COMPACTED MATHEMATICS
CHAPTER 4B
INEQUALITIES
TOPICS COVERED:



Solving inequalities with adding and subtracting
Solving inequalities with multiplying and dividing
Solving two-step inequalities
Activity 4-45: Describing the Ideal School with Inequalities
Name:
(Taken from The Language of Algebra)
These variables represent information about a particular school.
t = number of teachers at the school
m = number of math teachers at the school
b = the number of boys at the school
g = the number of girls at the school
p = number of class periods in one day
l = length of one class period, in minutes
1.
Add six more variables to the list above.
Using the variables above write at least five inequalities that describe the ideal school. Keep a record of
what the inequalities represent.
b  g  1,500
Ex.
There are less than or equal to 1,500 students.
2.
3.
4.
5.
6.
Suppose that g represents the number of girls in a particular class and b represents the number of boys.
Write each inequality using symbols. Then give the range for the number of girls if the number of boys
is 20.
The number of girls is less than or equal to
7.
half the number of boys.
The number of girls is less than 2 more than
8.
the number of boys.
The number of boys is at least 3 times the
9.
number of girls.
Write three inequalities of your own using variables g and b. Write each one using symbols and words.
Use each of the four operations at least once.
10.
11.
12.
Activity 4-46: Inequalities
Name:
The prefix “in” means not, therefore an inequality is something that is not equal. The following symbols
are often used with inequalities:

greater than

greater than or equal to

not equal

less than

less than or equal to
State whether each inequality is true or false for the given value. Show all work.
1.
b  10  12, b  4
2.
3  x  8, x  12
3.
6m  3  8, m  1
4.
12  2 p  6, p  9
5.
k  12  18, k  31
6.
13  4  c, c  9
7.
15  n  15, n  6
8.
2  t  3, t  3
9.
4t  4  20, t  7
10.
29  24  a, a  6
11.
10  2a  4, a  4
12.
5v  25, v  4
13.
r
21  , r  66
3
14.
s
 4, s  32
8
15.
5w  8  12, w  0
16.
2 y  7  41, y  16
17.
3z  z  6  11, z  4
18.
5 f  2 f  3  9, f  2
19.
6h  3  15, h  2
20.
81  3d  90, d  2
Activity 4-47: Inequalities
Name:
Evaluate each expression if a=2, b=4, and c=6. Then write >, <, or = to make each sentence true.
21.
bc
23.
5b  2a
25.
4c  5b
22.
c6
4b
24.
3c
ba
26.
ac
27. Applicants with less than 5 years’ experience must take a test.
28. The home team needs more than 6 points to win.
29. The minimum voting age is 18.
30. You must answer at least 10 questions correctly to stay in the game.
32. The cost including tax is no more than $75.
33. The maximum load for an elevator is 2900 pounds.
34. A car can seat up to 8 passengers.
35. No persons under the age of 18 are permitted.
36. You must be at least 13 years old to join.
37. You should not drive over the speed limit of 70 mi/h.
2b  4a  2
5c  3b  a 16
Write an inequality for each sentence.
31. A tip of no less than 10% is considered appropriate.
3a  2c
0
Activity 4-48: Addition and Subtraction with Inequalities
Name:
Write an inequality for each solution set graphed below.
1.
2.
3.
4.
5.
6.
7.
8.
5
4
3
2
1
0
1
2
3
4
5
5
4
3
2
1
0
1
2
3
4
5
5
4
3
2
1
0
1
2
3
4
5
5
4
3
2
1
0
1
2
3
4
5
5
4
3
2
1
0
1
2
3
4
5
5
4
3
2
1
0
1
2
3
4
5
5
4
3
2
1
0
1
2
3
4
5
5
4
3
2
1
0
1
2
3
4
5
Interpret the graph. This graph shows the range of
temperature in degrees Fahrenheit during a day in February.
9.
24
34
Solve each inequality showing all work on a separate sheet of paper. Check your solution. Then
graph the solution on the number line.
x  3 1
x  8  6
9.
10.
5 4 3 2
1
1  2  3  4  5
0
5 4 3 2
x  21  25
0
1  2  3  4  5
1
0
1  2  3  4  5
1
0
1  2  3  4  5
12  x  16
11.
12.
5 4 3 2
1
0
1  2  3  4  5
5 4 3 2
3  x  4
13.
1
5 4 3 2
1
0
1  2  3  4  5
14.
1
1
x 1  2
2
2
5 4 3 2
x  7  11
x  6  6
15.
16.
5 4 3 2
1
0
1  2  3  4  5
5 4 3 2
1
0
1  2  3  4  5
Activity 4-49: Multiplying and Dividing with Inequalities
Name:
Solve each inequality showing all work on a separate sheet of paper. Check your solution. Then
graph the solution on the number line.
5x  25
4x  8
1.
2.
5 4 3 2
1
1  2  3  4  5
0
5 4 3 2
b
2
2
1
0
1  2  3  4  5
1
0
1  2  3  4  5
1
0
1  2  3  4  5
1
0
1  2  3  4  5
x
1

