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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

Why it matters: Tolerance specifications (acceptable parts satisfy |measurement - target| <= tolerance) and 'within X of...' problems in physics, manufacturing, and quality control all use absolute value inequalities.

Worked example

Problem. Solve |x - 3| = 5.

  1. Split into two cases: x - 3 = 5 or x - 3 = -5.
  2. Solve each: x = 8 or x = -2.
  3. Connect the result back to Absolute Value Equations and Inequalities so the method and meaning are both clear.
  4. Check the result against the original representation before writing the final answer.

Answer: x = 8 or x = -2

Practice problems

1. Solve |x| = 7. Enter the positive solution.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve |x| = 7. Enter the positive solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. |x| = 7 means x = 7 or x = -7.
  4. The positive solution is 7.
  5. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

2. Solve |x - 4| = 6. Enter the larger solution.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve |x - 4| = 6. Enter the larger solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Cases: x - 4 = 6 or x - 4 = -6.
  4. x = 10 or x = -2. Larger is 10.
  5. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

3. Solve |x + 2| = 5. Enter the positive solution.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve |x + 2| = 5. Enter the positive solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Cases: x + 2 = 5 or x + 2 = -5.
  4. x = 3 or x = -7. Positive is 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

4. Solve |2x| = 10. Enter the positive solution.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve |2x| = 10. Enter the positive solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Cases: 2x = 10 or 2x = -10.
  4. x = 5 or x = -5. Positive is 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

5. Solve |x - 1| = 0.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve |x - 1| = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The only number with absolute value 0 is 0 itself.
  4. x - 1 = 0 means x = 1.
  5. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

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