Inequality Word Problems Study Guide (page 2)
Introduction to Inequality Word Problems
Still more astonishing is that world of rigorous fantasy we call mathematics.
—GREGORY BATESON (1904–1980)
This lesson details the basics of translating and solving inequalities and word problems with inequalities. Use the information in Lesson 1 for additional review of the basics of translating words into symbols.
Translating from Words to Inequalities
The key words and phrases are summarized in the following chart.
Read through the following examples for more help with translating inequalities. Let x = a number in each example. The key phrases are in italics.
|1.||Two is less than a number||2 < x|
|2.||Four more than a number is greater than or equal to five.||x + 4 ≥ 5|
|3.||The maximum value of a number is 28.||x ≤ 28|
|4.||The sum of a number and nine is at least 81.||x + 9 ≥ 81|
Solve inequalities by using the same golden rule as equation solving: Whatever you do to one side of the inequality, do to the other side.
Solve the inequality for x:
- 4x > 16
- Divide each side by 4.
- x > 4
When you are dividing or multiplying each side of an inequality by a negative value, the inequality symbol must switch directions. For example, to solve the inequality –5x > 10, divide each side by –5. The inequality becomes x < –2.
Solve the inequality for x:
+ 4 ≤ 8 + 4 – 4 ≤ 8 – 4 Subtract 4 from each side. ≤ 4 Simplify. × –2 ≥ 4 × –2 Multiply each side by –2. Switch the inequality symbol. x = –8 Simplify.
Inequality Word Problems
To solve inequality word problems, first translate from words into mathematical symbols. Then, solve the inequality using the same rules as equation solving with two differences. The solution will not be a single value, but usually a set of values. In addition, if you multiplied or divided each side of the equation by a negative number, you must switch the inequality symbol to the other direction.
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
- 2(20) + 5 = 45
- 40 + 5 = 45
- 45 = 45
This statement is true, so the solution is checking.
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.