98 Chapter 14 Linear Inequalities
Lesson
14.1
An inequality is a mathematical sentence that compares expressions.
It contains the symbols
<, >,
≤
,
or ≥ . To write an inequality, look for the
following phrases to determine where to place the inequality symbol.
Key Vocabulary
inequality
solution of an
inequality
solution set
graph of an inequality
EXAMPLE
Writing an Inequality
1
A number w minus 3.5 is less than or equal to 2. Write this sentence as
an inequality.
A number w minus 3.5 is less than or equal to 2.
w 3.5 ≤ 2
An inequality is w3.5 ≤ 2.
Write the word sentence as an inequality.
1. A number b is fewer than 30.4. 2. Twice a number k is at least
7
10
.
A solution of an inequality is a value that makes the inequality true. An
inequality can have more than one solution. The set of all solutions of an
inequality is called the solution set.
Exercises 6 – 9
Reading
The symbol ≥
/
means
“is not greater than or
equal to.”
Value of xx 5 ≥ 2 Is the inequality true?
6
6 5 ≥
?
2
1 ≥ 2
✓
yes
7
7 5 ≥
?
2
2 ≥ 2
✓
yes
8
8 5 ≥
?
2
3
≥
/
2
✗
no
Inequality Symbols
Symbol <>
≤ ≥
Key
Phrases
● is less
than
● is fewer
than
● is greater
than
● is more
than
● is less than or
equal to
● is at most
● is no more than
● is greater than
or equal to
● is at least
● is no less than
MSFL7WBAD_1401.indd 98 4/14/10 4:10:04 PM
Section 14.1 Writing and Graphing Inequalities 99
EXAMPLE
Checking Solutions
2
Tell whether 4 is a solution of the inequality.
a. x 8
<
3 b. 4.5x
>
21
x 8
<
3
Write the inequality. 4.5x
>
21
4 8
<
?
3 Substitute −4 for
x
. 4.5(4)
>
?
21
4
<
/ 3
✗
Simplify. 18
>
21 ✓
4 is not less than 3. 18 is greater than 21.
So, 4 is not a solution So, 4 is a solution
of the inequality. of the inequality.
Tell whether 6 is a solution of the inequality.
3. c 4
<
1 4. 5 m ≤ 10 5. 21 x ≥ 3.5
The graph of an inequality shows all of the solutions of the inequality on
a number line. An open circle
is used when a number is not a solution. A
closed circle
is used when a number is a solution. An arrow to the left or
right shows that the graph continues in that direction.
Exercises 11–16
EXAMPLE
Graphing an Inequality
3
Graph y ≤ 3.
Graph the inequality on a number line.
6. b
>
8 7. g ≤ 1.4 8. r
<
1
2
9. v ≥
0.09
Exercises 17–20
7 6 5 4 3 2 10123
Test a number to the left of 3.
y
4 is a solution.
Test a number to the right of 3.
y
0 is
not
a solution.
Use a closed circle
because 3 is a solution.
7 6 5 4 3 2 10123
Shade the number line on the side
where you found the solution.
MSFL7WBAD_1401.indd 99 4/14/10 4:10:05 PM
Exercises
14.1
9
+(-6)=3
3
+(-3)=
4
+(-9)=
9
+(-1)=
100 Chapter 14 Linear Inequalities
1. VOCABULARY Would an open circle or a closed circle be used in the graph of
the inequality k
<
250? Explain.
2. DIFFERENT WORDS, SAME QUESTION Which is different? Write “both” inequalities.
w is greater than or equal to 7.
w is no less than 7.
w is no more than 7.
w is at least 7.
3. REASONING Do x ≥ 9 and 9 ≥ x represent the same inequality? Explain.
Write an inequality for the graph. Then, in words, describe all the values of x that
make the inequality true.
4.
3 0 3 6 9 12 15 18
5.
7 6 5 4 3 2 1
Write the word sentence as an inequality.
6. A number x is no less than 4. 7. A number y added to 5.2 is less than 23.
8. A number b multiplied by 5 is at most
3
4
.
9. A number k minus 8.3 is greater than 48.
10. ERROR ANALYSIS Describe and correct the error in
writing the word sentence as an inequality.
