PROBLEMS ON LINEAR INEQUALITIES IN ONE VARIABLE WORD PROBLEMS

Problem 1 :

Amanda rented a bike from Shawna's Bikes. They charged her $2 per hour, plus a $10 fee. Amanda paid less than $27.

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

b) Solve the inequality to find the maximum number of hours Amanda rented the bike.

Solution :

Fee = $10

Charge per hour = $2

Let x be the number of hours rented.

2x + 10 < 27

2x < 27 - 10

2x < 17

x < 17/2

x < 8.5

So, the maximum number of hours is 8.5.

Problem 2 :

You need to buy some pencils and an eraser. You can spend no more than $5. The eraser costs $1 and the pencils cost $0.25 each.

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

b) Solve the inequality to find the maximum number of pencils you can buy.

Solution :

No more than 5 means, lesser than 5 only is possible.

Cost of each eraser = $1, cost of each pencil = $0.25

Let x be the number of pencils 

1 + 0.25x < 5

0.25x < 5 - 1

0.25x < 4

x < 4/0.25

x < 16

The maximum number of pencils he can buy is 16.

Problem 3 :

Mark's Canoes rents canoes for $20 plus $35 per hour. You do not want to spend more than $150. For how many hours can you afford to rent the canoe?

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

b) Solve the inequality and answer the question.

Solution :

No more than 150, means less than 150.

Let x be the number of hours.

20 + 35x < 150

35x < 150 - 20

35x < 130

x < 130/35

x < 3.7

So, the maximum number of hours is 3.7

Problem 4 :

For a field trip 18 students rode in cars and the rest filled five buses. How many students were in each bus if no more than 250 students went on the trip?

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

b) Solve the inequality and answer the question.

Solution :

Let x be the number of students travelled in each 5 buses.

18 + 5x ≤ 250

5x ≤ 250 - 18

5x ≤ 232

x ≤ 232/5

x ≤ 46.4

So, the maximum number of students in each bus is 46.

Problem 5 :

Charles is saving $5 each week. He earns an extra $15 by mowing his neighbor's lawn. How many weeks will he need to save in order to have at least $75?

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

b) Solve the inequality and answer the question.

Solution :

At least 75 means, more than 75 is also possible.

Let x be the number of weeks he is saving.

5x + 15 ≥ 75

5x ≥ 75 - 15

5x ≥ 60

x ≥ 60/5

x ≥ 12

So, he will need to save 12 weeks.

Problem 6 :

Write an inequality that represents the missing dimension x. The area is less than 42 square meters.

Solution :

Area of the triangle is less than 42, then A < 42 square meters

(1/2) x base x height < 42

(1/2) ⋅ x ⋅ 6 < 42

3x < 42

To solve for x, let us divide by 3 on both sides.

x < 42/3

x < 14

So, the value of x is lesser than 14 meter.

Problem 7 :

The area is greater than or equal to 8 square feet.

Solution :

Area of the triangle is less than 8, then A < 8 square meters

(1/2) x base x height < 10

(1/2) ⋅ x ⋅ 10 < 10

5x < 10

To solve for x, let us divide by 5 on both sides.

x < 10/5

x < 2

So, the value of x is lesser than 2 meter.

Problem 8 :

The area is less than 18 square centimeters.

Solution :

Area of the shape = Area of triangle + area of rectangle.

Length of shape = base of triangle + length of rectangle

8 = base of triangle + 4

Base of triangle = 4

(1/2) ⋅ 4 ⋅ x + 4 ⋅ x < 18

2x + 4x < 18

6x < 18

x < 18/6

x < 3

Problem 9 :

The area is greater than 12 square inches.

Solution :

Area of rectangle < 12 square inches

Length = 2 inches and width = x

2x < 12

x < 12/2

x < 6

Problem 10 :

You order the hardcover book shown from a website that offers free shipping on orders of $25 or more. Write and solve an inequality that represents how much more you must spend to get free shipping.

Solution :

Price of the book = $19.76

Let x be the difference amount.

19.76 + x > 25

x > 25 - 19.76

x > 5.24

To get the offer of free shipping you must spend more than $5.24.

Problem 11 :

You are riding a train. Your carry-on bag can weigh no more than 50 pounds. Your bag weighs 38 pounds.

a. Write and solve an inequality that represents how much weight you can add to your bag.

b. Can you add both a 9-pound laptop and a 5-pound pair of boots to your bag without going over the weight limit? Explain.

Solution :

No more than 50 pounds = lees than 50 pounds

Weight of my bag = 38 pounds

Let x be the weight to be added.

38 + x < 50

x < 50 - 38

x < 12

b) By adding 9 pounds laptop and 5-pound pair of boots

38 + 9 + 5 < 50

52 < 50

Since it is false, it is going to be over weight.