A system of linear inequalities is a set of two or more linear inequalities with the same variables.

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.
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 lt 6
2x y gt 1.5
For 2x y gt 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 50 2x 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
50 2x 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
20 20 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
2(5) 2.5(32) 90
5 32 40
37 40
?
?
90 90
14
HW pg. 202 s 15, 16, 17, 20, 24, 27 HW pg.
203 s 25, 26, 30, 32, 39