absolute value
Absolute value answers the question “how far from zero?” while ignoring which side you are on. Whether you walk 3 steps right or 3 steps left, you have walked a distance of 3. So both +3 and −3 have an absolute value of 3 — the sign is stripped away, only the size remains.
We write the absolute value of a number x as |x|. By definition it is never negative: |x| = x when x is zero or positive, and |x| = −x when x is negative (and −x of a negative number is positive). For instance |7| = 7, |−7| = 7, and |0| = 0.
Geometrically, |x| is the distance from x to 0 on the number line, and more generally |a − b| is the distance between a and b. This distance view is the key to solving absolute-value equations and inequalities: |x| = 5 means x is 5 units from 0, so x is 5 or −5, while |x| < 5 means x is within 5 units of 0.
Solve |x| = 5. The numbers exactly 5 units from 0 are x = 5 and x = −5, so there are two solutions.
|x| = 5 has two answers because distance ignores direction.
A common mistake is to think |x| = x always. That fails for negatives: |−4| is 4, not −4. Absolute value can never produce a negative result.