18 18
3.
4.
5 4 3 2
1
1  2  3  4  5
0
5 4 3 2
3x  3
2x  4
5.
6.
5 4 3 2
1
0
1  2  3  4  5
5 4 3 2
6x  0
c
 1
3
7.
8.
5 4 3 2
1
0
5 4 3 2
1  2  3  4  5
4x  16
w
 5
1
9.
10.
5 4 3 2
1
0
1  2  3  4  5
5 4 3 2
1
m

4 4
2
11.
1
0
1  2  3  4  5
t
1
12.
5 4 3 2
1
0
1  2  3  4  5
5 4 3 2
3x  6
1
0
1  2  3  4  5
1
0
1  2  3  4  5
1
0
1  2  3  4  5
1
n

8 32
13.
14.
5 4 3 2
1
0
1  2  3  4  5
5 4 3 2
1
x2
2
x
1

12 4
15.
16.
5 4 3 2
1
0
1  2  3  4  5
5 4 3 2
Activity 4-50: Why does the sign change?
Name:
Take the following example:
-3x + 5 > 17
+3x
+3x
5 > 17 + 3x
-17
-17
-12 > 3x
3
From this we see that x must be less than -4.
3
-4 > x
Therefore when we solve it the fastest way:
-3x + 5 > 17
-5
-5
-3x > 12
-3
-3
x < -4
The sign must switch in order to match our answer above.
Another way to think about it is as a balanced scale
2=2
2
2
A
an inequality can be thought of as an unbalanced scale:
2>1
1
2
A
Multiplying it by -1 is like reversing gravity, so that things fall up, and the heavier object is now pulled
more strongly up rather than down. If you do that, the scale will reverse:
-2 < -1
2
A
1
Activity 4-51: Solving Inequalities & Equations
Name:
Solve each inequality or equation showing all work on a separate sheet of paper. Check your
solution.
b
 5
1.
2.
8a  5.68
4.2
d
  4.7
3.
4.
27.44  4.9c
4
5.
12.6  3n
6.
5 f  35
5
125   m
8
t

 3.01
8.7
k
 40
1.5
8.
2.3n  0.805
10.
8.4r  5.88
12.
20.4  3.4d
13.
18.24  6x
14.

