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
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
Holt Algebra 2
3-3 Solving Systems of Linear Inequalities
Example 3 Continued
Holt Algebra 2
3-3 Solving Systems of Linear Inequalities
Example 1A: Graphing Systems of Inequalities
Graph the system of inequalities.
y<
–3
y ≥ –x + 2
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.
x – 3y < 6
2x + y > 1.5
For x – 3y < 6, graph the dashed
1
boundary line y = x – 2, and
3
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.
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0
y0
x + y ≤ 70
50 + 2x + 3y ≤ 215
Holt Algebra 2
.
3-3 Solving Systems of Linear Inequalities
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.
Holt Algebra 2
3-3 Solving Systems of Linear Inequalities
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 
Holt Algebra 2
150 ≤ 215 
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
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.
The system of inequalities is
d0
s0
d + s ≤ 40
2d + 2.5s ≥ 90
Holt Algebra 2
.
3-3 Solving Systems of Linear Inequalities
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.
Holt Algebra 2
3-3 Solving Systems of Linear Inequalities
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.
d + s ≤ 40
2d + 2.5s ≥ 90
5 + 32 ≤ 40
37 ≤ 40
2(5) + 2.5(32) ≥ 90
Holt Algebra 2

90 ≥ 90 
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
Holt Algebra 2