Inequalities, Word Problems, Algebra 1

1. Word Problems
with Inequalities
Don’t panic! Remember to:
• Read the questions carefully.
• Define the variable.
• Write an inequality.
• Solve the inequality.
• Check that your answer is reasonable.
• Answer in a complete sentence.

2. Keywords for Inequalities

3. Example 1
Eight less than the product of -3 and a number is
greater than -26. Write and solve an inequality to
represent this relationship. Graph the solution set, and
check your answer.
Let x = the number
-3x – 8 > -26
+8 +8
-3x > -18
-3 -3
x < 6
4 5 6 7 8
CHECK @ x = 5
-3x – 8 > -26
-3(5) – 8 > -26
-15 – 8 > -26
Don’t forget to reverse the inequality sign -23 > -26
when dividing by a negative!

4. Example 2
The sum of two consecutive integers is at least
negative fifteen. What are the smallest values of
consecutive integers that will make this true?
Let x = the first integer
x + 1 = the next consecutive integer
x + (x + 1) ≥ -15
2x + 1 ≥ -15
- 1 - 1
2x ≥ -16
2 2
x ≥ -8
x + 1 ≥ -7
The smallest values of
consecutive integers that
will add to at least -15 are
-8 and -7.

5. Example 3
Connor went to the carnival with $22.50. He bought a hot dog
and a drink for $3.75, and he wanted to spend the rest of his
money on ride tickets which cost $1.25 each. What is the
maximum number of ride tickets that he can buy?
Let r = the number of ride tickets he can buy
cost of food + cost of rides ≤ $22.50
3.75 + 1.25r ≤ 22.50
-3.75 -3.75
1.25r ≤ 18.75
1.25 1.25
r ≤ 15 tickets
Connor can buy a maximum
of 15 ride tickets.

6. Example 4
Stan earned $7.55 per hour plus an additional $100 in
tips waiting tables on Saturday evening. He earned
$160 in all. To the nearest hour, what is the least
number of hours Stan would have to work to earn this
much money?
Let h = the number of hours Stan will have to work
tips + hourly wages ≥ $160
100 + 7.55h ≥ 160
-100 -100
7.55h ≥ 60
7.55 7.55
x ≥ 7.9 hours
To the nearest hour, Stan
would have to work at
least 8 hours to earn $160.

7. Example 5
Brenda has $500 in her bank account. Every week, she
withdraws $40 for expenses. Without making any
deposits, how many weeks can she withdraw this money
if she wants to maintain a balance of at least $200?
Let w = the number of weeks Brenda withdraws money
starting account balance – money withdrawn ≥ $200
500 – 40w ≥ 200
-500 -500
– 40w ≥ -300
– 40 – 40
w ≤ 7.5 weeks
Don’t forget to reverse the inequality
sign when dividing by a negative!
Brenda can withdraw $40
from the account for 7 full
weeks and still have at
least $200 in the account.

8. Sean earns $6.70 per hour working after school. He needs
at least $155 (≥ 155) for a stereo system. Write an
inequality that describes how many hours he must work to
reach his goal.
$6.70 h ≥ 155
Per means to multiply
To figure your pay check:
You multiply rate times hours worked
$6.70 $6.70
h  23.1343
So 23 hours
is not enough
So Sean must
Work 24 hours
To earn enough
Money.
h  24

9. Sean earns $6.85 per hour working after school. He needs
at least $310 (≥ 310) for a stereo system. Write an
inequality that describes how many hours he must work to
reach his goal.
$6.85 h ≥ 310
Per means to multiply
To figure your pay check:
You multiply rate times hours worked
$6.85 $6.85
h  45.25547
So 45 hours
is not enough
So Sean must
Work 46 hours
To earn enough
Money.
h  46

10. The width of a rectangle is 11 cm. Find all possible
values for the length of the rectangle if the perimeter is
at least 300 cm. ( ≥ 300 cm)
• 2L + 2W = p
2L  2(11)  300
2L  22  300
22 22
2L  278
2 2
L 139

11. The width of a rectangle is 31 cm. Find all possible
values for the length of the rectangle if the perimeter is
at least 696 cm. ( ≥ 696 cm)
• 2L + 2W = p
2 2(31) L  696
2L  62  696
62 62
2L  634
2 2
L  317

12. The perimeter of a square is to be between 11 and 76 feet,
inclusively. Find all possible values for the length of its sides.
• 4 equal sides
• Perimeter – add up
sides s
s s
s
P = s +s+s+s=4s
11 p  76
11 4s  76
4 4 4
2.75  s 19

13. The perimeter of a square is to be between 14 and 72 feet,
inclusively. Find all possible values for the length of its sides.
• 4 equal sides
• Perimeter – add up
sides s
s s
s
P = s +s+s+s=4s
14  p  72
14  4s  72
4 4 4
3.5  s 18

14. Three times the difference of a number and 12 is at most
87. Let x represent the number and find all possible
values for the number.
3( 12) x  87
3x 36 87 
36 36
3x 123 
3 3
x  41

15. Marlee rented a paddleboat at the Park for a fixed charge of
$2.50 plus $1.50 per hour. She wants to stay out on the water
as long as possible. How many hours can she use the boat
without spending more than $7.00?
$2.50 $1.50h  $7.00
$2.50  $2.50 
$1.50h  $4.50
$1.50 $1.50
h  3