Absolute value measures distance, not direction — it answers 'how far from zero' rather than 'which way.' That simple idea covers four related tasks: evaluating |x| for a single number, solving absolute-value equations, solving absolute-value inequalities, and measuring the distance between two points on a number line.

Why absolute value is always non-negative

Absolute value is defined piecewise: |x| = x when x is already zero or positive, and |x| = −x (which flips a negative number positive) when x is negative. Either way, the output can never be negative — |0| = 0 is the only place the two definitions meet. Geometrically, |x| is just the length of the segment from 0 to x on the number line, and length is never negative.

Why absolute-value equations usually have two solutions

An equation like |ax + b| = c is really asking: 'what values of x make the expression inside the bars exactly c units from zero?' There are two such values in general — one where ax + b lands exactly on +c, and one where it lands on −c — which is why solving |x − 2| = 5 gives both x = 7 and x = −3. The exception is c = 0, which has exactly one solution (the bars close in on a single point), and c < 0, which has no solution at all, since an absolute value can never equal a negative number.

Why 'less than' and 'greater than' inequalities look so different

|ax + b| < c asks for every x whose expression stays within c units of zero — that describes a single continuous interval, like a band around a target. |ax + b| > c asks for every x whose expression strays further than c units away in either direction — that describes two separate rays heading off toward positive and negative infinity, with a gap in the middle. Recognizing which shape you're looking for makes it much easier to sanity-check a solution: an interval should be one continuous range, a 'greater than' answer should be two disconnected pieces.

Distance is just absolute value in disguise

When a problem asks for 'the distance between a and b,' it is asking for |a − b|. Because absolute value discards sign, |a − b| and |b − a| always give the same answer — distance doesn't care which point you call 'first.' This is the same idea used later in coordinate geometry (distance formula) and in statistics (mean absolute deviation), so getting comfortable with it here pays off elsewhere.