Absolute value on the ACT is distance from zero: |x| is always nonnegative, and solving |x| = a means x = a or x = -a. Inequalities follow the same split logic, and the sign-flip trap catches students who rush.
What |x| really means
|x| is how far x sits from 0 on the number line, so |5| = 5 and |-5| = 5. That is why |x| can never be negative. When you see |x| = 7, x could be 7 or -7—both are seven units from zero.
Solving |x| = a
If a is positive, split into two equations: x = a and x = -a. Example: |2x - 3| = 5 gives 2x - 3 = 5 (so x = 4) or 2x - 3 = -5 (so x = -1). If a is zero, you get one solution: x = 0. If a is negative, there is no solution—distance from zero cannot be negative.
Inequalities: |x| < a
"Closer to zero than a" means the middle band: -a < x < a. Example: |x - 2| < 5 becomes -5 < x - 2 < 5, so -3 < x < 7. This is an AND region—a single interval on the number line. Review ACT inequalities for the algebra moves.
Inequalities: |x| > a
"Farther from zero than a" means two outer regions: x < -a OR x > a. Example: |3x + 1| > 4 gives 3x + 1 < -4 or 3x + 1 > 4, so x < -5/3 or x > 1. Draw a quick number line if OR vs. AND feels fuzzy.
Watch the negatives
The biggest slip is treating |x| < -3 as solvable—it is not. Also check your split when the expression inside the bars is complicated; isolate carefully before splitting. Absolute value sits inside the broader algebra block on the ACT—see how much algebra is on the ACT. Pair this with linear equations and domain and range when graphs or restrictions show up.
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