LESSON
5.4
Name
Solving Systems of
Linear Inequalities
Class
5.4
Date
Solving Systems of Linear
Inequalities
Essential Question: How can you represent the solutions of a system of two or more linear
inequalities?
Texas Math Standards
A2.3.F Solve systems of two or more linear inequalities in two variables. Also A2.3.E, A2.3.G
The student is expected to:
Explore
A2.3.F
Turning a System of Equations into a System
of Inequalities
Solve systems of two or more linear inequalities in two variables.
Also A2.3.E, A2.3.G
What does the graph of a system of inequalities
Mathematical Processes
look like? Start with a system of equations. ⎨
8
⎧y = x +2
A2.1.D

⎩
x
Graph the lines on the coordinate plane.
-8
-4
0

How many regions do the two lines divide the graph into?
Language Objective

Consider what it means to replace the equals sign with an inequality. Replace the top
equation by the inequality y ≥ x + 2.
2.C.4, 2.D.1, 2.E.3, 2.I.4
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo
and how to use a system of linear inequalities to
represent how much a person can spend on presents.
Then preview the Lesson Performance Task.
© Houghton Mifflin Harcourt Publishing Company
Graph the solutions of each individual inequality;
then find the region where all solution areas
overlap.
Four

-8
Now replace the second equation with the inequality y < -2x + 1.
How is this different? It means that for any x-value, the solution includes every point in
the coordinate plane that has a lesser y-value than does the point on the line with the same
x-value. Indicate this by lightly shading the two regions below the line.
However, because this inequality is non-inclusive, the solution will not contain the points on
the line itself. Indicate this by converting the solid line to a dashed line. Answer shown on graph in (A).
One

How many regions of the graph were shaded in both steps?

Pick a point that is in the darkest region of the graph and check that it agrees with both
inequalities.
2 ≥ -2 + 2
True or False? True
2 < -2 -2 + 1 True of False? True
Module 5
ges must
EDIT--Chan
DO NOT Key=TX-B
Correction
be made
through “File
Lesson 4
289
info”
Date
Class
5.4
you
How can
?
Question:
Essential
inequalities
A2.3.F Solve
A2_MTXESE353930_U2M05L4.indd 289
ear
stems of Lin
Solving Sy s
Inequalitie
Name
Explore
What does
the graph
Start with

y
g Compan

systems of
ons of a
the soluti
represent
two or more
lities in two
linear inequa
a System
Turning ties
of Inequali
of a system
a system
of inequa
two or more
variables.
A2.3.G
⎧ y = x +2
⎨
⎩
plane.
8
y
4
+1
-8
HARDCOVER PAGES
PAGES 205–212
System
lities
coordinate
Resource
Locker
linear
Also A2.3.E,
ns into a
of Equatio
x
of equations. y = -2
on the
the lines
system of
-4
Turn to these pages to
find this lesson in the
hardcover student
edition.
x
0
4
8
-4
-8
Four
graph into?
divide the
e the top
the two lines
lity. Replac
regions do
an inequa
sign with
the equals
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the
not only
what it means y ≥ x + 2.
n includes
Consider
lity
plane
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by the inequa
coordinate
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for any x-valu every point in the
the line.
that
point on
It means
but also
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ented by
different?
y = x + 2,
line repres
than the corres
How is this
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r y-value
s above the
ng point
has a greate
two region
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x-value that
shading the
with the same the graph by lightly
on
this
Indicate
(A).
y = x + 2.
graph in
on
n
+ 1.
y < -2x
Answer show
in
inequality
es every point same
on with the
includ
equati
n
solutio
with the
e the second
x-value, the
on the line
Now replac
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It means
e than does
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How is this
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in (A).
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solution will
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se this inequa converting the solid
becau
by
One
However,
Indicate this
steps?
the line itself.
d in both
were shade
with both
the graph
that it agrees
regions of
and check
How many
the graph
of
t region
the darkes
that is in
Pick a point
True
inequalities.
True or False?
Lesson 4
+2
2 ≥ -2
True
True of False?
-2 + 1
289
2 < -2
Graph

