How to Solve Word Problems Involving One-step Inequalities?

One-step inequality word problems appear all the time on the GED Math test. Once you know how to translate a real-world situation into an inequality and solve it in a single step, these problems become straightforward. This lesson walks you through the rules, shows you four worked examples, and gives you practice problems to build confidence.

What Is a One-Step Inequality?

An inequality is a mathematical statement that two expressions are not necessarily equal; instead one is greater than, less than, or equal to the other. A one-step inequality is solved by performing exactly one operation—addition, subtraction, multiplication, or division—to isolate the variable. The four inequality symbols are:

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• \(\color{blue}{>}\) — greater than
• \(\color{blue}{<}\) — less than
• \(\color{blue}{\ge}\) — greater than or equal to
• \(\color{blue}{\le}\) — less than or equal to

Rules for Solving One-Step Inequalities

Addition and Subtraction

Add or subtract the same number from both sides. The inequality symbol does not flip.

• \(\color{blue}{x + 5 > 12 \rightarrow x > 7}\)
• \(\color{blue}{x – 4 \ge 10 \rightarrow x \ge 14}\)

Multiplication and Division by a Positive Number

Multiply or divide both sides by the same positive number. The symbol does not flip.

• \(\color{blue}{3x \le 18 \rightarrow x \le 6}\)
• \(\color{blue}{x \div 2 < 6 \rightarrow x < 12}\)

Multiplication and Division by a Negative Number

Multiply or divide both sides by the same negative number. The symbol flips direction.

• \(\color{blue}{-2x > 10 \rightarrow x < -5}\)  (symbol flipped)

Step-by-Step Summary

1. Read the problem carefully and identify the key number and inequality phrase (at least, more than, no more than, fewer than, etc.).
2. Write the inequality using a variable.
3. Perform one inverse operation to isolate the variable. If you divide or multiply by a negative, flip the symbol.
4. Write the answer in words and check by substituting a value from the solution set.

Watch: How to Solve One-Step Inequalities (Video Lesson)

Math with Mr. J walks through every type of one-step inequality with step-by-step examples:

Worked Examples

Example 1: Maria needs more than $12 for a book. She already has $5. Write and solve an inequality for the additional amount x she needs.

Inequality: \(\color{blue}{x + 5 > 12}\)
Subtract 5 from both sides: \(\color{blue}{x > 7}\)
Maria needs more than $7 more.

Example 2: A box can hold at most 18 kg. Each item weighs 3 kg. How many items x can fit?

Inequality: \(\color{blue}{3x \le 18}\)
Divide both sides by 3: \(\color{blue}{x \le 6}\)
At most 6 items can fit.

Example 3: A temperature must be at least 14°F above the freezing point. If freezing is 0°F, what temperatures x are acceptable?

Inequality: \(\color{blue}{x \ge 14}\)
The temperature must be 14°F or higher.

Example 4: A delivery driver can drive fewer than 12 hours in two days. Write an inequality for the hours per day x.

Inequality: \(\color{blue}{2x < 12}\)
Divide both sides by 2: \(\color{blue}{x < 6}\)
The driver must work fewer than 6 hours per day.

More Practice: Inequality Word Problems (Video)

Khan Academy shows additional inequality word problem examples from 6th-grade math:

Exercises

1. A student needs at least 70 points to pass. She already has 45 points. Write and solve an inequality for the additional points p needed.
2. A bag can hold no more than 24 pounds. Each book weighs 4 pounds. How many books b can fit?
3. A runner wants to run more than 15 miles this week. She has already run 9 miles. Write and solve an inequality for miles m still needed.
4. A car travels at most 60 mph. Write an inequality for the distance d it can cover in 3 hours.
5. A store needs to sell more than 50 items per day. On day one, it sold 32 items. How many more items n must it sell?
6. A tank holds fewer than 12 gallons. If each fill adds 3 gallons, how many fills f can be done?

Answers

1. \(\color{blue}{p + 45 \ge 70 \rightarrow p \ge 25}\); at least 25 more points.
2. \(\color{blue}{4b \le 24 \rightarrow b \le 6}\); at most 6 books.
3. \(\color{blue}{m + 9 > 15 \rightarrow m > 6}\); more than 6 miles.
4. \(\color{blue}{3d \le 60 \rightarrow d \le 20}\); at most 20 miles per hour (distance in 1 \(\color{blue}{\text{ hr } \le 60}\) mi total in 3 \(\color{blue}{\text{ hrs } \le 180}\)).
5. \(\color{blue}{n + 32 > 50 \rightarrow n > 18}\); more than 18 items.
6. \(\color{blue}{3f < 12 \rightarrow f < 4}\); fewer than 4 fills.

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Frequently Asked Questions

When do I flip the inequality symbol?

Flip the symbol only when you multiply or divide both sides by a negative number. Adding, subtracting, or multiplying/dividing by a positive number never changes the direction of the symbol.

What words signal an inequality in a word problem?

At least / no less than signal ≥; at most / no more than signal ≤; more than / greater than signal >; less than / fewer than signal <.

How do I check my inequality answer?

Pick a number from the solution set and substitute it back into the original inequality. For example, if the answer is \(\color{blue}{x > 7}\), try \(\color{blue}{x = 8}\): \(\color{blue}{8 + 5 = 13 > 12}\). True — the answer is correct.

Related Topics

• Graphing Single Variable Inequalities
• Writing Inequalities From Number Lines
• Solve Integers Inequalities Involving Absolute Values
• How to Solve Word Problems Involving the One Step Equation
• Identifying Expressions and Equations