Absolute Value Inequalities

To solve an absolute value inequality, get the absolute value alone, then rewrite it without the bars. A “less than” inequality becomes an “and” inequality: |x| \lt 3 means -3 \lt x \lt 3. A “greater than” inequality becomes an “or” inequality: |x| \gt 3 means x \lt -3 or x \gt 3.

Before this lesson: You should know how to solve absolute value equations and compound inequalities.

What Does an Absolute Value Inequality Mean?

Absolute value is distance from 0. So |x| \lt 3 means “x is less than 3 units from 0.” Those numbers are all between -3 and 3.

|x| \gt 3 means “x is more than 3 units from 0.” Those numbers are outside: below -3 or above 3.

A memory trick: “less thand” and “greator.” Less than gives an “and” inequality. Greater than gives an “or” inequality.

How to Solve Absolute Value Inequalities

1. Get the absolute value alone on one side.
2. Check the number on the other side. If it is negative, see the special cases below.
3. Rewrite the inequality using the table below.
4. Solve the compound inequality.
5. Graph the answer and check a number from the shaded part.

Here A is whatever is inside the bars, and c is a positive number. Notice that the sign turns around on the -c side of an “or” inequality.

Worked Examples

Example 1: Less than

Solve |x - 1| \lt 4.

Step 1. It is “less than,” so write an “and” inequality.

Step 2. Add 1 to all three parts.

\begin{aligned}
-4 &\lt x - 1 \lt 4 \\
-3 &\lt x \lt 5
\end{aligned}

Check: Try x = 0. |0 - 1| = 1, and 1 \lt 4. It works.

Answer: -3 \lt x \lt 5. The graph has open circles at -3 and 5, shaded between.

Example 2: Greater than or equal to

Solve |x + 2| \ge 5.

Step 1. It is “greater than,” so write an “or” inequality. Turn the sign around on the negative side.

Step 2. Subtract 2 in each part.

\begin{aligned}
x + 2 \le -5 \quad &\text{or} \quad x + 2 \ge 5 \\
x \le -7 \quad &\text{or} \quad x \ge 3
\end{aligned}

Check: Try x = 4. |4 + 2| = 6, and 6 \ge 5. Try x = 0: |0 + 2| = 2, which is not at least 5, so 0 is not a solution.

Answer: x \le -7 or x \ge 3

Example 3: Get the absolute value alone first

Solve 3|2x - 1| + 4 \le 25.

Step 1. Subtract 4, then divide by 3.

\begin{aligned}
3|2x - 1| + 4 &\le 25 \\
3|2x - 1| &\le 21 \\
|2x - 1| &\le 7
\end{aligned}

Step 2. Write an “and” inequality. Add 1, then divide by 2.

\begin{aligned}
-7 &\le 2x - 1 \le 7 \\
-6 &\le 2x \le 8 \\
-3 &\le x \le 4
\end{aligned}

Check: Try x = 4. 3|2(4) - 1| + 4 = 3(7) + 4 = 25, and 25 \le 25. It works.

Answer: -3 \le x \le 4

Example 4: A negative coefficient inside

Solve |5 - 2x| \gt 9.

Step 1. Write an “or” inequality.

Step 2. Subtract 5. Then divide by -2 and flip each sign.

\begin{aligned}
5 - 2x \lt -9 \quad &\text{or} \quad 5 - 2x \gt 9 \\
-2x \lt -14 \quad &\text{or} \quad -2x \gt 4 \\
x \gt 7 \quad &\text{or} \quad x \lt -2
\end{aligned}

Check: Try x = 8. |5 - 16| = 11, and 11 \gt 9. Try x = -3: |5 + 6| = 11 \gt 9. Both work.

Answer: x \lt -2 or x \gt 7

Example 5: Word problem

A machine cuts boards that should be 48 inches long. A board passes if its length is within 0.25 inch of 48 inches. What lengths pass?

Step 1. Let L = the length. “Within 0.25 of 48” means the distance from 48 is at most 0.25.

|L - 48| \le 0.25

Step 2. Write an “and” inequality and add 48 to all three parts.

\begin{aligned}
-0.25 &\le L - 48 \le 0.25 \\
47.75 &\le L \le 48.25
\end{aligned}

Answer: Boards from 47.75 inches to 48.25 inches long pass.

Common Mistakes

Special Cases

When the number on the other side is negative or zero, think about distance instead of splitting.

Absolute value inequalities are useful for tolerances, like a part that must be within a set amount of its target size. To review the steps for “and” and “or” graphs, see compound inequalities. To review the sign-flip rule, see one-step inequalities.

Practice Problems

Solve each inequality. Then describe or sketch its graph.

1. |x| \lt 6
2. |x| \ge 2
3. |x - 3| \le 5
4. |x + 4| \gt 1
5. |2x + 1| \lt 9
6. |3x - 6| \ge 12
7. 2|x - 5| + 1 \le 9
8. |4 - x| \gt 3
9. |x + 9| \lt -1
10. A cereal box should hold 20 ounces. A box passes if its weight is within 0.5 ounce of 20 ounces. Write and solve an inequality for the weights that pass.

Answers

Absolute Value Inequalities FAQ

How do you solve absolute value inequalities?

Get the absolute value alone. If it is less than a number, write an “and” inequality. If it is greater than a number, write an “or” inequality. Then solve and graph.

When is an absolute value inequality “and” or “or”?

Less than (\lt or \le) gives “and,” because the answers are between two numbers. Greater than (\gt or \ge) gives “or,” because the answers are outside.

What if the absolute value is less than a negative number?

Then there is no solution, because an absolute value is never negative. If it is greater than a negative number, every number is a solution.

Do you flip the sign in absolute value inequalities?

For an “or” inequality, the part with -c uses the opposite sign. You also flip a sign any time you multiply or divide by a negative number.

Summary

• An absolute value inequality compares a distance to a number.
• Get the absolute value alone before you rewrite it.
• |A| \lt c means -c \lt A \lt c: an “and” inequality, shaded between.
• |A| \gt c means A \lt -c or A \gt c: an “or” inequality, shaded outside.
• If the number is negative, the answer is either no solution or all real numbers.

Previous lesson: Absolute Value Equations

Next lesson: Relations and Functions (coming soon)

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