Tell whether the given value is a solution of the inequality.
11. s 6 ≤ 12; s 4 12. 15n
>
3; n 2 13. a 2.5 ≤ 1.6; a 4.1
14. 3.3q
>
13; q 4.6 15.
4
5
h ≥ 4; h 15 16.
1
12
p
<
1
3
; p
1
6
Graph the inequality on a number line.
17. g ≥ 6 18. q
>
1.25 19. z
<
11
1
4
20. w ≤
289
21. DRIVING When you are driving with a learner’s license, a licensed driver
who is 21 years of age or older must be with you. Write an inequality that
represents this situation.
Twice a number c is
at least
4
9
.
2c ≤
4
9
✗
1
2
3
MSFL7WBAD_1401.indd 100 4/14/10 4:10:05 PM
Section 14.1 Writing and Graphing Inequalities 101
Solve the equation. Check your solution.
28. r 12 3 29. 4.2 p 2.5 30. n 3
7
31. MULTIPLE CHOICE Which linear function relates y to x ?
A y 0.5x 3 B y 2x 3
C y 0.5x 3 D y 2x 3
Tell whether the given value is a solution of the inequality.
22. 3p
>
5 p; p 4 23.
y
2
≥ y 11; y 18
24. VIDEO GAME RATINGS Each rating is matched with the inequality that
represents the recommended ages of players. Your friend is old enough to
play “E 10+” games. Is your friend old enough to play “T” games? Explain.
25. SCUBA DIVING Three requirements for a scuba diving training
course are shown.
a. Write and graph three inequalities that represent the requirements.
b. You can swim 10 lengths of a 25-yard pool. Do you satisfy the
swimming requirement of the course? Explain.
26. LUGGAGE On an airplane, the maximum sum of
the length, width, and height of a carry-on bag is
45 inches. Find three different sets of dimensions
that are reasonable for a carry-on bag.
27.
A number m is less than another
number n. The number n is less than or equal
to a third number p.
a. Write two inequalities representing these relationships.
b. Describe the relationship between m and p.
c. Can m be equal to p ? Explain.
x 1012
y 5 3 11
x q3 x q6 x q10 x q13 x q17
6
3
13
17
10
h
w
The ESRB rating icons are registered trademarks
of the Entertainment Software Association.
MSFL7WBAD_1401.indd 101 4/14/10 4:10:06 PM
102 Chapter 14 Linear Inequalities
Lesson
14.2
Addition Property of Inequality
Words If you add the same number to each side of an inequality,
the inequality remains true.
Numbers
3
<
2
Algebra x
3
>
10
4 4 3 3
1
<
6 x
>
7
Subtraction Property of Inequality
Words If you subtract the same number from each side of an
inequality, the inequality remains true.
Numbers
3
<
1
Algebra x
7
>
20
5 5 7 7
8
<
4 x
>
27
These properties are also true for ≤ and ≥ .
Study Tip
You can solve
inequalities the
same way you solve
equations. Use inverse
operations to get the
variable by itself.
Study Tip
To check a solution, you
check some numbers
that are solutions and
some that are not.
EXAMPLE
Solving an Inequality Using Addition
1
Solve x 6 ≥ 10. Graph the solution.
x 6 ≥ 10
Write the inequality.
6 6 Add 6 to each side.
x ≥ 4 Simplify.
The solution is x ≥ 4.
8 7 6 5 4 3 2 1012
x ≥ 4
Check:
x
5 is
not
a solution.
Check:
x
0 is a solution.
Solve the inequality. Graph the solution.
1. b 2
>
9 2. m 3.8 ≤ 5 3.
1
4
>
y
1
4
Undo the subtraction.
MSFL7WBAD_1402.indd 102 4/14/10 4:09:42 PM
Section 14.2 Solving Inequalities Using Addition or Subtraction 103
EXAMPLE
Solving an Inequality Using Subtraction
2
Solve 8
>
1.4 x. Graph the solution.
8
>
1.4 x
Write the inequality.
1.4 1.4 Subtract 1.4 from each side.
9.4
>
x Simplify.
The solution is x
<
9.4.
