lauren wants to paint no more than 70 plates for the art show. It costs her at least $50 plus $2 per item to produce red plates and $3 per item to produce gold plates. She wants to spend no more than $215. Write and graph a system of inequalities that can be used to determine the number of each plate that Lauren can make. Example: Application

Let x represent the number of red plates, and let y represent the number of gold plates. The total number of plates Lauren is willing to paint can be modeled by the inequality x + y ≤ 70. The amount of money that Lauren is willing to spend can be modeled by x + 3y ≤ 215. The system of inequalities is. x + y ≤ x + 3y ≤ 215 x  0 y  0 Example: continued

Graph the solid boundary line x + y = 70, and shade below it. Graph the solid boundary line x + 3y ≤ 215, and shade below it. The overlapping region is the solution region. Example: continued

Check Test the point (20, 20) in both inequalities. This point represents painting 20 red and 20 gold plates. x + y ≤ x + 3y ≤ ≤ ≤ (20) + 3(20) ≤ ≤ 215 Example: continued

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. Your Turn:

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. The system of inequalities is. d + s ≤ 40 2d + 2.5s ≥ 90 d  0 s  0 Your Turn: Solution

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. Your Turn: Solution

Check Test the point (5, 32) in both inequalities. This point represents selling 5 hot dogs and 32 sausages. d + s ≤ 40 2d + 2.5s ≥ ≤ ≤ 40 2(5) + 2.5(32) ≥ ≥ 90 Your Turn: Solution

Essential Question Big Idea: Solving Equations and Inequalities How do you know which region on a graph is the solution of a system of linear inequalities? Graph each inequality. The overlapping region is the solution of the system. To check your answer, choose a point in the region and test it in both inequalities.

Assignment Section 3-3, Pg. 165 – 167; #1 – 3 all, 5 – 7 all, 12 – 36 even.