15.
1.27 y  0.0381
16.
17.
0.16s  9.6
18.
19.
3.4 j  0.816
20.
21.
48.374  1.34w
22.
0.3u  2.73
24.
0.025x  5.25
26.
8
 k  0.16
9
7.
9.
11.
23.
25.
3
11
 v  1
4
16
z
  7.6
50
z
 100
2.1
r
 3
0.5
2
8
 t
3
9
r
 0.5
7.4
Solving Two-Step Inequalities
Solve 3 y  12  78 .
We are solving for y so we need to get y on one side of the equation all by itself.
We must undo both the multiplication and the subtraction. Since we are undoing each operation, we
work backwards through the order of operations: add first, then divide.
Example:
3 y  12  78
12 =  12
3 y  66
3  3
y  22
Activity 4-52: Solving Two-Step Inequalities
Name:
Reminder: When multiplying or dividing by a negative, reverse
the order of the inequality.
Use the distributive property to simplify each expression.
2( w  2)
1.
2.
6( x  10)
3.
5(a  16)
4.
(c  9)5
5.
(t  8)(3)
6.
(r  12)
Solve each inequality showing all work on a separate sheet of paper. Check your solution.
7  6 y  55
7.
8.
2a  5  13
9.
5x  9  21
10.
10 g  15  95
11.
9  3z  72
12.
3x  5  x  27
13.
5( w  10)  105
14.
2(k  6)  38
15.
6t 1  8t  15
16.
a
 7  11
3
17.
n
  12  20
2
18.
5 g  ( g  1)  19
Write an inequality for each problem and solve showing all work on a separate sheet of paper.
Wade wants to buy two shirts and a tie and must spend no more than $60.00. The tie he likes
19.
costs $12.50. If the two shirts are the same price, how much can he pay for each shirt?
In a new school auditorium, folding chairs 16 in. wide are to be installed side by side to form a
20.
row that is no longer than 25 ft. How many chairs will fit into this row?
Patio blocks are packaged in bundles containing enough blocks to cover 16 ft.2 Sophie wants to
21. build a patio having dimensions 18 ft by 13 ft. What is the least number of bundles of blocks
she should order?
22. An integer k divided by -3 is greater than 1. What is the greatest possible value of the integer?
23. The sum of three consecutive even integers is less than -25. Find the 3 largest possible integers.
Mr. Mangham started with $300 in his savings account. Each month he has to pay for his voice
24. lesson, which is a $40 monthly fee. With x equal to the number of months of lessons, write an
equation to represent the balance in Mr. Mangham’s savings account each month.
In the problem above, the bank will close his account if the balance drops below $50. To keep
25. the account open, the balance must be greater than or equal to $50. Write an inequality to
represent this situation.
The Golden Rule of Inequalities
Whenever you multiply or divide both sides of an inequality by a negative number,
you must flip the inequality symbol.
1.
2.
3.
4.
Isolate the variable by solving the inequality.
Check the order. Variable on left side if preferred.
Circle the number on the number line. Open or closed circle?
Shade appropriately.
Open Circle
, , 
Example: Solve and Graph
5  3x  13  x
Closed Circle
, , 
The Golden Rule of Inequalities
Whenever you multiply or divide both sides of an inequality by a negative number,
you must flip the inequality symbol.
1.
2.
3.
4.
Isolate the variable by solving the inequality.
Check the order. Variable on left side if preferred.
Circle the number on the number line. Open or closed circle?
Shade appropriately.
Open Circle
, , 
Closed Circle
, , 
Example: Solve and Graph
5  3x  13  x
Inequalities 1
Name:
An inequality statement can be read from left to right or from right to left. We usually use the variable
as our starting point.
Example: x  5 can be read “x is less than 5” or “5 is greater than x”.
Example: 3  y can be read “y is less than or equal to 3 ” or “ 3 is greater than or equal to y”.
1.
2.
3.
4.
5.
Which inequality symbols would be used for
word problems with the following?
A. At most
B. At least
C. Cannot exceed
D. Must exceed
E. Less than
F. No less than
G. More than
H. No more than
Write an inequality that represents the statement: To get a driver’s license, a driver,
d, should be at least 16 years old.
The inequality 4  5 is a true number sentence. Will it remain true if you subtract
7 from both sides? Explain.
Write an inequality that represents the statement: According to the forecast the
temperature, t, will not exceed 82 degrees.
The inequality 4  5 is a true number sentence. Will it remain true if you multiply
both sides by 7 ? Explain.
6.
Solve: 2 x  12
7.
Solve: 2x  12
Solve and graph each inequality. Select one value from the shaded portion of the number line and verify
that it is a solution to the inequality.
Problem A
Problem B
3x  9  18
6w1  13
Problem C
Problem D
1
0.3 p  2.1  3.6
2 b 5
2
8.
How are problem A and problem D similar?