10.09.9 9.8 9.7 9.6 9.5 9.4 9.3 9.2 9.1 9.0
x < 9.4
Solve the inequality. Graph the solution.
4. k 5 ≤ 3 5.
5
6
≤ z
2
3
6. p 0.7
>
2.3
Reading
The inequality
9.4
>
x is the same
as x
<
9.4.
Exercises 6 –17
EXAMPLE
Real-Life Application
3
On a train, carry-on bags can weigh no more than 50 pounds. Your bag
weighs 24.8 pounds. Write and solve an inequality that represents the
amount of weight you can add to your bag.
Words
Weight of
your bag
plus
amount of weight
you can add
is no
more than
the weight
limit.
Variable Let w be the possible weight you can add.
Inequality 24.8 w ≤ 50
24.8 w ≤ 50
Write the inequality.
24.8 24.8 Subtract 24.8 from each side.
w ≤ 25.2 Simplify.
You can add no more than 25.2 pounds to your bag.
7. WHAT IF? Your carry-on bag weighs 32.5 pounds. Write and
solve an inequality that represents the possible weight you
can add to your bag.
Undo the addition.
MSFL7WBAD_1402.indd 103 4/14/10 4:09:43 PM
Exercises
14.2
9
+(-6)=3
3
+(-3)=
4
+(-9)=
9
+(-1)=
104 Chapter 14 Linear Inequalities
1. REASONING Is the inequality r 5 ≤ 8 the same as 8 ≤ r 5? Explain.
2. WHICH ONE DOESN’T BELONG? Which inequality does not belong with the
other three? Explain your reasoning.
c
7
2
≤
3
2
c
7
2
≥
3
2
3
2
≥ c
7
2
c
3
2
≤
7
2
Graph the inequality.
3. x ≥ 6 4. p
<
2 5. 4 ≤ w
Solve the inequality. Graph the solution.
6. y 3 ≥ 7 7. t 8
>
4 8. n 11 ≤ 20
9. a 7
>
1 10. 5
<
v
1
2
11.
1
5
>
d
4
5
12.
2
3
≤ g
1
3
13. m
7
4
≤
11
4
14. 11.2 ≤ k 9.8
15. h 1.7
<
3.2 16. 0
>
s
17. 5 ≥ u 4.5
18. ERROR ANALYSIS Describe and
correct the error in graphing the
solution of the inequality.
19. PELICAN The maximum volume of a great white pelican’s
bill is about 700 cubic inches.
a. A pelican scoops up 100 cubic inches of water.
Write and solve an inequality that represents the
additional volume the bill can contain.
b. A pelican’s stomach can contain about one-third
the maximum amount that its bill can contain.
Write an inequality that represents the volume
of the pelican’s stomach.
1
2
5 ≥ x 5
10 ≥ x
✗
8 9 10 11 12 13
MSFL7WBAD_1402.indd 104 4/14/10 4:09:43 PM
Section 14.2 Solving Inequalities Using Addition or Subtraction 105
Solve the equation.
28. 6 3x 29.
r
5
2 30. 4c 15 31. 8
2
3
b
32. MULTIPLE CHOICE Which fraction is equivalent to 3.8?
A
5
19
B
19
5
C
12
15
D
12
5
Write and solve an inequality that represents the value of x.
20. The perimeter is less 21. The base is greater 22. The perimeter is less
than 16 feet. than the height. than or equal to 5 feet.
4 ft
4 ft
x
x 2
10 m
x
12 in.12 in.
10 in. 10 in.
23. REASONING The solution of w c ≤ 8 is w ≤ 3. What is the value of c ?
24. FENCE The hole for a fence post is 2 feet deep. The top of the fence post
needs to be at least 4 feet above the ground. Write and solve an inequality
that represents the required length of the fence post.
25. VIDEO GAME You need at least 12,000 points to advance
to the next level of a video game.
a. Write and solve an inequality that represents the
number of points you need to advance.
b. You fi nd a treasure chest that increases your score
by 60%. How does this change the inequality?
26. POWER A circuit overloads at 1800 watts of electricity. A microwave
that uses 1100 watts of electricity is plugged into the circuit.
a. Write and solve an inequality that represents the
additional number of watts you can plug in without
overloading the circuit.
b. In addition to the microwave, what two appliances in the
table can you plug in without overloading the circuit?
