Absolute Value and Its Properties

GRE Quantitative · Axiom Academy

LESSON Absolute Value and Its Properties One idea — distance from zero — drives every absolute value equation, inequality, and GRE shortcut you'll see. 1. Absolute Value Is a Distance Watch a point slide along the number line. At every position, two bars grow from zero out to the point's mirror image on both sides — that shared length is |x| . Whether the point sits left or right of zero, the distance reads the same. Definition: flip the sign only when x is negative for every x — a distance can't be negative. — the sizes multiply; the sign question is handled separately. — combining first can only shorten (or match) the trip. 2. Solving |x| = a : Two Mirror Points If a distance from zero equals a (with ), there are exactly two points that far out — one on each side. That's the entire reason absolute value equations split into two cases. Case 1 (inside is already non-negative): Case 2 (inside is negative, so flip its sign): Solution: x = 10 or x = -4 — both really are 7 away from 3 on the number line. 3. Inequalities: a Window or Two Rays Now let the target distance become a boundary instead of an exact hit. "Closer than a " fills in a window around zero; "farther than a " leaves two rays escaping outward. These are the two patterns the GRE tests over and over. One connected interval, centered at zero. Two separate rays, opening away from zero. The distance from x to 2 must exceed 3 , so or . Solution: or — everything strictly outside the window (-1,5) .

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