How many
© Houghto
n Mifflin
Harcour t
Publishin



Module 5
L4.indd
0_U2M05
SE35393
A2_MTXE
Lesson 5.4
8
Answer shown on graph in (A).
look like?
289
-4
4
How is this different? It means that for any x-value, the solution includes not only the
corresponding point on the line y = x + 2, but also every point in the coordinate plane
with the same x-value that has a greater y-value than the corresponding point on the line.
Indicate this on the graph by lightly shading the two regions above the line represented by
y = x + 2.
Explain to a partner what a system of linear inequalities in two variables
is and how this system’s solutions differ from those of a system of linear
equations.
Essential Question: How can you
represent the solution of a system of
two or more linear inequalities?
y
4
y = -2x + 1
Communicate mathematical ideas, reasoning, and their implications
using multiple representations, including symbols, diagrams, graphs, and
language as appropriate.
ENGAGE
Resource
Locker
289
1/14/15
9:45 PM
1/14/15 9:45 PM
Reflect
1.
EXPLORE
Discussion Why did a dashed line replace the solid line in only the second inequality?
The second inequality is non-inclusive: points on the line are NOT solutions to the second
Turning a System of Equations into
a System of Inequalities
inequality (but they were to the original equation) while the first inequality is inclusive
and points on the line are solutions to it.
Explain 1
Graphing a System of Linear Inequalities
in Two Variables
INTEGRATE TECHNOLOGY
Students can graph a system of equations on their
graphing calculators and then decide which area
should be shaded for a corresponding system of
inequalities.
Graphing a system of linear inequalities is similar to graphing a single inequality, but every point in the solution
region must make all the inequalities in the system true. The boundary line of each inequality in the system is the
graph of the related equation for the inequality, using a solid line if the inequality is inclusive (≥ or ≤) or a dashed
line if the inequality is exclusive (> or <).
Graph the system of inequalities on the grid. Give the coordinates of two
points in the solution set.
1x + 2
⎧ y ≥ - _
2
⎩ y < 2x
Example 1

⎨
AVOID COMMON ERRORS
Students may revert to using all solid lines when
graphing boundary lines in systems of inequalities.
Remind students that the line representing the
boundary of an inequality with a < or > sign is
dotted, while the line representing the boundary of
an inequality with a ≤ or ≥ sign is solid.
Graph the boundary line y = -__12 x + 2. Since the inequality is inclusive, the line should be solid.
Shade the half-plane that represents the solutions to the inequality y ≥ -__12 x + 2 (above and to the right
of the line).
Graph the boundary line y = 2x. Since the inequality is exclusive, the line should be dashed.
Shade the half-plane that represents the solutions to the inequality y < 2x (below and to the right
of the line).
y
8
The region that is in both half-planes will be shaded the darkest.
This represents the solutions of the system. The regions that were
4
shaded only once are useful to help find the solution but they do not
Points: (2, 2) and (4, 2) are in the solution set.
-8
-4
0
-4
4
x
8
-8
EXPLAIN 1
© Houghton Mifflin Harcourt Publishing Company
represent valid solutions of the system; they are regions that represent
solutions to only one of the two inequalities.
Graphing a System of Linear
Inequalities in Two Variables
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
Shade below the line for an inequality with a < or ≤
sign. Shade above the line for an inequality with
a > or ≥ sign.
Module 5
290
Lesson 4
PROFESSIONAL DEVELOPMENT
A2_MTXESE353930_U2M05L4 290
Learning Progressions
15-01-11 4:47 AM
Students have already learned to graph linear equations, linear inequalities, and
systems of linear equations. This lesson expands upon their ability to graph linear
systems. Eventually, students will use graphs of multiple linear inequalities as
constraints in linear programming optimization problems.
Solving Systems of Linear Inequalities
290
⎧
x>3
B ⎨x + y < 2
QUESTIONING STRATEGIES
⎩
Graph a [solid/dashed] boundary line for x = 3.
How can you find points in the solution
set? Pick any points that lie within the
dark-shaded region where the lightly-shaded
regions overlap.
The half-plane that represents the solutions to the inequality x > 3 lies [above/below/left of/right of]
the line.
Graph a [solid/dashed] boundary line for y = -x + 2.
The half-plane that represents the solutions to the inequality x + y < 2 lies [above/below] the line.
Shade each half-plane. The region that is in both half-planes
will be double-shaded.
8
Sample Points: (4, -4), (6, -6)
4
y
x
-8
-4
0
4
-4
8
-8
` Your Turn
Graph the system of inequalities on the grid. Give the coordinates of two points in the
solution set.
⎧x + y < 4
⎧2y + 2x - 3 < 0
2. ⎨
3. ⎨
y - 3x < 1
⎩ y ≥ 2x - 2
⎩
8
y
8
4
y
4
© Houghton Mifflin Harcourt Publishing Company
x
-8
-4
0
-4
4
x
-8
8
-8
0
-4
4
8
-8
Possible points: (0, 0), (-2, 2)
Module 5
-4
Possible points: (0, 0), (2, -2)
291
Lesson 4
COLLABORATIVE LEARNING
A2_MTXESE353930_U2M05L4.indd 291
Peer-to-Peer Activity
Have students work in pairs. Have one student in each pair write a system of linear
inequalities, and have the second student graph the solution. Then, have students
change roles and repeat the process.
291
Lesson 5.4
4/10/15 4:43 PM
Explain 2
Example 2

