Example 1 — a modulus equation with two cases
Solve .
- 1 means or .The modulus discards the sign, so both possibilities must be considered.
- 2Case 1: , so and .
- 3Case 2: , so and .
- 4Check : ✓
- 5Check : ✓Both are valid, so there are two solutions.
The Modulus Function
Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 2 — two moduli, solved by squaring
Solve .
- 1Both sides are non-negative, so squaring is safe and removes both moduli at once..
- 2
- 3Expand both sides fully.
- 4Collect everything on one side.
- 5, so or
- 6Check : and ✓And gives on both sides ✓
Example 3 — a 'greater than' modulus inequality
Solve .
- 1 means or .Two separate intervals — points further than 3 from zero.
- 2Case 1: , so .
- 3Case 2: , so .The inequality flips direction with the sign of the right side.
- 4Solution: or .This cannot be written as a single inequality — the values between are excluded.
- 5Check : ✓ and : , not ✓Testing one value inside and one outside confirms the direction.