Holt Algebra 2

3-3 Solving Systems 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 the region where the shadings overlap is the solution region.

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

Graph the system of inequalities, and classify the figure created by the solution region.

Example 3: Geometry Application

x ≥ –2

y ≥ –x + 1

x ≤ 3

y ≤ 4

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

Example 3 Continued

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

Graph the system of inequalities.Example 1A: Graphing Systems 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.

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

Check It Out! Example 1a

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.

1

3x

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.

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

Example 2: Art Application

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.

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

Example 2 Continued

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 50 + 2x + 3y ≤ 215.

The system of inequalities is . x + y ≤ 70

50 + 2x + 3y ≤ 215

x 0

y 0

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

Graph the solid boundary line x + y = 70, and shade below it.

Graph the solid boundary line 50 + 2x + 3y ≤ 215, and shade below it. The overlapping region is the solution region.

Example 2 Continued

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

Check Test the point (20, 20) in both inequalities. This point represents painting 20 red and 20 gold plates.

x + y ≤ 70 50 + 2x + 3y ≤ 215

20 + 20 ≤ 70

40 ≤ 70

50 + 2(20) + 3(20) ≤ 215

150 ≤ 215

Example 2 Continued

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

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.

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

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

Check It Out! Example 2 Continued

d 0

s 0

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

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

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

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 ≥ 90

5 + 32 ≤ 4037 ≤ 40

2(5) + 2.5(32) ≥ 90

90 ≥ 90

Check It Out! Example 2 Continued

Holt Algebra 2

3-3 Solving Systems of Linear Inequalities

HW pg. 202

#’s 15, 16, 17, 20, 24, 27

HW pg. 203

#’s 25, 26, 30, 32, 39