# How to Write Inequalities for Word Problems

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Inequalities can be written in any direction, but if there is a variable, like x, the statement usually starts with the variable, followed by the inequality symbol, then a number. If reading from left to right, > means "greater than" and < means "less than"; if the > symbol is underlined, it signifies greater than or equal to; an underlined < means less than or equal to. The open part of the symbol always faces the larger number. Now you are ready to tackle inequality word problems. The trick is to identify some clue words within the text.

• Read through the word problem and underline clue words that suggest the inequality you need to write down. According to a Cuesta College online math course: "is at least" means greater than or equal to; "is at most" and "cannot exceed" mean less than or equal to; "must exceed" and "is more than" signify greater than; and "is less than" means less than. For example, read the following word problem: "Samantha makes \$10 an hour and receives a weekly bonus of \$25. How many hours does she need to work this week to make more than \$250?" The clue words "more than \$250" suggests this is a "greater than" problem. You would write the > symbol to solve it.

• Write down the inequality statement with the applicable symbol in the middle, the variable or unknown and accompanying numbers to the left of the symbol, and the known benchmark number to the right of the symbol. Continuing with the example word problem in Step 1, the inequality statement would be: 10x + 25 > 250. The variable "x" stands for the number of hours. It is multiplied by 10 because \$10 is the hourly rate. 10x is added to 25 because \$25 dollars is the weekly bonus. 10x + 25 must be greater than 250 for Samantha to make more than \$250 this week.

• Solve the equality statement. According to Bard College, you can add, subtract, multiply or divide to isolate the variable as long as you do the same operation to both sides of the inequality. To solve 10x + 25 > 250, start by subtracting 25 from each side of the inequality statement to get 10x > 225. Now divide each side by 10 to get x > 22.5. Thus, Samantha must work more than 22.5 hours to make more than \$250 in a week.

## Tips & Warnings

• As in any word problem, you can often find clue words that tell you whether you need to add, subtract, multiply or divide. Check out the University of Arkansas at Monticello page in the Resources section for a table of key words.
• If you multiply or divide both sides of an inequality statement by a negative number, you have to switch the direction of the inequality symbol, according to Bard College.

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