Solving a Real-World Problem
EXPLAIN 2
Use the four-step problem solving method (Analyze, Formulate a Plan,
Solve, Justify and Evaluate) to solve the problem.
Solving a Real-World Problem
Sandy makes $2 profit on every cup of juice that she sells and $1 on every fruit bar that she
sells. She wants to sell at least 5 fruit bars per day and at least 5 cups of juice per day. She
wants to earn at least $25 per day. Show and describe all the possible combinations of juice
and fruit bars that Sandy needs to sell to meet her goals, and pick two possible combinations
that meet her goals.
AVOID COMMON ERRORS
Students may extend real world solutions into more
than one quadrant of the graph. Remind students
that most real-world situations require that x ≥ 0 and
y ≥ 0, so the solution set exists only in the first
quadrant.
Analyze Information
Identify the important information
• Profit on juice is $2 per cup.
• Profit on fruit bars is $1 each.
• Profit should be $25 or more.
• She wants to sell at least 5 fruit bars.
• She wants to sell at least 5 cups of juice.
Formulate a Plan
You want to find an equation relating profit to number of fruit bars and number of
cups of juice sold. Then convert Sandy’s sales and profit goals into inequalities and
graph them to see where the solutions are.
Solve
Let c represent the number of juice cups sold, and let b represent the number of fruit
bars sold. Profit (p) is given by the equation
p = 2c + b.
28
© Houghton Mifflin Harcourt Publishing Company
The inequalities that represent Sandy’s daily sales and profit goals are given by
c≥5
Sell at least 5 cups of juice.
b≥ 5
Sell at least 5 fruit bars.
2c + b ≥ 25
Earn at least $25 profit.
c
Cups of Juice
24
20
16
12
8
4
b
0
Module 5
4
8
12 16 20
Fruit Bars
292
24
28
Lesson 4
DIFFERENTIATE INSTRUCTION
A2_MTXESE353930_U2M05L4 292
Visual Cues
1/12/15 10:09 PM
Show students that the solution set for a system of inequalities consists of the
points in one of the four regions into which the boundary lines divide the
coordinate plane. Illustrate that the boundary rays for a given region are either
included in the solution set or not, depending on whether the inequalities include
or exclude equality.
Solving Systems of Linear Inequalities
292
Justify and Evaluate
QUESTIONING STRATEGIES
This solution seems reasonable because the solution region includes areas of large
sales and profits increase with increasing sales. Sandy can meet her sales goal if she
sells at the points (8, 9) or (6, 12), corresponding to $8 from sales of fruit bars and
$18 from sales of juice, or $6 from sales of fruit bars and $24 from sales of juice.
This type of solution is called an unbounded solution. The solution region is not
contained in a finite area on the graph.
How do you know whether to use a
< , ≤ , > , or ≥ when writing a system of
linear inequalities to model a real-world
problem? Pay attention to certain words and
phrases. “At most” and “no more than” indicate
that you should use ≤ . “At least” and “no less
than” indicate that you should use ≥.

