ACT Math: Solve Absolute Value Problems Using the Split-Case Method

Absolute value |x| means distance from zero. When you see |x|=5, two values work: x=5 or x=-5. The split-case method handles all absolute value problems: write "|expression|=number" as two separate equations, one positive and one negative, then solve both. For |2x-3|=7, case 1: 2x-3=7 gives x=5. Case 2: 2x-3=-7 gives x=-2. Check both in the original equation. Never skip the check step; absolute value problems only accept values that satisfy the original equation.

Example: Solve |x+4|=10. Case 1: x+4=10 means x=6. Check: |6+4|=|10|=10. ✓ Case 2: x+4=-10 means x=-14. Check: |-14+4|=|-10|=10. ✓ Both solutions are valid. If you only found x=6, you would miss half the points on a multi-answer problem.

Mistake 1: Forgetting the negative case. Many students solve |2x|=8 as 2x=8 (x=4) and stop, missing x=-4. Mistake 2: Failing to check answers. An algebraic answer might not satisfy the absolute value equation. Mistake 3: Confusing |x|<5 (range, means -55 (means x<-5 or x>5). Write a one-line reminder: "Absolute value always has two cases." Post it above your desk. Practice this split-case method on ten problems until it feels automatic and you never skip either case.

When you see an absolute value symbol on test day, pause and think "split case." This one habit prevents the errors that cost most students 2-3 points per test.

Problem 1: |x-2|=8. Case 1: x-2=8 means x=10. Case 2: x-2=-8 means x=-6. Problem 2: |3x+1|=16. Case 1: 3x+1=16 means x=5. Case 2: 3x+1=-16 means x=-17/3. Problem 3: |4-x|=12. Case 1: 4-x=12 means x=-8. Case 2: 4-x=-12 means x=16. For each problem, solve both cases, check both answers in the original equation, and write your solutions as "x=a or x=b."

Time yourself: can you solve and check all five in under 10 minutes? If yes, you own this skill. If no, slow down and check every step. Speed comes after accuracy.

Absolute value questions appear on most ACT Math sections, typically in the medium-difficulty range. Many students skip them or guess because they feel uncertain. The split-case method is so reliable that you can answer absolute value questions faster and more accurately than you answer linear equations.

Master this method this week, and you will confidently earn 3-5 points on test day that other students leave on the table.