27.
The maximum surface area of the solid is
15
square millimeters. Write and solve an inequality
that represents the height of the cylinder.
Appliance Watts
Clock radio 50
Blender 300
Hot plate 1200
Toaster 800
TIME LEFT: 1 min.
CURRENT SCORE: 4500
h
2 mm
MSFL7WBAD_1402.indd 105 4/14/10 4:09:44 PM
106 Chapter 14 Linear Inequalities
14
Study Help
Make a four square to help you study these
topics.
1. inequality (
>
) 2. inequality ( ≤ )
3. inequality ( ≥ )
4. solving an inequality using addition
5. solving an inequality using subtraction
After you complete this chapter, make four
squares for the following topics.
6. solving an inequality using multiplication
7. solving an inequality using division
You can use a four square to organize information about a topic. Each of the four squares
can be a category, such as defi nition, vocabulary, example, non-example, words, algebra, table,
numbers, visual, graph, or equation. Here is an example of a four square for an inequality.
“Sorry, but I have limited space in my
four square. I needed pet names with
only three letters.”
INEQUALITY
(<)
Graph:
To graph x < 2, draw an open circle at
x = 2. Then draw an arrow pointing to
the left.
Definition:
A mathematical sentence that uses
an inequality symbol (<).
Example:
Words: A number x is less than 2.
Symbols: x < 2
Vocabulary:
Key phrases that can represent
an inequality:
• i s less than
• i s fewer than
−3 −2 −1 0 1 2 3
MSFL7WBAD_1400_mc.indd 106 4/14/10 4:10:27 PM
Sections 14.1– 14.2 Quiz 107
Quiz
14.1– 14.2
Write the word sentence as an inequality.
1. A number x plus 1 is less than −13. 2. A number t minus 1.6 is at most 9.
Tell whether the given value is a solution of the inequality.
3. 12n
<
−2; n = −1 4. y + 4
<
−3; y = −7
Graph the inequality on a number line.
5. x
>
−10 6. y ≤
3
—
5
7. w
<
6.8
Solve the inequality.
8. x − 2
<
4 9. g + 14 ≥ 30 10. h − 1 ≤ −9
11. s + 3
>
−7 12. v −
3
—
4
<
0 13.
3
—
2
<
p +
1
—
2
14. WATERCRAFT You must be at least 14 years
old to operate a personal watercraft in
Florida. Write an inequality that
represents this situation.
15. REASONING The solution of x − a
>
4
is x
>
11. What is the value of a ?
16. MP3 PLAYER Your MP3 player can store up to 8 gigabytes of media.
You transfer 3.5 gigabytes of media to the MP3 player. Write and solve
an inequality that represents the amount of memory available on the
MP3 player.
17. LIFEGUARD Three requirements for a lifeguard
training course are shown.
a. Write and graph three inequalities that
represent the requirements.
b. You can swim 350 feet. Do you satisfy the
swimming requirement of the course?
Explain.
Lifeguard Training Requirements
Swim at least 100 yards
Tread water for at least
5 minutes
Swim 10 yards or more
underwater without taking
a breath
LIFEGUARDS NEEDED
LIFEGUARDS NEEDED
Take Our Training
Course NOW!!!
MSFL7WBAD_1400_mc.indd 107 4/14/10 4:10:32 PM
108 Chapter 14 Linear Inequalities
Lesson
14.3
Remember
Multiplication and
division are inverse
operations.
EXAMPLE
Solving an Inequality Using Multiplication
1
Solve
x
8
>
5. Graph the solution.
x
8
>
5 Write the inequality.
8
x
8
>
8
(5) Multiply each side by 8.
x
>
40 Simplify.
The solution is x
>
40.
Solve the inequality. Graph the solution.
1. a 2
<
4 2.
n
7
≥ 1 3. 6.4 ≥
w
5
Undo the division.
90 80 70 60 50 40 30 20 10
0 10
x > 40
Check:
x
0 is a solution.Check:
x
80 is
not
a solution.