Vance wants to fence in a rectangular area for his dog. He wants the length of the rectangle
to be at least 30 feet and the perimeter to be no more than 150 feet. Graph all possible
dimensions of the rectangle.
Analyze
Identify the important information
• Rectangular area.
• The length is at least 30 feet.
• The perimeter is at most 150 feet.
Formulate a Plan
You are looking for a solution to the dimensions of the rectangle (width and length).
To find the limits on width you will need to relate width and length to perimeter
with an equation.
perimeter = 2 × length + 2 × width
Solve
Using l and w for length and width, write the inequalities that represent Vance’s
requirements for the fence:
l ≥ 30
180
© Houghton Mifflin Harcourt Publishing Company
2l + 2w ≤ 150
w
160
Width (feet)
140
120
100
80
60
40
20
0
Module 5
ℓ
20
293
40
60
80 100 120 140 160 180
Length (feet)
Lesson 4
LANGUAGE SUPPORT
A2_MTXESE353930_U2M05L4 293
Communicate Math
Have students work in pairs, using cards with the equations y = x and y = x + 3
on them. Give each pair a card with ≤, ≤, ≥, and ≥ on it. Have one student circle
any two symbols and have the other student replace the = signs with the symbols,
then describe the resulting graph. Then have the other student switch the
placement of the inequality symbols and describe that graph. Pairs can compare
their results with others who chose different symbols to see all of the possible
solutions.
293
Lesson 5.4
1/12/15 10:10 PM
Justify and Evaluate
Check two points to see if the solution makes sense.
(l, w) = (50, 20) or (40, 30) are both in the solution region.
Reflect
4.
Think about the difference between a real-world quantity like an amount of flour and one like the balance
of a bank account. There is a boundary on cups of flour that is not stated explicitly but does limit the
choice of real numbers that can represent cups of flour. How is this different from an account balance?
You cannot have negative cups of flour, but an account may be overdrawn and thus have
negative dollars.
5.
In the first example, think of a statement that would make the solution bounded or limit how much profit
Sandy could make.
Possible answer: Sandy can only purchase 50 items (fruit bars or cups of juice) per day or
less.
Your Turn
6.
Olivia is painting a logo for a billboard and wants to use a combination of blue paint and red paint to
completely cover a 5000 square foot billboard. A gallon of paint can cover 500 square feet. Red paint costs
$20 per gallon and blue paint costs $30 per gallon. She only has $250 to spend. Make a graph showing all
the possible combinations of paint that meet her goals.
b
10
8
6
© Houghton Mifflin Harcourt Publishing Company
Blue Paint (gallons)
12
4
2
r
0
20r + 30b ≤ 250
-20r
250
b≤
+
30
30
25
-2
+
b≤
3r
3
_ _
_ _
Module 5
A2_MTXESE353930_U2M05L4 294
2
4
6
8
10
Red Paint (gallons)
12
500r + 500b ≥ 5000
-500r
5000
b≥
+
500
500
_ _
b ≥ -r + 10
294
Lesson 4
1/12/15 10:11 PM
Solving Systems of Linear Inequalities
294
7.
ELABORATE
Dustin decides to try selling whole-wheat muffins and bagels at his farm stand. He has $60 to spend on
ingredients and eight hours to prepare the muffins and bagels that he will sell on opening day. One dozen
muffins costs $10 to make and takes 1 hour, while one dozen bagels costs $15 to make and takes 2 hours.
Make a graph showing all the possible combinations of food that meet his goals.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
8
m
7
6
Muffins (dozen)
The area that represents the solution set can be
identified by choosing a point in each of the four
areas formed by the intersecting lines and
substituting it into both inequalities.
5
4
3
2
1
SUMMARIZE THE LESSON
0
How do you graph the solution of a system of
linear inequalities? Graph the corresponding
system of linear equations, using dotted lines for
inequalities with < or > and solid lines for
inequalities with ≥ or ≤. Shade the regions
indicated by the inequality signs. The solution set, if
one exists, is the region where the shaded regions
