Absolute Value

ACT Math · Axiom Academy

Understand distance from zero, evaluate |x| expressions, and solve absolute value equations — a fast, frequent ACT topic. 1. Absolute Value Is Distance From Zero The absolute value of a number is how far it sits from 0 on the number line — and distance is never negative. That's why |5| and |-5| come out the same : 5 and −5 are mirror images, each exactly 5 units from zero. The definition: distance from a to 0 Always non-negative, for every real x 2. Solving |x| = a — Why There Are Two Answers An equation like |x| = 7 asks: "which numbers are exactly 7 units from zero?" There are always two such numbers when a>0 — one on each side — because distance doesn't care which direction you measure it. |x| = 7 : splits into x=7 or x=-7 — both valid. |x| = 0 : only x=0 is zero units from zero — one solution. |x| = -3 : no solution — absolute value can never equal a negative number. 3. Solving |ax+b| = c — Split Into Two Cases When the expression inside the bars isn't just x , use the same idea: whatever's inside the bars must equal c or -c . Solve both cases as ordinary linear equations, then check each answer. Check: |2(5)-3|=7 ✓ and |2(-2)-3|=7 ✓ 4. Absolute Value Inequalities — Interval vs. Two Rays Inequalities with absolute value split into two very different shapes. |x| < a traps x inside a single interval around zero. |x| > a pushes x out onto two separate rays , one on each side. "Less than" → one interval, close to zero "Greater than" → two rays, far from zero

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