9.
How are problem B and problem C similar?
10. How are problem A and problem B different from problem C and problem D?
Inequalities 2
Name:
1. Solve 0.5x  2  5.5 and graph the solution on a number line.
2. Solve 1.2x  7.8  9.6 and graph the solution on a number line.
3. Solve 2.6x  4.7  12.5 and graph the solution on a number line.
4. Solve 2.2x  5.85  13 and graph the solution on a number line.
5. Solve 0.4x  4  2.4 and graph the solution on a number line.
6.
Mrs. Smith wrote “Eight less than three times a number is greater than fifteen”
on the board. If x represents the number, which inequality is a correct
translation of this statement?
3x  8  15
3x  8  15
8  3x  15
8  3x  15
The sign shown below is posted in front of a rollercoaster ride at Six Flags. If h
represents the height of a rider in inches, what is the correct translation of the
statement on this sign?
7.
Inequalities 3
1.
2.
3.
4.
Which value of x is in the solution set of the inequality 2x  5  17 ?
8
6
 4 12
Which value of x is in the solution set of the inequality 4x  2  10 ?
2 2 3
4
4
Which value of x is in the solution set of the inequality x  5  17 ?
3
8 9 12 16
Which value of x is in the solution set of the inequality 2( x  5)  4 ?
0 2 3 5
5.
What is the solution of 3(2m  1)  4m  7 ?
6.
What is the solution of 6x 17  8x  25 ?
7.
8.
9.
Roger is having a picnic for 78 guests. He plans to serve each guest at least one
hot dog. If each package, p, contains eight hot dogs, which inequality could be
used to determine how many packages of hot dogs Roger will need to buy?
p  78
8 p  78
8  p  78
78  p  8
The ninth grade class at a local high school needs to purchase a park permit for
$250 for their upcoming class picnic. Each ninth grader attending the picnic pays
$0.75. Each guest pays $1.25. If 200 ninth graders attend the picnic, which
inequality can be used to determine the number of guests, x, needed to cover the
cost pf the permit?
0.75 x  (1.25)(200)  250.00
0.75 x  (1.25)(200)  250.00
(0.75)(200)  1.25 x  250.00
(0.75)(200)  1.25 x  250.00
Solve the inequality: 5n  25
10. Solve the inequality: 42  6d
r
3
6
y
 10
12. Solve the inequality:
6
1
13. Solve the inequality: n  8
4
Write an inequality for the graph.
11. Solve the inequality:
14.
Name:
Inequalities 4
1.
Name:
A road has a speed limit of 30 mph. Write an inequality that describes the legal
speeds for motor vehicles.
Write an inequality for the sentence: c is not less than zero. Graph the solution on a number line.
2.
Solve the inequality: a  4  8 Graph the solution.
3.
Solve the inequality: q  7  0 Graph the solution.
4.
Solve the inequality: h  2  1 Graph the solution.
5.
Together Louisa and Jill scored more than 30 points in the basketball game. If Jill
6. scored 12 points, how many points did Louisa score?
p  18
p  42
p  42
p  185
Four times the sum of a number and 15 is at least 120. Let x represent the number.
7. Find all possible values for x.
x  26 x  15 x  15 x  26
Represent each of the following as an algebraic inequality.
8.
1) x is at most 30
2) the sum of 5x and 2x is at least 14
3) the product of x and y is less than or equal to 4
4) 5 less than a number y is under 20
Inequalities 5
Name:
1.
The sum of twice a number and 5 is at most 15. What are the possible values
for the number?
2.
Two-thirds of a number plus 5 is greater than 12. Find the number.
In order to be admitted for a certain ride at an amusement park, a child must be
greater than or equal to 36 inches tall and less than 48 inches tall. Which graph
represents these conditions?
3.
Which statement can be modeled by x + 3  12?
4.
A. Sam has 3 bottles of water. Together, Sam and Dave have at most 12 bottles of
water.
B. Jennie sold 3 cookbooks. To earn a prize, Jennie must sell at least 12 cookbooks.
C. Peter has 2 baseball hats. Peter and his brothers have fewer than 12 baseball hats.
D. Kathy swam 3 laps in the pool this week. She must swim more than 12 laps.
Julia has $80. She wants to purchase a nail set for $16 and earrings. She spends
5. the rest of the money on earrings. Each pair of earrings costs $8. Write an
inequality for the number of pairs of earrings she can purchase.
Liam brought $28 to the arcade and played games that cost 75 cents each.
6. Laney brought $12 to the arcade and spent $4.25 on a slice of pizza and a Coke.
How many games can Liam play so that he leaves with less money than Laney?
Christina goes to the market with $50. She buys some papayas for $20 and
7. spends the rest of the money on bananas. Each banana cost $0.60. Write an
inequality for the number of bananas she can purchase.