intersect.
m + 2b ≤ 8
10m + 15b ≤ 60
b≤
b≤
8
-m _
_
+
2
2
-m
b≤_+4
2
3
4
5
6
Bagels (dozen)
7
8
60 _
10m
_
15
15
-2m
b≤_+4
3
© Houghton Mifflin Harcourt Publishing Company
8.
When graphing inequalities, how can you check that you shaded the correct areas?
Pick a point in the shaded area and check it in the inequality.
9.
Will the point where the two lines intersect always be a solution? Why or why not?
No; if at least one of the inequalities is not inclusive than the point will not be a solution.
It will only be a solution of both inequalities are inclusive.
10. Essential Question Check-In What is the difference between the way a “less than” inequality is graphed
and the way a “less than or equal to” inequality is graphed?
Graph the boundary of each inequality as a dashed line for exclusive inequalities (< or >)
and a solid line for inclusive inequalities (≤ or ≥).
A2_MTXESE353930_U2M05L4 295
Lesson 5.4
2
Elaborate
Module 5
295
b
1
295
Lesson 4
1/12/15 10:12 PM
EVALUATE
Evaluate: Homework and Practice
• Online Homework
• Hints and Help
• Extra Practice
Graph the system of inequalities on the grid. Give the coordinates
of two points in the solution set.
1x - 4
1. y ≥ _
2. y < 2x - 2
2
y ≤ -5x + 5
y < -2x + 4
8
y
8
y
ASSIGNMENT GUIDE
4
4
x
x
-8
-4
0
4
-4
-8
8
-4
4
-4
8
-8
-8
Possible points: (0, 0), (-2, 2)
3.
0
Possible points: (0, -4), (1, -8)
x-y>2
4.
2x + 3y ≤ 4
-3y ≤ 6x
-x - 2y - 2 > 0
8
y
8
4
y
-4
0
4
-4
-4
0
4
-4
8
-8
Possible points: (2, -2), (4, -4)
6.
y ≥ 2x - 5
3x - 2y ≤ 12
3x - 2y > -6
y
8
y
4
4
x
x
-8
-4
0
-4
4
-8
8
Possible points: (0, 0), (1, 1)
Module 5
-4
0
-4
4
8
Exercises 5–16
Example 2
Solving a Real-World Problem
Exercises 17–21
Possible points: (0, 0), (1, 1)
Lesson 4
296
Exercise
Example 1
Graphing a System of Inequalities in
Two Variables
-8
-8
A2_MTXESE353930_U2M05L4 296
Exercises 1–4
In graphing an inequality involving a vertical line,
the area to the left or to the right of the line is shaded.
For inequalities in the forms x < a and x ≤ a , the
region to the left is shaded. For inequalities in the
forms x > a and x ≥ a , the region to the right is
shaded.
© Houghton Mifflin Harcourt Publishing Company
Possible points: (4, -4), (6, -6)
y ≤ 2x + 5
8
Explore
Turning a System of Equations into a
System of Inequalities
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
x
-8
8
-8
5.
Practice
4
x
-8
Concepts and Skills
Depth of Knowledge (D.O.K.)
Mathematical Processes
1–16
1 Recall of Information
1.E Create and use representations
17–21
2 Skills/Concepts
1.A Everyday life
22
2 Skills/Concepts
1.D Multiple representations
23
3 Strategic Thinking
1.B Problem solving model
24–25
3 Strategic Thinking
1.D Multiple representations
1/12/15 10:13 PM
Solving Systems of Linear Inequalities
296
7.
AVOID COMMON ERRORS
y ≥ -2x + 2
8.
2x + y < 10
Students may not recall the rules for multiplying and
dividing inequalities by negative numbers. Remind
students that when they multiply or divide both sides
an inequality by a negative number, the direction of
the inequality sign changes.
9x + 3y ≤ 10
3x + y > -5
8
y
8
4
y
4
x
-8
-4
0
4
-4
x
-8
8
-4
-8
4
-4
8
-8
Possible points: (4, 0), (2, 2)
9.
0
Possible points: (0, 0), (-1, -1)
1x - 4
y≤_
2
1x + 6
y≥_
2
10. 3x - 5y > 6
5y - 3x > 10
8
y
8
y
4
4
x
x
-8
-4
0
4
-4
-8
8
-4
No Solution
© Houghton Mifflin Harcourt Publishing Company
8
4
8
No Solution
11. x > 3
12. y < 2x + 1
y ≥ -2
2x + 3y <18
x+y<6
y - x ≥ -4
8
y
8
4
y
4
x
-8
-4
0
-4
4
x
-8
8
-8
Module 5
A2_MTXESE353930_U2M05L4 297
-4
0
-4
-8
Possible points: (2, 2), (0, -2)
Possible points: (4, 0), (4, 1)
Lesson 5.4
-4
4
-8
-8
297
0
297
Lesson 4
15-01-11 4:47 AM
Write the system of inequalities shown by each graph.
13.
8
14.
y
8
4
CONNECT VOCABULARY
y
4
x
-8
-4
0
4
-4
x
-4
8
-8
y > 2x - 4
4
-4
8
x≤3
1
y≤ x
5
_
8
0
-8
_
y ≤ -1x + 2
2
15.
Compare and contrast a system of linear inequalities
in two variables with a system of three linear equations
in three variables and the linear-quadratic systems
