Absolute Value Equations and Inequalities
A free Algebra I lesson from the “Algebra Foundations” unit, with a worked example and practice problems including step-by-step solutions.
An absolute value equation |expression| = k (with k positive) means the expression is k units from zero, so split into two cases: expression = k or expression = -k. For inequalities, |x| < k means -k < x < k (between), and |x| > k means x < -k or x > k (outside). Absolute value can never equal a negative number, so |expression| = -5 has no solution.
What you'll learn
- Solve absolute value equations by splitting into two cases
- Solve absolute value inequalities (less-than gives 'between', greater-than gives 'outside')
- Recognize when an absolute value equation has no solution
Worked example
Problem. Solve |x - 3| = 5.
- Split into two cases: x - 3 = 5 or x - 3 = -5.
- Solve each: x = 8 or x = -2.
- Connect the result back to Absolute Value Equations and Inequalities so the method and meaning are both clear.
- Check the result against the original representation before writing the final answer.
Answer: x = 8 or x = -2
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Solve |x| = 7. Enter the positive solution.
Show solution
- |x| = 7 means x = 7 or x = -7.
- The positive solution is 7.
Answer: 7
2. Solve |x - 4| = 6. Enter the larger solution.
Show solution
- Cases: x - 4 = 6 or x - 4 = -6.
- x = 10 or x = -2. Larger is 10.
Answer: 10
3. Solve |x + 2| = 5. Enter the positive solution.
Show solution
- Cases: x + 2 = 5 or x + 2 = -5.
- x = 3 or x = -7. Positive is 3.
Answer: 3
4. Solve |2x| = 10. Enter the positive solution.
Show solution
- Cases: 2x = 10 or 2x = -10.
- x = 5 or x = -5. Positive is 5.
Answer: 5
5. Solve |x - 1| = 0.
Show solution
- The only number with absolute value 0 is 0 itself.
- x - 1 = 0 means x = 1.
Answer: 1
Practice this interactively with instant feedback and an AI tutor.
Practice Absolute Value Equations and Inequalities Take the free placement check