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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.

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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