Billy goes to the store. He has $90. He wants to purchase a leather jacket for
8. $45, a hat for $10, and the rest on jeans. Each pair of jeans costs $35. Write an
inequality for the number of jeans he can purchase.
Rebecca bought one goldfish ($32) and one starfish ($12). She spends the rest
9. of her money on guppy fish. She starts with $80. Each guppy costs $6. Write
and solve inequality for the number of guppies she can purchase.
Erin has $50. She wants to purchase a cell phone ($20) and spend the rest on
10. music CSs. Each music CD costs $8. Write and solve an inequality for the
number of music CDs she can purchase.
Inequalities 6
Name:
Tamara has a cell phone that charges $0.07 per minute plus a monthly fee of
$19.00. She budgets $29.50 per month for total cell phone expenses without
1.
taxes. What is the maximum number of minutes Tamara could use her phone
each month in order to stay within her budget?
An online music club has a one-time registration fee of $13.95 and charges
2. $0.49 to buy each song. If Emma has $50 to join the club and buy songs, what
is the maximum number of songs she can buy?
Peter begins his kindergarten year able to spell 10 words. He is going to learn
to spell 2 new words every day. Write an inequality that can be used to
3. determine how many days, d, it takes Peter to be able to spell at least 75 words.
Use this inequality to determine the minimum number of whole days it will
take for him to be able to spell at least 75 words.
Chelsea has $45 to spend at the fair. She spends $20 on admission and $15 on
snacks. She wants to play a game that costs $0.65 per game. Write an
4. inequality to find the maximum number of times, x, Chelsea can play the game.
Using this inequality, determine the maximum number of times she can play
the game.
Mandy and 2 friends bought some mechanical pencils at a special sale. They
divided some of the pencils equally among themselves and then gave 3 to
5. Mandy’s little brother. At that time they had 19 pencils left. Solve the
p
equation  3  19 to find the number of pencils p that they bought at the sale.
3
Melissa wants to spend no more than $300 on school clothes. She spends $75
on a coat and then wants to buy some sweaters that are on special for $10 each.
6.
Solve the inequality 75 10s  300 to find the greatest number of sweaters s
she can buy.
A small airplane can carry less than 1,050 pounds of luggage and mail. The
7. mail for the day weighs 490 pounds. If each passenger brings 70 pounds of
luggage, what is the greatest possible number of passengers that can be taken?
You rent a car and are offered 2 payment options. You can pay $25 a day plus
15 cents a mile or you can pay $10 a day plus 40 cents a mile. For what
8.
amount of daily miles will Option A be the cheaper plan? Write and solve an
inequality to determine the answer.
Keith and Michelle went out to dinner. The total cost of the meal, including the
9. tip, came to $53.70. If the combined tip came out to $9.60, and each friend
spent an equal amount, how much did each friend pay not including the tip?
Jason is saving up to buy a digital camera that costs $490. So far, he saved
$175. He would like to buy the camera 3 weeks from now. What is the equation
10.
used to represent how much he must save every week to have enough money to
purchase the camera?
Adrian works in New York City and makes $42 per hour. She works in an
11. office and must get her suit dry cleaned every day for $75. If she wants to make
more than $260 a day, at least how many hours must she work?
LESSON
13-1
Writing Inequalities
Practice and Problem Solving: A/B
Complete the graph for each inequality.
1. a  3
2. r  –2
Graph the solutions of each inequality. Check the solutions.
3. w  0
Check: ___________________________
4. b  4
Check: ___________________________
5. a  1.5
Check: ___________________________
Write an inequality that represents each phrase. Draw a graph
to represent the inequality.
6. the sum of 1 and x is less than 5
_____________________________________
7. 3 is less than y minus 2
_____________________________________
Write and graph an inequality to represent each situation.
8. The temperature today will be at least 10F.
_________________________
9. Ben wants to spend no more than $3.
_________________________
Write an inequality that matches the number line model.
10. _________________________
11. _________________________
LESSON
13-1
Writing Inequalities
Practice and Problem Solving: C
Circle the values that are solutions for each inequality.
1. a  2
3.5
2. r  2
1
0
4
1
4
3.5
1
0
4
1
4
Graph the solutions of each inequality. Check the solutions.
