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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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 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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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