WORD PROBLEMS ON LINEAR INEQUALITIES IN ONE VARIABLE

Problem 1 :

Brian wants to earn at least $20 this week to go to the fair. His father said he will pay him $9 for mowing the lawn and $2.75 an hour to weed the flower bed. Brian has decided to do both chores. What is the minimum number of hours he will need to weed to earn at least $20?

Solution :

He will get $9 for moving the lawn

Let x be the number of hours he is working to weed the flower bed.

9 + 2.75x ≥ 20

2.75x ≥ 20 - 9

2.75x ≥ 11

x  ≥ 11/2.75

x ≥ 4

So, minimum number of hours he has to work is 4.

Problem 2 :

An employee earns $2 for every magazine sold and a salary of $10 a week. How many magazines will the employee need to sell in order to earn at least $40 in one week?

Solution :

Let x be the number of magazines he sold.

Weekly salary = 10

10 + 2x ≥ 40

2x ≥ 40 - 10

2x ≥ 30

x ≥ 15

He has to sell minimum 15 magazines.

Problem 3 :

Juan spent $2.50 on apples and oranges. He bought 5 apples at $0.36 each. What is the most he spent on the oranges?

Solution :

Cost of each apple = 0.36 and number of apples he bought = 5

Cost of 5 apples = 5(0.36) ==> 1.8

Cost of apples and oranges = 2.50

Cost spent only for orange = 2.50 - 1.8

= 0.7

Amount spent for oranges = 0.7

Problem 4 :

The cashier in a movie box office sold 200 more adult admission tickets at $11.00 each than children’s admission tickets at $8.00 each. What is the minimum number of each type of ticket that the cashier had to sell for the total receipts to be at least $5000

Solution :

Let x be the number of children tickets sold.

x + 200 will be the number of adult tickets.

8 x + 11(x + 200) ≥ 5000

8x + 11x + 2200 ≥ 5000

19x ≥ 5000 - 2200

19x ≥ 3400

x ≥ 147.3

x = 148, x + 200 ==> 348 adult tickets.

So, minimum number of children ticket sold is 148 and 348 adult tickets.

Problem 5 :

You must have an average score of at least 80 to get a B- on your report card. You have scores of 61, 70, 99 and 70. What is the minimum score you must get on the last test to get a B-on your report card? 

Solution :

Average score must be at least 80

Average = sum of all quantities/total number of quantities

Let x be the 5th quantity.

(61 + 70 + 99 + 70 + x)/5 ≥ 80

300 + x ≥ 80(5)

300 + x ≥ 400

x ≥ 100

The minimum score is 100.

Problem 6 :

Elisa won 40 lollipops playing basketball at the school fair. She gave two to every student in her math class. She has at least 7 lollipops left.

a) Write an inequality to represent the situation. Be sure to define your variable.

b) Solve the inequality to find the maximum number of students in her class.

Solution :

a)  Let x be the number of students in her math class

40 - 2x ≥ 7

b)  

2x ≥ 40 - 7

2x ≥ 33

x ≥ 33/2

x ≥ 16.5

When x = 16

The maximum number of lollipops = 40 - 2(16)

= 40 - 32

= 8 (is greater than 7)

When x = 17

The maximum number of lollipops = 40 - 2(17)

= 40 - 34

= 6 (is not greater than 7)

So, maximum number of students in the class is 16.

Problem 7 :

More than 450 students went on a field trip. Ten buses were filled and 5 more students traveled in a car.

a) Write an inequality to represent the situation. Be sure to define your variable.

b) Solve the inequality to find the minimum number of people on each bus.

Solution :

Let x be the number of people filled in each bus.

Ten buses were filled. So, number of people filled in 10 buses = 10x

10x + 5 ≥ 450

b)  10x + 5 ≥ 450

Subtracting 5 on both sides.

10x ≥ 450 - 5

10x ≥ 445

x ≥ 445/10

x ≥ 44.5

When x = 45

10x + 5 ==> 10(45) + 5 ==> 455

450 + 5 (5 more students were travelling)

Problem 8 :

Bill spent less than $26 on a magazine and five composition books. The magazine cost $4.

a) Write an inequality to represent the situation. Be sure to define your variable.

b) Solve the inequality to find the maximum cost of each composition book.

Solution :

Let x be the number of composition books.

Cost of a magazine book + cost of x composition books = 26

a)  4 + 5x < 26

b)

5x < 26 - 4

5x < 22

x < 22/5

x < 4.4

When x = 4

4 + 5(4) < 26

4 + 20 < 26

24 < 26

True

The maximum cost for each composition book is $4.