3. 4  y
Check: ___________________________
4. b  0.5
Check: ___________________________
5. a  1  3
Check: ___________________________
Write and graph an inequality to represent each situation. Then determine if
36 is a possible solution. Write yes or no.
6. The temperature today will be at least 35F. ________________________
7. Monica wants to spend no more than $35. _________________________
Write an inequality that matches the number line model. Then write
a situation that the inequality could represent.
8. _________________________
________________________________________________________________________________________
________________________________________________________________________________________
9. _________________________
________________________________________________________________________________________
________________________________________________________________________________________
LESSON
13-1
Writing Inequalities
Practice and Problem Solving: D
Complete the graph for each inequality. The first one is done for you.
1. a  2
2. r  1
Graph the solutions of each inequality. Check the solutions. The first one is
done for you.
3. m  2
0  2; this is true
Check: ___________________________
4. d  3
Check: ___________________________
5. s < 3
Check: ___________________________
Write an inequality that represents each phrase. Draw a graph to represent
the inequality. The first one is done for you.
6. x is less than 4
x4
_____________________________________
7. 1 is greater than y
_____________________________________
Write and graph an inequality to represent each situation. The first one is
done for you.
t0
8. Today’s temperature is greater than 0F. _________________________
9. Lyle paid more than $2 for lunch.
_________________________
LESSON
13-2
Model and Solve Addition and Subtraction Inequalities
Practice and Problem Solving: A/B
Solve each inequality. Graph and check the solution.
1. 9  n  2 _________________
2. x  8  10 _________________
3. m  18  36 _________________
4. 112  b  128 _________________
5. 83  p  183 _________________
6. c  93  17 _________________
Write an inequality to solve each problem.
7. DeShawn has $53. He needs at least $76 to buy the jacket he wants. How
much more money does he need for the jacket?
________________________________________________________________________________________
8. Kristen needs at least 500 points to earn the next trophy on her video game.
She has 418 points. How many points must she earn to win the trophy?
________________________________________________________________________________________
9. Abha is hosting a party at a place that can hold up to 125 people. Seventyeight people have said they are coming. How many more people can Abha
invite?
________________________________________________________________________________________
Solve.
10. Write a real-world problem that can be represented by the inequality
r  32  56. Solve the inequality.
________________________________________________________________________________________
________________________________________________________________________________________
LESSON
13-2
Model and Solve Addition and Subtraction Inequalities
Practice and Problem Solving: C
Write an inequality to solve each problem.
The table shows Meagan’s family budget for three months of the year.
June
July
August
Mortgage
$1210
$1210
$1210
Utilities
$403
$455
$461
Food
$680
$572
$905
Savings
$500
$500
$500
Entertainment
$214
$290
$614
Total
$3,007
$3,027
$3,690
1. Meagan had $12,000 to spend for the months of June – September. What is
the most Meagan could spend for September?
________________________________________________________________________________________
2. In June and July combined, Meagan spent less on food than she plans to in
August and September combined. What is the least that Megan could plan to
spend on food in September so that the total of August and September is
greater than that of July and August?
________________________________________________________________________________________
3. Meagan does not want to spend more on entertainment in June through
September than she does on utilities. If she spends $412
on utilities in September, what is the most she can spend on entertainment?
________________________________________________________________________________________
4. Meagan tries to spend less on food, savings, and entertainment than she does
on her mortgage and utilities. How much less could she have spent on
entertainment in August to meet that goal?