Multiplication and Division Properties of Inequality (Case 1)
Words If you multiply or divide each side of an inequality by the
same positive number, the inequality remains true.
Numbers
6
<
8 6
>
8
2
(6)
<
2
8
6
2
>
8
2
12
<
16 3
>
4
Algebra
x
2
<
9 4x
>
12
2
x
2
<
2
(9)
4x
4
>
12
4
x
<
18 x
>
3
These properties are also true for ≤ and ≥ .
MSFL7WBAD_1403.indd 108 4/14/10 4:09:15 PM
Section 14.3 Solving Inequalities Using Multiplication or Division 109
EXAMPLE
Solving an Inequality Using Division
2
Solve 3x ≤ 24. Graph the solution.
3x ≤ 24
Write the inequality.
3x
3
≤
24
3
Divide each side by 3.
x ≤ 8 Simplify.
The solution is x ≤ 8.
Solve the inequality. Graph the solution.
4. 4b ≥ 36 5. 2k
>
10 6. 18
>
1.5q
Undo the multiplication.
Exercises 10 –18
A negative sign in an
inequality does not
necessarily mean you
must reverse the
inequality symbol.
Only reverse the
inequality symbol when
you multiply or divide
both sides by a
negative number.
Common Error
16 14 12 10 8 6 4 20
24
x ≤ 8
Check:
x
0 is
not
a solution.Check:
x
10 is a solution.
Multiplication and Division Properties of Inequality (Case 2)
Words If you multiply or divide each side of an inequality by the
same negative number, the direction of the inequality symbol
must be reversed for the inequality to remain true.
Numbers
6
<
8 6
>
8
(2)
(6)
>
(2)
8
6
2
<
8
2
12
>
16 3
<
4
Algebra
x
6
<
3 5x
>
30
6
x
6
>
6
3
5x
5
<
30
5
x
>
18 x
<
6
These properties are also true for ≤ and ≥ .
MSFL7WBAD_1403.indd 109 4/14/10 4:09:15 PM
110 Chapter 14 Linear Inequalities
EXAMPLE
Solving an Inequality Using Multiplication
3
Solve
y
3
>
2. Graph the solution.
y
3
>
2 Write the inequality.
3
y
3
< 3
2
Multiply each side by 3. Reverse the
inequality symbol.
y
<
6 Simplify.
The solution is y
<
6.
Undo the division.
EXAMPLE
Solving an Inequality Using Division
4
Solve 7y ≤ 35. Graph the solution.
7y ≤ 35
Write the inequality.
7y
7
≥
35
7
Divide each side by 7. Reverse the inequality symbol.
y ≥ 5 Simplify.
The solution is y ≥ 5.
101234567
89
y ≥ 5
Check:
y
6 is a solution.
Check:
y
0 is
not
a solution.
Solve the inequality. Graph the solution.
7.
p
4
<
7 8.
x
5
≤ 5
9. 1 ≥
1
10
z 10. 9m
>
63
11. 2r ≥ 22 12. 0.4y ≥ 12
Undo the multiplication.
Exercises 27–35
9 8 7 6 5 4 3 2 1
01
y < 6
Check:
y
9 is a solution.
Check:
y
0 is
not
a solution.
MSFL7WBAD_1403.indd 110 4/14/10 4:09:16 PM
Section 14.3 Solving Inequalities Using Multiplication or Division 111
Exercises
14.3
9
+(-6)=3
3
+(-3)=
4
+(-9)=
9
+(-1)=
1. VOCABULARY Explain how to solve
x
6
<
5.
2. WRITING Explain how solving 2x
<
8 is different from solving 2x
<
8.
3. OPEN-ENDED Write an inequality that is solved using the Division Property of
Inequality where the inequality symbol needs to be reversed.
Use a table to solve the inequality.
4. 4x
<
4 5. 2x ≤ 2 6. 5x
>
15
7.
x
3
≥ 1 8.
x
2
>
5
2
9.
x
4
≤
3
8
Solve the inequality. Graph the solution.
