Quick answer

Remove the sign of negative inputs; keep non-negative inputs. The result is always ≥ 0.

Formula

  • |x| ≥ 0
  • |x| = −x when x < 0

Introduction

Each scenario lists x, notation, and the final magnitude so you can copy the pattern into your notes. Examples are where rules become reflexes.

Work a few by hand, then verify with the calculator. Alternating manual and tool work catches sign errors early.

The Absolute Value Calculator checks any example below. Theory lives in the formula guide; for step order, see how to calculate absolute value.

Why examples matter

The workflow repeats even when units change: temperature deltas, elevations, bank balances, or coordinates. Only the numbers change; the sign rule does not.

Seeing many inputs builds confidence before you tackle equations where the unknown sits inside the bars.

Group examples by type: integers, decimals, fractions, zero, and expressions like |a − b|. Mixed practice beats repeating one easy case.

When examples feel easy, move on to equations where |x| equals a constant and you solve for x.

Template line

  • |x| = non-negative magnitude
  • If x < 0, |x| = −x
  • If x ≥ 0, |x| = x

Copy this line into your notebook and fill in values underneath each problem. Write the sign check in words, not only symbols.

For |a − b|, compute the subtraction first, then apply | | if the intermediate result is negative.

Distance-style examples appear when a story mentions two locations; subtract first, then take magnitude if needed.

Pattern in every example

  1. Integer: x = −15 → |−15| = 15. The input is negative, so flip.
  2. Decimal: x = 2.01 → |2.01| = 2.01. Non-negative inputs stay.
  3. Fraction: x = −3/8 → |−3/8| = 3/8. Treat fractions like any other signed number.
  4. Zero: x = 0 → |0| = 0. Zero is neither positive nor negative, but |0| is defined as 0.
  5. Two-step: x = 7 − 19 = −12 → |−12| = 12. Simplify inside the bars when the problem gives an expression.

More worked cases

Find |7 − 19|. First compute 7 − 19 = −12, then |−12| = 12.

Find |−0.004|. The value is negative, so the magnitude is 0.004.

Find |√2 − 2|. The inside is negative because √2 ≈ 1.414 is less than 2, so the result is 2 − √2 (about 0.586).