________________________________________________________________________________________
5. Use the table above to write a problem that can be solved using an inequality.
Then find the solution to your problem.
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
LESSON
13-2
Model and Solve Addition and Subtraction Inequalities
Practice and Problem Solving: D
Solve each inequality. Graph and check the solution. The first one is done
for you.
r3
1. r  1  2 _________________
2. m  3  6 _________________
3. x  4   1 _________________
4. k  2  5 _________________
5. a  7  2 _________________
6. h  9  3 _________________
7. v  8   16 _________________
8. 14  y   7 _________________
Write an inequality to solve each problem. The first one is done
for you.
9. A small car averages up to 29 more miles per gallon of gas than an SUV. If a
small car averages 44 miles per gallon, what is the average miles per gallon
for an SUV?
Write and solve an inequality. _____________________________________
44  29  s
15  s
Write the solution.
An SUV gets at least 15 miles per gallon.
________________________________________________________________________________________
10. Carlos is taking a car trip that is more than 240 miles. He has already driven
135 miles. How much farther does he have to go?
Write and solve an inequality. _____________________________________
Write the solution in words.
________________________________________________________________________________________
LESSON
13-3
Multiplication and Division Inequalities with Positive Numbers
Practice and Problem Solving: A/B
Solve each inequality. Graph and check the solution.
y
 8 _________________
5
1. 9 x  270 _________________
2.
3. b  4 _________________
4
4. 5c  125 _________________
Solve each inequality.
5. a  12 _________________
20
6.
r  2 _________________
13
7. 6b  720 _________________
8.
s  15 _________________
4.2
Write and solve an inequality for each problem.
9. It cost $660 to put on the school play. How many tickets must be sold at $6
apiece in order to make a profit?
________________________________________________________________________________________
10. Jorge’s soccer team is having its annual fund raiser. The team hopes to earn
at least three times as much as it did last year. Last year the team earned
$87. What is the team’s goal for this year?
________________________________________________________________________________________
11. Alicia earns $9.00 per hour working at a part-time job. She wants to earn more
than $180 this week. How many hours does Alicia have to work?
________________________________________________________________________________________
12. Marc wants to buy a set of 6 antique chairs for his dining room. He has decided
to spend no more than $360. How much can he spend per chair?
________________________________________________________________________________________
LESSON
13-3
Multiplication and Division Inequalities with Positive Numbers
Practice and Problem Solving: C
Solve each inequality.
1.
a  120 _________________
2.5
3. 6b  73.2 _________________
2. 13r  19.5 _________________
4.
s  15 _________________
42
Write and solve an inequality for each problem.
5. It cost $660 to put on the school play. Tickets sell for $6 each. The drama club
has already sold 25 tickets. How many more tickets must be sold at $6 apiece
in order to make a profit? Explain.
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
6. Jorge’s soccer team is having its annual fundraiser. The team will sell T-shirts
at $10 each. The team hopes to earn at least three times as much as it did
last year. Last year the team earned $80 selling hats for $5 each. To meet its
goal for this year, how many more T-shirts will the team have to sell than hats
it sold last year? Justify your answer.
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
7. Akiko earns $12.00 per hour babysitting. She also receives a weekly
allowance of $10 from her parents. She wants to make more than $190 this
week. How many hours does Akiko have to babysit?
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
8. Marc wants to buy a set of 6 leather chairs for his dining room. He has
decided to spend no more than $360. He receives coupons good for $20 off
each chair. How much can he spend on each chair? Explain.
________________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
LESSON
13-3
Multiplication and Division Inequalities with Positive Numbers
Practice and Problem Solving: D
Use the situation below to complete Exercises 1–3.
Alicia is buying melons. She buys 5 melons and spends more than $10.
How much did she spend on each melon?