10. 3n
>
18 11.
c
4
≤ 9 12. 1.2m
<
12
13. 14
>
x 2 14.
w
5
≥ 2.6 15. 5
<
2.5k
16. 4x ≤
3
2
17. 2.6y ≤ 10.4 18. 10.2
>
b
3.4
19. ERROR ANALYSIS Describe and correct
the error in solving the inequality.
Write the word sentence as an inequality.
Then solve the inequality.
20. The quotient of a number and 3 is at most 4.
21. A number divided by 8 is less than 2.
22. Four times a number is at least 12.
23. The product of 5 and a number is greater than 20.
24. CAMERA You earn $9.50 per hour at your
summer job. Write and solve an inequality
that represents the number of hours you need
to work in order to buy a digital camera that
costs $247.
x
2
<
5
2
x
2
>
2
(5)
x
>
10
✗
1
2
2
4
.
MSFL7WBAD_1403.indd 111 4/14/10 4:09:16 PM
112 Chapter 14 Linear Inequalities
25. COPIES You have $3.65 to make copies. Write and solve
an inequality that represents the number of copies you
can make.
26. SPEED LIMIT The maximum speed limit for a school bus
is 55 miles per hour. Write and solve an inequality that
represents the number of hours it takes to travel
165 miles in a school bus.
Solve the inequality. Graph the solution.
27. 2n ≤ 10 28. 5w
>
30 29.
h
6
≥ 7
30. 8
<
1
3
x 31. 2y
<
11 32. 7d ≥ 56
33. 2.4
>
m
5
34.
k
0.5
≤ 18 35. 2.5
>
b
1.6
36. ERROR ANALYSIS Describe and correct the
error in solving the inequality.
37. CRITICAL THINKING Are all numbers greater
than zero solutions of x
>
0? Explain.
38. TRUCKING By law, the maximum height (including freight) of a vehicle in
Florida is 13.5 feet.
a. Write and solve an inequality that represents the number of crates that can
be stacked vertically on the bed of the truck.
b. Five crates are stacked vertically on the bed of the truck. Is this legal? Explain.
Write and solve an inequality that represents the value of x.
39. Area ≥ 102 cm
2
40. Area
<
30 ft
2
12 cm
x
10 ft
x
4m ≥ 1 6
4m
4
≥
16
4
m ≥
4
✗
43
h
3.5 ft
28 in.
Not drawn to scale
MSFL7WBAD_1403.indd 112 4/14/10 4:09:17 PM
Section 14.3 Solving Inequalities Using Multiplication or Division 113
Solve the equation.
48. 4w 5 11 49. 4(x 3) 21
50.
v
6
7 4 51.
m 300
4
96
52. MULTIPLE CHOICE Which measure can have more than one value for a given
data set?
A mean B median C mode D range
41. TRIP You and three friends are planning a trip. You want to keep the cost
below $80 per person. Write and solve an inequality that represents the
total cost of the trip.
42. REASONING Explain why the direction of the inequality symbol must be
reversed when multiplying or dividing by the same negative number.
43. PROJECT Choose two musical artists to research.
a. Use the Internet or a magazine to complete the table.
b. Find the average number of copies sold per month
for each CD.
c. Use the release date to write and solve an
inequality that represents the minimum
average number of copies sold per month
for each CD.
d. In how many months do you expect the
number of copies of the second top selling
CD to surpass the current number of copies
of the top selling CD?
Artist Name of CD Release Date
Current Number
of Copies Sold
1.
2.
Describe all numbers that satisfy both inequalities. Include
a graph with your description.
44. 3m
>
12 and 2m
<
12 45.
n
2
≥ 3 and
n
4
≥ 1
46. 2x ≥ 4 and 2x ≥ 4 47.
m
4
>
5 and
m
4
<
10
g
h
.
the table.
mont
h
s
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114 Chapter 14 Linear Inequalities
Lesson
14.4
You can solve multi-step inequalities the same way you solve
multi-step equations.
EXAMPLE
Solving Two-Step Inequalities
1
a. Solve 5x 4 ≥ 11. Graph the solution.
5x 4 ≥ 11
Write the inequality.
4 4 Add 4 to each side.
5x ≥ 15 Simplify.
5x
5
≥
15
5
Divide each side by 5.
x ≥ 3 Simplify.
The solution is x ≥ 3.