discussed in the previous lessons. Have students
complete a chart with the similarities and differences
among these three kinds of systems.
16.
y
8
y
4
4
x
-8
-4
0
-4
4
8x
-8
-4
8
-8
-8
_
_
y < -1x + 4
3
y ≤ -1x - 2
3
y > -4
y≤x
y < -5x + 20
21
w
Wakeboards
18
15
12
9
s≥3
6
w≥6
3
150s + 100w ≥ 2000
0
2000
150s
3
6
9
12 15
w≥+
100
100
Surfboards
3
w ≥ - s + 20
2
Check point (10, 10): 150(10) + 100(10) = 2500, so the owner sells at
least 3 surfboards and 6 wakeboards and earns more than $2000.
_ _
_
Module 5
A2_MTXESE353930_U2M05L4 298
298
s
18
21
© Houghton Mifflin Harcourt Publishing Company
17. A surf shop makes profits of $150 for each surfboard
and $100 for each wakeboard. The owner sells at least
3 surfboards and at least 6 wakeboards per month.
The shop owner wants to earn at least $2000 per
month. Graph all possible combinations of surfboard
and wakeboard sales that would satisfy the store
owner’s earnings goal. Use a check point to justify the
reasonableness of the solution.
Lesson 4
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Solving Systems of Linear Inequalities
298
Cheddar Cheese (lb)
18. Alice is serving pepper jack cheese and cheddar cheese on a platter. She wants to have
more than 2 pounds of each. Pepper jack cheese costs $4 per pound and cheddar
cheese costs $2 per pound. Alice wants to spend at most $20 on cheese. Graph all
possibile combinations of the two cheeses Alice could buy. Use a check point to justify
the reasonableness of the solution.
p> 2
c
8
c> 2
6
4p + 2c ≤ 20
4p
20
c≤ +
2
2
c ≤ -2p + 10
_ _
4
2
p
0
2
4
6
8
Pepper Jack Cheese (lb)
Check point (3, 3): 4(3) + 2(3) = 18, so Alice has at least
2 pounds of each cheese and spends less than $20.
19. In one week, Ed can mow at most 9 lawns and rake at most 7 lawns. He charges $20
for mowing and $10 for raking. He needs to earn more than $120 in one week. Graph
all the possible combinations of mowing and raking that Ed can do to meet his goal.
Use a check point to justify the reasonableness of the solution.
Raking Jobs
16
r
m≤ 9
12
r≤ 7
8
4
m
24
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Lesson 5.4
Check point (6, 6): 20(6) + 10(6) = 180, so Ed mows
fewer than 9 lawns, rakes fewer than 7 lawns, and earns
more than $120.
4
8
12 16
Mowing Jobs
b
15p + 10b > 90
15p
90
+
p + b ≤ 20
b> 10
10
3p
+9
b ≤ - p + 20 b > 2
Check point (9, 6): 15(9) + 10(6) = 195, so Linda works
no more than 20 hours and earns more than $90.
_ _
_
18
12
6
p
0
299
_ _
20. Linda works at a pharmacy for $15 an hour. She also baby-sits for $10 an hour. Linda
needs to earn more than $90 per week, but she does not want to work more than
20 hours per week. Graph the number of hours Linda could work at each job to meet
her goals. Use a check point to justify the reasonableness of the solution.
Hours Babysitting
© Houghton Mifflin Harcourt Publishing Company
0
20m + 10r > 120
120
20m
+
r > - 10
10
r > -2m + 12
6
12 18 24
Hours at Pharmacy
299
Lesson 4
1/28/15 9:02 AM
Soybeans (acres)
21. Tony wants to plant at least 40 acres of corn and at least 50 acres of soybeans. He has
200 acres on which to plant. Graph all the possible combinations of the number of
acres of corn and of soybeans Tony could plant. Use a check point to justify the
reasonableness of the solution.
200
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
c ≥ 40
s
Students graph the solutions of linear systems by
graphing each linear inequality and shading the
overlapping region. Shading the individual
inequalities is helpful but not essential, since none of
the region outside the overlap areas is a solution. The
solution of two inequalities whose equations form
vertical angles could be shown as the V-shaped
region and its boundary only.
s ≥ 50
s + c ≤ 200 → s ≤ -c + 200
150
Check point (80, 80): 80 + 80 = 160, so Tony can plant
at least 40 acres of corn, at least 50 acres of soybeans,
and less than 200 total acres.
100
50
c
0
50 100 150 200
Corn (acres)
22. Match each set of inequalities with the correct graph.
⎧y < x
⎨
⎩ y ≤ -2x + 1
