Name
Class
Date
Reteaching
3-1
Inequalities and Their Graphs
You use the following symbols for inequalities.
> is greater than
≥ is greater than or equal to
< is less than
≤ is less than or equal to
Problem
What inequality represents “5 plus a number y is less than –10”?
5 plus a number y
5+y
is less than
–10
<
–10
The inequality 5 + y < –10 represents the phrase.
Exercises
Write an inequality that represents each verbal expression.
1. p is greater than or equal to 5
2. a is less than or equal to –4
3. 2 times d is less than 10
4. r divided by 5 is greater than 0
Problem
Is –2 a solution of 3t + 10 ≥ 5?
3t + 10 ≥ 5
?
3(–2) + 10  5
?
Original inequality
Substitute –2 for t.
–6 + 10  5
Simplify.
4
–2 is not a solution.
5
Exercises
Determine whether each number is a solution of the given inequality.
5. 5b – 7 > 13
a. –4
b. 4
c. 8
6. 2(m + 1) < –6
a. –6
b. –4
c. –2
a. 6
b. 8
c. 10
7.
8h
2
8
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Reteaching (continued)
Inequalities and Their Graphs
When graphing an inequality on a number line, an open circle means the number
is not included in the inequality. A closed circle means the number is included in
the inequality.
Problem
What is the graph of w ≥ –1?
Since w is greater than or equal to –1, place a closed circle at –1.
Draw a dark line with an arrow to the right of the closed circle to show the
numbers greater than – 1.
Exercises
Graph each inequality.
8. y ≤ 0
9. p > –4
10. a ≥ –2
Problem
What inequality represents the graph?
The circle is open so 4 is not included in the inequality.
The dark line and arrow are to the left indicating less than.
The graph represents “x is less than 4” or x < 4.
Exercises
Write an inequality for each graph.
11.
12.
13.
14.
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Reteaching
Solving Inequalities Using Addition or Subtraction
You can add the same number to each side of an equation. You can also add the
same number to each side of an inequality.
Problem
What are the solutions of b – 4 > –2? Graph and check the solutions.
Original inequality.
b – 4 > –2
b – 4 + 4 > –2 + 4
b>2
Add 4 to each side.
Simplify.
To graph b > 2, place an open circle at 2 and shade to the right.
To check the endpoint of b > 2, make sure that
2 is the solution of the related equation
b – 4 = –2.
b – 4 = –2
Then check to see if a number greater
than 2 is a solution of the inequality 5 is
greater than 2.
b – 4 > –2
?
?
2 – 4  –2
5–4
2=2
 –2
1 > –2 
Exercises
Solve each inequality. Graph and check your solutions.
1. m – 14 ≥ –10
2. t – 2 < 4
3. y – 3 ≤ 4
4. d – 9 ≥ –12
5. w – 17 > 13
6. a – 22 < –7
7. Writing Explain how you would solve t – 15 ≤ 5.
8. Anita is baking dinner rolls and pumpkin bread. She needs 4 cups of flour for
the rolls. She needs at least 7 cups of flour left for the pumpkin bread. Write
and solve an inequality to determine how much flour Anita needs before she
starts baking.
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Reteaching (continued)
Solving Inequalities Using Addition or Subtraction
You can subtract the same number from each side of an equation. You can also
subtract the same number from each side of an inequality.
Problem
What are the solutions of h + 7 ≤ 4? Graph and check the solutions.
h+7≤4
h+7–7≤4–7
h ≤ –3
Original inequality.
Subtract 7 from each side.
Simplify.
To graph h ≤ –3, place a closed circle at –3 and shade to the left.
To check the endpoint of h ≤ –3, make
sure that –3 is the solution of the related
equation h + 7 = 4.
Then check to see if a number less than
–3 is a solution of the inequality. –4 is
less than –3.
h7  4
h7  4
?
?
3  7  4
4  7  4
44
34 
Exercises
Solve each inequality. Graph and check your solutions.
9. s + 7 ≥ 12
10. p + 3 < –1
11. b + 5 ≤ –4
12. n + 1 ≥ 8
13. v + 18 > –12
14. k + 26 < 6
15. A boat can hold up to 1000 pounds. Two friends get in the boat. Together they
weigh 285 pounds. Write and solve an inequality to determine how much
more weight can be added to the boat.
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3-3
Solving Inequalities Using Multiplication or Division
You can solve inequalities using multiplication or division using these two
important rules.
• You can multiply or divide each side of an inequality by a positive number.
• You can multiply or divide each side of an inequality by a negative number
only if you reverse the inequality sign.
Problem
What are the solutions of
c
5
 2
c
5
 2 ? Graph the solutions.
Original inequality
 c
5    (2)
 5
Multiply each side by 5. Keep the inequality symbol the same.
c ≤ –10
Simplify.
To graph c ≤ –10, place a closed circle at –10 and shade to the left.
Problem
2
3
What are the solutions of  t  4 ? Graph the solutions.
2
 t4
3
3 2 
3
  – t   – (4)
2 3 
2
t < –6
Original inequality
Multiply each side by 
3
2
. Reverse the inequality symbol.
Simplify.
To graph t < –6, place an open circle at –6 and shade to the left.
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3-3
Solving Inequalities Using Multiplication or Division
Problem
What are the solutions of –6h ≤ –39? Graph the solutions.
6h  39
6h 39