3 2 101234567
x ≥ 3
Check:
x
0 is
not
a solution.
Check:
x
4 is a solution.
b. Solve
y
6
7
<
9. Graph the solution.
y
6
7
<
9 Write the inequality.
7 7 Subtract 7 from each side.
y
6
<
2 Simplify.
6
y
6
>
6
2
Multiply each side by 6. Reverse the
inequality symbol.
y
>
12 Simplify.
The solution is y
>
12.
24681012141618 0 2
y > 12
Solve the inequality. Graph the solution.
1. 4b 1
<
7 2. 8 9c ≥ 28 3.
n
2
11
>
12
Step 1: Undo the subtraction.
Step 2: Undo the multiplication.
Exercises 5–10
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Section 14.4 Solving Multi-Step Inequalities 115
EXAMPLE
Standardized Test Practice
2
Which graph represents the solution of 7(x 3) ≤ 28?
7(x 3) ≤ 28
Write the inequality.
7x 21 ≤ 28 Use Distributive Property.
21 21 Add 21 to each side.
7x ≤ 49 Simplify.
7x
7
≥
49
7
Divide each side by 7. Reverse the
inequality symbol.
x ≥ 7 Simplify.
The correct answer is B .
EXAMPLE
Real-Life Application
3
You need a mean score of at least 90 to advance to the next round of
the trivia game. What score do you need on the fi fth game to advance?
Use the defi nition of mean to write and solve an inequality. Let x be the
score on the fi fth game.
95 91 77 89 x

5
≥ 90
352 x
5
≥ 90 Simplify.
5
352 x
5
≥ 5
90 Multiply each side by 5.
352 x ≥ 450 Simplify.
352 352 Subtract 352 from each side.
x ≥ 98 Simplify.
You need at least 98 points to advance to the next round.
Solve the inequality. Graph the solution.
4. 2(k 5)
<
6 5. 4(n 10)
<
32 6. 3 ≤ 0.5(8 y)
7. WHAT IF? In Example 3, you need a mean score of at least 88 to
advance to the next round of the trivia game. What score do you
need on the fi fth game to advance?
Remember
The mean in Example
3 is equal to the sum
of the game scores
divided by the number
of games.
Exercises 12–17
The phrase “at least” means
greater than or equal to.
A
10 9 8 7 6 5 4
B
10 9 8 7 6 5 4
C
45678910
D
45678910
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Exercises
14.4
9
+(-6)=3
3
+(-3)=
4
+(-9)=
9
+(-1)=
116 Chapter 14 Linear Inequalities
1. WRITING Compare and contrast solving multi-step inequalities and solving
multi-step equations.
2. OPEN-ENDED Describe how to solve the inequality 3(a 5)
<
9.
3. For what values of k will the 4. For what values of h will the
perimeter of the octagon be surface area of the solid be
less than or equal to 64 units? greater than 46 square units?
k
k
k
1
2
k
1
2
4
4
4
4
5
3
h
Solve the inequality. Graph the solution.
5. 7b 4 ≥ 11 6. 2v 4
<
8 7. 1
m
3
≤ 6
8.
4
5
<
3w
11
5
9. 1.8
<
0.5 1.3p 10. 2.4r 9.6 ≥ 4.8
11. ERROR ANALYSIS Describe and correct
the error in solving the inequality.
Solve the inequality. Graph the solution.
12. 6( g 2) ≤ 18 13. 2( y 5) ≤ 16 14. 10 ≥
5
3
(h 3)
15.
1
3
(u 2)
>
5 16. 2.7
>
0.9(n 1.7) 17. 10
>
2.5(z 3.1)
18. ATM Write and solve an inequality that
represents the number of $20 bills you can
withdraw from the account without going
below the minimum balance.
1
2
x
4
6 ≥ 3
x
6 ≥ 1 2
x ≥
6
✗
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74 ft
8 ft
Section 14.4 Solving Multi-Step Inequalities 117
Find the area of the circle.
26.
10 mm
27.
25 in.
28.
66 m
29. MULTIPLE CHOICE What is the volume of the cube?
A 8 ft
3
B 16 ft
3
C 24 ft
3
D 32 ft
3
Solve the inequality. Graph the solution.
