ACT Math Absolute Value Equations: Solve Using the Two-Case Method

Absolute value equations produce two cases because the expression inside the bars could be positive or negative. For |x+3|=5, Case 1 (positive): x+3=5, so x=2. Case 2 (negative): x+3=-5, so x=-8. Always test both solutions in the original equation. Check x=2: |2+3|=|5|=5 ✓. Check x=-8: |-8+3|=|-5|=5 ✓. Both are valid. This two-case method is the only reliable way to solve absolute value equations on the ACT, so master it and you'll never guess on these problems.

Another example: |2x-4|=6. Case 1: 2x-4=6→2x=10→x=5. Case 2: 2x-4=-6→2x=-2→x=-1. Check both: |2(5)-4|=|6|=6 ✓ and |2(-1)-4|=|-6|=6 ✓. Both solutions are valid.

Mistake 1: Forgetting the negative case. If you only solve the positive case, you've missed half the problem. Absolute value always produces two cases; remember this ritual. Mistake 2: Forgetting to check your solutions. If a check fails, the solution is extraneous (invalid). Always verify before submitting your answer. Mistake 3: Treating absolute value bars like parentheses. |x+3| is not (x+3); the bars mean distance from zero, which requires expanding to two cases. Always set up both cases before you solve.

Create a two-step checklist: (1) Set up two cases, (2) Check both solutions. Use this checklist on every absolute value problem until it becomes automatic habit.

Problem 1: |x|=7. Solutions: x=7 or x=-7. Problem 2: |x-2|=4. Case 1: x-2=4→x=6. Case 2: x-2=-4→x=-2. Both valid. Problem 3: |3x+1|=10. Case 1: 3x+1=10→x=3. Case 2: 3x+1=-10→x=-11/3. Both valid. Problem 4: |x+5|=0. Only one solution (distance of zero): x=-5. Problem 5: |2x-3|=7. Case 1: 2x-3=7→x=5. Case 2: 2x-3=-7→x=-2. Both valid. Problem 6: |x-4|=-2. No solution (absolute value is never negative). Solve all six, showing both cases and verifying each answer in the original equation.

Find five absolute value questions from a practice test and solve them using the two-case method. Time yourself. Once you solve all five correctly within 2 minutes, you're ready for harder problems.

Absolute value equations appear on most ACT Math tests, usually in questions 25-45. Once you know the two-case method, these problems are mechanical and predictable. Students who master this method pick up 1-2 points because absolute value is straightforward once you follow the ritual of setting up two cases and checking both.

Drill the two-case method daily this week. Spend 10 minutes solving five absolute value problems until both cases become automatic. By test day, you should solve any absolute value equation, find both solutions, and verify them within 90 seconds.