Inequality Word Problems Study Guide

Example 1

Five more than twice a number is at least 45. What is the minimum value of the number?

Read and understand the question. This question is looking for a number when clues about this number are given.

Make a plan. Translate the statement into inequality form. Then, solve for x using the inequality solving steps.

Carry out the plan. Let x = a number. The key phrase more than means addition, and twice a number is written as 2x. The first part of the statement translates to 2x + 5. The key phrase is at least means greater than or equal to. The entire inequality is 2x + 5 ≥ 45. Subtract 5 from each side of the inequality to get 2x ≥ 40. Next, divide each side by 2 to get the variable alone.

x = 20

Check your answer. Check your solution by substituting the answer into the inequality.

2x + 5 = 45

becomes

2(20) + 5 = 45
40 + 5 = 45
45 = 45

This statement is true, so the solution is checking.

Example 2

There are dogs and cats in a kennel. The number of cats is twice the number of dogs. What is the greatest number of cats in the kennel if there are at most a total of 48 animals in the kennel?

Read and understand the question. This question is looking for a number of cats in the kennel. There are at most a total of 48 animals, so there are 48 animals or less in the kennel.

Make a plan. Translate the statement into inequality form using the clues. Then, solve for x using the inequality solving steps.

Carry out the plan. Let x = the number of dogs. Therefore, 2x = the number of cats. Because the number of animals is at most 48, add the number of cats and dogs together and set it less than or equal to 48. The inequality is x + 2x ≤ 48. Combine like terms to get 3x ≤ 48. Next, divide each side by 3 to get the variable alone.

x = 16

Thus, the number of cats is at most 2(16) = 32.

Check your answer. Check your solution by adding the number of cats and dogs: 16 + 32 = 48, which was the most the number of animals could be. This solution is checking.

Find practice problems and solutions for these concepts at Inequality Word Problems Practice Questions.