Cost of a melon

Total cost of melons
1. Let x represent the cost of each melon. Write an inequality.
Number of melons


2. The model shows the inequality from
Exercise 1.
There are _____ long  tiles, so draw circles to
separate the one tiles into _____ groups.
How many units are in each group? _____
3. What values make the inequality you wrote in Exercise 1 true? _____
Graph the solution of the inequality.
Solve each inequality. Graph and check the solution.
4. 9 x  27 _________________
5. 5y  20 _________________
6. 4b  12 _________________
7. 7c  28 _________________
Solve each inequality. The first one is done for you.
a  24
8. a  12 _________________
2
9. 3r  5 _________________
10. 6b  24 _________________
11. s  14 _________________
2
LESSON
13-4
Multiplication and Division Inequalities with Rational Numbers
Practice and Problem Solving: A/B
Solve each inequality. Graph and check the solution.
y
 4 _________________
2
1. 3x  18 _________________
2. 
3. 11b  55 _________________
4.  c  1.5 _________________
2
Solve each inequality.
5.  x  1 _________________
5
20
6. 0.9  r _________________
7. 2b  3 _________________
8.  a  70 _________________
0.5
Solve each problem.
9. Melissa’s family is on vacation. They spend $125 per day on food
and entertainment. They have budgeted only up to $750 for food
and entertainment. How many days of vacation can they afford?
________________________________________________________________________________________
10. A marine biologist is doing research on a shellfish that lives 80 feet below sea
level. The marine biologist dives into the sea at a rate that
is no less than 5 feet per second. How long will it take the marine biologist to
reach the shellfish?
________________________________________________________________________________________
11. Kevin is playing a game at the state fair. For each ball he can toss
into a jar, he gets 2 points. To win a stuffed animal, he must
earn a score of 24 or less. How many times does he have
to toss a ball into a jar?
________________________________________________________________________________________
Multiplication and Division Inequalities with Rational Numbers
Practice and Problem Solving: C
LESSON
13-4
Solve each inequality.


1.  x 52  1  36 _________________
4


3. 8   42  16 a _________________


2.  1 18  32 y  2 _________________
9


4. 7  1 52  81 b _________________
8
Solve each inequality.
5. Guadalupe’s family is on a business trip. Guadalupe doesn’t want to spend
more than $125 per day on hotels and $85 per day for food and client
entertainment. Her total trip budget is less than $1,900. How many days can
she afford to go on this trip?
________________________________________________________________________________________
6. A diver is diving to reefs that are 480 feet below sea level. He starts at a diving
station 30 feet below sea level. From there, he dives into the sea at a rate that
is no less than 6 feet per second. How long will it take the diver to reach the
reef?
________________________________________________________________________________________
7. Cara is playing a game. For each dart she can toss onto a board, she gets
20 points. To win a stuffed animal, she must earn a score of 240 or less.
She gets 5 minutes to play the game. To win a stuffed animal, how many
seconds does she have to toss a dart onto the board?
________________________________________________________________________________________
8. Saira’s bank deducts $1.75 each time she withdraws money from
an out-of-state bank machine. The bank also charges a $10 maintenance fee
for each week that Saira’s bank balance is less than $500. Saira’s current
bank balance is $580.25. She is visiting her aunt in another state and wants to
withdraw $25 per week for expenses. For how many weeks can she do this
without being charged the maintenance fee?
________________________________________________________________________________________
Multiplication and Division Inequalities with Rational Numbers
Practice and Problem Solving: D
LESSON
13-4
Complete the table. The first one is done for you.
Inequality
Do this to each
side:
New
inequality
1.  1  3
Multiply by 2
1  6
2. 4  12
Divide by 4
1 3
3. 8  16
Divide by 8
1 2
4.  1  21
Multiply by 3
1  63
2
3
New inequality
is true or false?
false
If false, what can
you do to make
the inequality true?
reverse the
inequality symbol
Solve each inequality. Graph and check the solution. The first one
is done for you.
x  4
5. 4x  16 _________________
6. 
7. 5a  20 _________________
8.  b  1 _________________
2
y
 6 _________________
2
Solve each inequality. The first one is done for you.
a  36
9.  a  18 _________________
2
10. 4b  24 _________________
11. 7c  14 _________________
12.  d  9 _________________
3