How to Solve Integers Inequalities involving Absolute Values?

Absolute value inequalities combine two key ideas: the meaning of absolute value and the rules for solving inequalities. On the GED, these problems ask you to find all integer values that make a statement like \(\color{blue}{|x| < 5}\) or \(\color{blue}{|x - 2| \ge 3}\) true. Once you know the two standard cases, every problem follows the same pattern.

What Is an Absolute Value Inequality?

The absolute value of a number is its distance from zero on the number line, always non-negative. So \(\color{blue}{|5| = 5}\) and \(\color{blue}{|-5| = 5}\). An absolute value inequality is an inequality that contains an absolute value expression, such as \(\color{blue}{|x| \le 4}\) or \(\color{blue}{|x + 1| > 2}\).

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The Two Cases

Case 1: |expression| < k (or ≤ k) — “And” compound inequality

When the absolute value is less than a positive number k, the solution lies between −k and k:

|A| < k  ⇒  −k < A < k

• \(\color{blue}{|x| < 5 \rightarrow -5 < x < 5}\)
• \(\color{blue}{|x + 2| \le 4 \rightarrow -4 \le x + 2 \le 4 \rightarrow -6 \le x \le 2}\)

Case 2: |expression| > k (or ≥ k) — “Or” compound inequality

When the absolute value is greater than a positive number k, the solution lies outside the interval:

|A| > k  ⇒  A < −k  or  A > k

• \(\color{blue}{|x| > 3 \rightarrow x < -3 \text{ or x } > 3}\)
• \(\color{blue}{|x – 1| > 3 \rightarrow x – 1 < -3 \text{ or x } - 1 > 3 \rightarrow x < -2 \text{ or x } > 4}\)

Step-by-Step Summary

1. Isolate the absolute value expression on one side.
2. Determine the case: is the absolute value less than (Case 1) or greater than (Case 2) a number?
3. Write the compound inequality and solve each part.
4. Check: substitute a test value from each part of the solution.

Watch: Absolute Value Inequalities (Video Lesson)

Khan Academy explains both cases with clear number-line illustrations:

Worked Examples

Example 1: Solve \(\color{blue}{|x| \le 6}\). List the integer solutions.

Case 1: \(\color{blue}{-6 \le x \le 6}\).
Integer solutions: −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6

Example 2: Solve \(\color{blue}{|x + 2| \le 4}\).

Case 1: \(\color{blue}{-4 \le x + 2 \le 4}\).
Subtract 2 from all parts: \(\color{blue}{-6 \le x \le 2}\).
Check: \(\color{blue}{|-6 + 2| = |-4| = 4 \le 4}\) ✓; \(\color{blue}{|2 + 2| = 4 \le 4}\) ✓

Example 3: Solve \(\color{blue}{|x – 1| > 3}\).

Case 2: \(\color{blue}{x – 1 < -3 \text{ or x } - 1 > 3}\).
\(\color{blue}{x < -2 \text{ or x } > 4}\).
Check: try \(\color{blue}{x = -3}\): \(\color{blue}{|-3 – 1| = 4 > 3}\) ✓; try \(\color{blue}{x = 5}\): \(\color{blue}{|5 – 1| = 4 > 3}\) ✓

Example 4: Solve \(\color{blue}{|2x| < 10}\).

Case 1: \(\color{blue}{-10 < 2x < 10}\).
Divide by 2: \(\color{blue}{-5 < x < 5}\).
Integer solutions: −4, −3, −2, −1, 0, 1, 2, 3, 4

More Practice: Solving Absolute Value Equations and Inequalities (Video)

The Organic Chemistry Tutor provides an in-depth walkthrough with interval notation and number line graphs:

Exercises

1. Solve \(\color{blue}{|x| < 4}\) and list all integer solutions.
2. Solve \(\color{blue}{|x| \ge 5}\).
3. Solve \(\color{blue}{|x + 3| \le 5}\).
4. Solve \(\color{blue}{|x – 2| > 4}\).
5. Solve \(\color{blue}{|3x| \le 9}\) and list all integer solutions.
6. Solve \(\color{blue}{|x + 1| < 2}\) and list integer solutions.

Answers

1. \(\color{blue}{-4 < x < 4}\); integers: −3, −2, −1, 0, 1, 2, 3
2. \(\color{blue}{x \le -5 \text{ or x } \ge 5}\)
3. \(\color{blue}{-8 \le x \le 2}\)
4. \(\color{blue}{x < -2 \text{ or x } > 6}\)
5. \(\color{blue}{-3 \le x \le 3}\); integers: −3, −2, −1, 0, 1, 2, 3
6. \(\color{blue}{-3 < x < 1}\); integers: −2, −1, 0

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Frequently Asked Questions

What is the key difference between the two cases?

If the absolute value is less than a number, the solution is a single interval (between two values). If it is greater than a number, the solution is two separate rays pointing outward.

What if the right side is negative?

If you have \(\color{blue}{|x| < -3}\), there is no solution (absolute value is \(\color{blue}{\text{ always } \ge 0}\)). If you have \(\color{blue}{|x| > -3}\), the solution is all real numbers.

Do I flip the inequality when solving absolute value problems?

In Case 1, when you write \(\color{blue}{-k \le A \le k}\), both inequalities keep their direction. When solving each part of Case 2, you flip only if you divide or multiply by a negative coefficient inside the absolute value expression.

Related Topics

• Absolute Value Definition
• Absolute Value of Rational Numbers
• Solving Word Problems Involving One Step Inequalities
• Graphing Single Variable Inequalities
• Solve Word Problems of Absolute Value and Integers