A.
8
⎧y ≥ x
⎨
⎩ y > -2x + 1
C.
B.
y
8
B.
y
x
-8
-4
4
-4
x
-8
8
-4
-8
⎧y ≤ x
⎨
⎩ y > -2x + 1
D.
D.
y
8
A.
y
x
-8
-4
-4
4
x
-8
8
-8
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A2_MTXESE353930_U2M05L4 300
© Houghton Mifflin Harcourt Publishing Company
8
-4
8
-8
⎧y > x
⎨
⎩ y ≥ -2x + 1
C.
4
-4
-4
4
8
-8
300
Lesson 4
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Solving Systems of Linear Inequalities
300
JOURNAL
H.O.T. Focus on Higher Order Thinking
Have students write a system of linear inequalities in
two variables and graph the solution.
23. Explain the Error Two students wrote a system of linear inequalities to describe the
graph. Which student is incorrect? Explain the error.
8
y
4
x
-8
-4
0
-4
4
8
Student A
Student B
1x + 4
y<_
2
y ≥ -3x - 6
1x + 4
y≥ _
2
y < -3x - 6
Student B is incorrect. The signs of the inequalities
are switched.
-8
24. Critical Thinking Can the solutions of a system of linear inequalities be the points
on a line? Give an example or explain why not.
Yes; if the inequalitites in each system are based on the same line and
that line is included in the system, then the solutions of the system are
the points on the line. For example, the solutions of the system
⎧y ≥ x + 5
are represented by all the ordered pairs on the line y = x + 5.
⎨
⎩y ≤ x + 5
25. Make a Conjecture What must be true of the boundary lines in a system of two
linear inequalities if there is no solution of the system? Explain.
© Houghton Mifflin Harcourt Publishing Company
There are two possibilities. One is that the boundary lines are parallel
and the solutions of each inequality go in opposite directions. The other
is that the boundary lines are the same line but that the inequalities are
non-inclusive and their solutions go in opposite directions.
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301
Lesson 5.4
301
Lesson 4
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Lesson Performance Task
QUESTIONING STRATEGIES
What do you notice about the graphs of the
boundary lines? They are two pairs of
parallel lines.
Ingrid has 6 nephews and 4 nieces and is going to buy them all presents. She wants to buy the
same present for each of the nephews and the same present for each of the nieces. Ingrid plans
to spend at least $180 but no more than $240. She wants the prices of the presents to be within
$4 of each other. Find and graph the solution set. What do you notice about the solution region
on the graph?
Nieces’ Presents
60
Suppose all of the inequalities in the system
were reversed? Describe the solution region.
Explain. There would be no solution; for each pair
of parallel lines, the solution regions would face
away from each other and would not intersect.
⎧6x + 4y ≥ 180
⎪
⎪ 6x + 4y ≤ 240
.
This gives the solution set . ⎨
⎪y ≤ x + 4
⎪
⎩y≥x-4
y
48
36
The solution region on the graph is a parallelogram.
24
12
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
x
0
10
20
30
40
Nephews’ Presents
Have students consider the simplest quadrilateral that
they could graph using inequalities and what types of
inequalities could be used. Students should see that a
set of inequalities like 2 ≤ x ≤ 5 and 1 ≤ y ≤ 4,
which represent four simple inequalities, would result
in a solution that is a rectangle.
© Houghton Mifflin Harcourt Publishing Company
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Lesson 4
302
EXTENSION ACTIVITY
A2_MTXESE353930_U2M05L4.indd 302
Systems of inequalities can be used to create other geometric shapes. Have
students graph the given system.
Students should find that they have shaded
a pentagon. Encourage students to come up
with different patterns based on different
systems of inequalities. They might also
consider how to make three-dimensional
shapes such as tetrahedrons by using systems
of inequalities in three variables.
20/02/14 4:25 AM
⎧ y ≥ - 3x + 6
⎪ y≤
4 x+6
__
5
⎪
4 x + 14
⎨ y ≤ - __
5
⎪
⎪ y ≥ 3x - 24
⎩ y≥0
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Solving Systems of Linear Inequalities
302