6
6
1
h6
2
To graph h  6
1
2
Original inequality
Divide each side by –6. Reverse the inequality symbol.
Simplify.
, place closed circle at 6
1
2
and shade to the right.
Exercises
Solve each inequality. Graph and check your solutions.
1.
3.
x
7
2
5
2. 8p ≤ 32
 2
r6
4. 
5. –3f ≥ 12
k
2
 5
3
6. t  9
5
7. –2w > –8
9. 
3
4
d 
8. 
z
5
4
10. –4n ≥ 14
3
8
11. A bus company charges $2 for each trip. It also sells monthly passes for $50.
Write and solve an inequality to find how many trips you could make before
the monthly pass is cheaper.
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Reteaching
3-4
Solving Multi-Step Inequalities
Solving inequalities is similar to solving equations. However, if you multiply or
divide each side of an inequality by a negative number, the direction of the
inequality sign is reversed.
What are the solutions of 6 – 3k > 45?
6 – 3k > 45
Original inequality
6 – 3k – 6 > 45 – 6
Subtract 6 from each side.
–3k > 39
Simplify.
3k
Divide each side by –3 and reverse the sign.
3

39
3
Simplify.
k < –13
What are the solutions of 6(n – 3) + 4n ≤ 42?
Original inequality
6(n – 3) + 4n ≤ 42
6n – 18 + 4n ≤ 42
Distributive Property
10n – 18 ≤ 42
Combine like terms.
10n – 18 + 18 ≤ 42 + 18
Add 18 to each side.
10n ≤ 60
Simplify.
10n
Divide each side by 10.
10

60
10
n≤6
Simplify.
What are the solutions of 7p + 12 > 6p – 15?
7p + 12 > 6p – 15
7p + 12 – 6p > 6p – 15 – 6p
p + 12 > –15
Subtract 6p from each side.
Simplify.
p + 12 – 12 > –15 – 12
p > –27
Original inequality
Subtract 12 from each side.
Simplify.
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Reteaching (continued)
Solving Multi-step Inequalities
Exercises
Solve each inequality.
1. 8w + 9 < –31
2. 5h – 6 ≥ 24
3. 17 – 2a ≤ 29
4. 5 – 3t > –7
5.
d
7
 4  2
6. 4 
2x
3
 8
7. 5(y – 2) – 2y ≥ 5
8. 8(2f + 3) + 4f ≤ –16
9. 3(p – 2) – 7p < 6
10. 2(3b + 5) – 10b > 30
11. 7z – 4 ≤ 6z + 18
12. 8m + 7 ≥ 6m – 9
13. 12c + 6 > 9c – 15
14. 7d + 2 < 17 – 3d
15. A student had $45 when she went to the mall. She spent $9 on a pair of earrings.
Then she wants to buy some CDs that cost $12 each. Write and solve an
inequality to determine how many CDs she can buy.
16. A friend needs at least $125 to go on the class trip. He has saved $45. He makes
$20 for each lawn he mows. Write and solve an inequality to determine how many
lawns he needs to mow to go on the trip.
17. You have earned 85, 92, 95, and 88 on tests this grading period. You have one last
test and want an average of at least 90. Write and solve an inequality to determine
what scores you can earn to achieve your goal.
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Reteaching
Compound Inequalities
A compound inequality with the word or means one or both inequalities must be true.
The graph of the compound inequality a < –4 or a ≥ 3 is shown below.
A compound inequality with the word and means both inequalities must be true. The
graph of the compound inequality b ≤ 4 and b > –1 is shown below.
To solve a compound inequality, solve the simple inequalities from which it is
made.
Problem
What are the solutions of 17 ≤ 2x + 7 ≤ 29? Graph the solutions.
17 ≤ 2x + 7 ≤ 29 is the same as 17 ≤ 2x + 7 and 2x + 7 ≤ 29. You can solve it as two
inequalities.
17 ≤ 2x + 7
and
2x + 7 ≤ 29
17 – 7 ≤ 2x + 7 – 7
and
2x + 7 – 7 ≤ 29 – 7
10 ≤ 2x
10 2 x

