Solving Systems of Linear Inequalities

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Solving Systems of Linear Inequalities
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up
1. Graph 2x y gt 4.
Determine if the given ordered pair is a
solution of the system of equations.
2. (2, 2)
yes
3. (4, 3)
no
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Objective
Solve systems of linear inequalities.
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Vocabulary
system of linear inequalities
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When a problem uses phrases like greater than
or no more than, you can model the situation
using a system 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 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.
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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.
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• Check Test a point from each region on the
  graph.

Region Point
Left (0, 0)
Right (5,2)
Top (0, 3)
Bottom (0,4)
y x 2
0 (0) 2
0 2
x
x
2 (5) 2
?
2 3
?
3 (0) 2
3 lt (0)3
3 2
x
?
2 lt 3
4 (0) 2
4 lt (0)3
4 lt 3
4 2
?
x
Only the point from the overlapping (right)
region satisfies both inequalities.
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Example 1B Graphing Systems of Inequalities
Graph each system of inequalities.
y lt 3x 2
y 1
For y lt 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.
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Example 1B Continued
Check Choose a point in the solution region,
such as (0, 0), and test it in both inequalities.
y lt 3x 2
y 1
0 lt 3(0) 2
0 1
0 lt 2
0 1
?
?
The test point satisfies both inequalities, so
the solution region is correct.
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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.
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• Check Test a point from each region on the
  graph.

Region Point
Left (0, 0)
Right (4,2)
Top (0, 3)
Bottom (0,4)
x 3y lt 6
2x y gt 1.5
0 3(0)lt 6
2(0) 0 gt1.5
x
x
0 gt 1.5
0 lt 6
4 3(2)lt 6
2(4) 2 gt1.5
?
6 gt 1.5
x
10 lt 6
0 3(3)lt 6
2(0) 3 gt1.5
?
?
9 lt 6
3 gt 1.5
0 3(4)lt 6
2(0) 4 gt1.5
?
x
12 lt 6
4 gt 1.5
Only the point from the overlapping (top) region
satisfies both inequalities.
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Check It Out! Example 1b
Graph each system of inequalities.
y 4
2x y lt 1
For y 4, graph the solid boundary line y 4,
and shade below it.
For 2x y lt 1, graph the dashed boundary line
y 3x 2, and shade below it.
The overlapping region is the solution region.
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Check It Out! Example 1b Continued
Check Choose a point in the solution region,
such as (0, 0), and test it in both directions.
2x y lt 1
y 4
2(0) 0 lt 1
0 4
0 4
0 lt 1
?
?
The test point satisfies both inequalities, so
the solution region is correct.
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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.
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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
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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.
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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
?
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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.
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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
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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.
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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
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Systems of inequalities may contain more than two
inequalities.
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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
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Example 3 Continued
Graph the solid boundary line x 2 and shade to
the right of it. Graph the solid boundary line x
3, and shade to the left of it. Graph the
solid boundary line y x 1, and shade above
it. Graph the solid boundary line y 4, and
shade below it. The overlapping region is the
solution region.
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Check It Out! Example 3a
Graph the system of inequalities, and classify
the figure created by the solution region.
x 6
y x 1
y 2x 4
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Check It Out! Example 3a Continued
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Check It Out! Example 3b
Graph the system of inequalities, and classify
the figure created by the solution region.
y 4
y 1
y x 8
y 2x 2
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Check It Out! Example 3b Continued
Graph the solid boundary line y 4 and shade to
the below it. Graph the solid boundary line y
1, and shade to the above it. Graph the solid
boundary line y x 8, and shade below it.
Graph the solid boundary line y 2x 2, and
shade below it. The overlapping region is the
solution region.
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Check It Out! Example 3b Continued
The solution region is a four-sided figure, or
quadrilateral. Notice that the boundary lines y
4 and y 1 are parallel, horizontal lines. The
boundary lines y x 8 and y 2x 2 are not
parallel since the slope of the first is 1 and
the slope of the second is 2.
A quadrilateral with one set of parallel sides is
called a trapezoid. The solution region is a
trapezoid.
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Lesson Quiz Part I
1. Graph the system of inequalities and classify
the figure created by the solution region.
y x 2
y 2x 2
x 4
x 1
trapezoid
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Lesson Quiz Part II
2. The cross-country team is selling water
bottles to raise money for the team. The
price of the water bottle is 3 for students
and 5 for everyone else. The team needs to
raise at least 400 and has 100 water
bottles. Write and graph a system of
inequalities that can be used to determine
when the team will meet its goal.
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Lesson Quiz Part II Continued
x y 100
3x 5y 400