1. 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.

2. 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

3. Example 3 Continued

4. 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.

5. Check It Out! Example 1a
Graph the system of inequalities.
x – 3y < 6
2x + y > 1.5
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.

6. 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.

7. 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 x + 3y ≤ 215.
x  0
y  0
The system of inequalities is
x + y ≤ 70
50 + 2x + 3y ≤ 215

8. Example 2 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.

9. Example 2 Continued
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
≤ 70
50 + 2(20) + 3(20) ≤ 215
40 ≤ 70 
150 ≤ 215 

10. 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.

11. 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

12. 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.

13. 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
≤ 40
2(5) + 2.5(32) ≥ 90
37 ≤ 40 
90 ≥ 90 

14. HW pg. 202
#’s 15, 16, 17, 20, 24, 27
HW pg. 203
#’s 25, 26, 30, 32, 39