2
2
5≤x
and
2x ≤ 22
22

2
2
x ≤ 11
2x
and
and
To graph the compound inequality, place closed circles at 5 and 11. Shade
between the two circles.
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3-6
Compound Inequalities
Problem
What are the solutions of 3t – 5 < –8 or 2t + 5 > 17? Graph the solutions.
Solve each inequality.
3t – 5 < –8
or
2t + 5 > 17
3t – 5 + 5 < –8 + 5
or
2t + 5 – 5 > 17 – 5
3t < –3
or
2t > 12
3t 3

3 3
or
2t 2

2
2
t < –1
or
t>6
To graph the compound inequality, place open circles at –1 and at 6. Shade to the left
of –1 and to the right of 6.
Exercises
Solve each compound inequality. Graph the solutions.
1. h – 7 ≥ –5 and h + 4 < 10
2. r – 2 ≤ –1 or r – 3 > 2
3. –7 < w – 4 < 2
4. 2 
5. 5p + 3 ≤ –2 or 3p – 6 ≥ 3
6. –2n – 5 ≥ 1 or 5n + 7 > 2
7.
3
2
a Ğ6  0 and a  4  2
4
3
y
1
2
8. –4 ≤ 4d + 24 ≤ 4
9. 5m – 2 < 8 or 6m – 2 > 6 + 5m
10.
w
 1  2 and w  5  1
2
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Extra Practice
Chapter 3
Lessons 3-1 to 3-4
Solve each inequality. Graph and check your solution.
1.
–8w < 24
2. 9 + p ≤ 17
3. r
4
 1
4.
7y + 2  28
5. t – 5 ≥ –13
6. 9h > –108
7.
8w + 7 > 5
8. s
9. 6c
 12
5
10
–8ℓ + 3.7  31.7
11. 9 – t ≤ 4
12. m + 4 ≥ 8
13
y + 3 < 16
14. n – 6 ≤ 8.5
15. 12b – 5 > –29
16
4 – a > 15
17. 4 – x ≤ 3
18. 1 – 4d ≥ 4 – d
20. s
21.
6
19. n + 7  3n – 1
2
22
8r 
r 1
 8
6 6
25. 2(m – 5) + 4m ≤ 56
3
1  s  2
3
2x
5
3
23. 1.4 + 2.4x < 0.6
24. x – 2 < 3x – 4
26. 6(c + 3) – 9 ≥ 27
27. –3(2t – 1) + 5t > 7
Define a variable and write an inequality for each situation.
28. A car dealership sells at least 35 cars each week.
29. No more than 425 tickets to a musical will be sold.
30. You must be at least 18 years old to vote.
31. The party store sold more than 720 balloons in July.
32. The booster club raised $102 in their car wash. They want to buy $18 soccer balls for
the soccer team. Write and solve an inequality to find how many soccer balls they
can buy.
33. You earn $7.50 per hour and need to earn $35. Write and solve an inequality to find
how many hours you must work.
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Extra Practice (continued)
Chapter 3
Write and solve an inequality for each situation.
34. Suppose you are trying to increase your coin collection to at least 500 coins. How
many more coins do you need if you already have a collection of 375 coins?
35. Janet has a balance of $125 on a credit card. On her next statement, she wants to reduce
her balance to no more than $60. How much does she need to pay off ?
36. A homeroom class with 25 students is holding a fund-raiser to support school sports.
Their goal is to raise at least $200. On average, how much money does each student
need to contribute to meet or exceed the goal?
37. You are reading a book with 19 chapters. How many chapters should you read
each week if you want to finish the book in 5 weeks or less?
38. The sophomore class is putting on a variety show to raise money. It costs $700 to rent
the banquet hall they are going to use. If they charge $15 for each ticket, how many
tickets do they need to sell in order to raise at least $1000?
39. A technical-support company charges $10 per month plus $35 per hour of phone
support. If you need to spend less than $100 per month on support, how many
hours can you get?
Lesson 3-6
Solve each compound inequality.
51. 8 < w + 3 < 10
52. –6 < t – 1 < 6
53. 6m – 15 ≤ 9 or 10m > 84
54. 9j – 5j ≥ 20 and 8j > –36
55. 37 < 3c + 7 < 43
56. 3 < 5 + 6h < 10
57. 1+ t < 4 < 2 + t
58. 2 + 3w < –1 < 3w + 5
59. 2x – 3 ≤ x and 2x + 1 ≥ x + 3
60. 3n – 7 > n + 1 or 4n – 5 < 3n – 3
Write each interval as an inequality. Then graph the solutions.
61. (–∞,5)
62. [2,9)
63. (–∞, 1] or [6, ∞)
Write a compound inequality for each situation. Graph your solution.
64. Water will not be in liquid form when it is colder than 32°F or warmer than 212°F.
65. The width of a parking space needs to be at least 8 feet and no more than 11 feet.
66. A car salesman has been told to sell a particular car for more than $14,500 and up to
the sticker price of $15,755.
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