Absolute value
SAT glossary · Math
Definition
|a| is the distance from a to zero, and is never negative. |x| = c has two solutions when c > 0, one when c = 0, and none when c < 0. Less-than inequalities give a band (-c < x < c); greater-than inequalities give two rays joined by or.
Absolute value in plain English
Absolute value just means 'how far is this number from zero, ignoring direction' — so both 5 and -5 have an absolute value of 5, written |5|=5 and |-5|=5. It can never be negative, because distance is never negative. When you solve an equation like |x|=c, you're really asking 'what numbers are exactly c away from zero on the number line' — if c is positive there are two answers (one on each side of zero), if c=0 there's only one answer (zero itself), and if c is negative there's no answer at all, because you can't have a negative distance. For inequalities, |x|<c means you're inside a safe zone between -c and c, while |x|>c means you've wandered past c in EITHER direction, giving you two separate ranges instead of one connected one.
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