19. 5x 2x 7 ≤ 15 10 20. 7b 12b 1.4
>
8.4 22
21. TYPING One line of text on a page uses about
3
16
of an inch.
There are 1-inch margins at the top and bottom of a page.
Write and solve an inequality to fi nd the number of lines
that can be typed on a page that is 11 inches long.
22. WOODWORKING A woodworker builds a cabinet
in 20 hours. The cabinet is sold at a store for $500.
Write and solve an inequality that represents the
hourly wage the store can pay the woodworker
and still make a profi t of at least $100.
23. FIRE TRUCK The height of one
story of a building is about 10 feet.
The bottom of the ladder on the
fi re truck must be at least 24 feet
away from the building. Write and
solve an inequality to fi nd the
number of stories the ladder
can reach.
24. DRIVE-IN A drive-in movie theater charges $3.50 per car.
The drive-in has already admitted 100 cars. Write and
solve an inequality to fi nd the number of cars the drive-in
needs to admit to make at least $500.
25.
For what values of r will the area of the shaded
region be greater than or equal to 9(
2)?
r
2 ft
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Quiz
14.3 – 14.4
Solve the inequality. Graph the solution.
1. x ÷ 4
>
12 2.
n
—
−6
≥ −2
3. −4y ≥ 60 4. −2.3 ≥
p
—
5
Write the word sentence as an inequality. Then solve the inequality.
5. The quotient of a number and 6 is more than 9.
6. Five times a number is at most −10.
Solve the inequality. Graph the solution.
7. 2m + 1 ≥ 7 8.
n
—
6
− 8 ≤ 2
9. 2 −
j
—
5
>
7 10.
5
—
4
>
−3w −
7
—
4
11. FLOWERS A soccer team needs to raise $200 for new uniforms.
The team earns $0.50 for each fl ower sold. Write and solve an
inequality to fi nd the number of fl owers it must sell to meet or
exceed its fundraising goal.
12. PARTY You buy lunch for guests at a party. You can
spend no more than $100. You will spend $20 on beverages
and $10 per guest on sandwiches. Write and solve an
inequality to fi nd the number of guests you can invite to
the party.
13. BOOKS You have a gift card worth $50. You want to buy several
paperback books that cost $6 each. Write and solve an inequality
to fi nd the number of books you can buy and still have at least $20
on the gift card.
14. GARDEN The area of the triangular garden must
be less than 35 square feet. Write and solve an
inequality that represents the value of b.
118 Chapter 14 Linear Inequalities
10 ft
b
ms.
a
n
t or
eral
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Chapter Test 119
Chapter Test
14
Write the word sentence as an inequality.
1. A number j plus 20.5 is greater than or equal to 50.
2. A number r multiplied by
1
—
7
is less than −14.
Tell whether the given value is a solution of the inequality.
3. v − 2 ≤ 7; v = 9 4.
3
—
10
p
<
0; p = 10 5. −3n ≥ 6; n = −3
Solve the inequality. Graph the solution.
6. n − 3
>
−3 7. x −
7
—
8
≤
9
—
8
8. −6b ≥ −30
9.
y
—
−4
≥ 13 10. 3v − 7 ≥ −13.3 11. −5(t + 11)
<
−60
12. VOTING U.S. citizens must be at least 18 years of age on Election Day
to vote. Write an inequality that represents this situation.
13. GARAGE The vertical clearance for a hotel
parking garage is 10 feet. Write and solve an
inequality that represents the height (in feet)
of the vehicle.
14. LUNCH BILL A lunch bill, including tax, is
divided equally among you and fi ve friends.
Everyone pays less than $8.75. Write and solve
an inequality that describes the total amount
of the bill.
15. TRADING CARDS You have $25 to buy trading cards
online. Each pack of cards costs $4.50. Shipping costs
$2.95. Write and solve an inequality to fi nd the
number of packs of trading cards you can buy.
16. SCIENCE QUIZZES The table shows your scores on four science quizzes.
What score do you need on the fi fth quiz to have a mean score of at least 80?
Test
1234
5
Score (%) 76 87 73 72
?
15 in.
h
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