ACT Math Absolute Value: Solve Any Equation in 90 Seconds

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

Another example: |2x-4|=6. Case 1: 2x-4=6, so 2x=10, x=5. Case 2: 2x-4=-6, so 2x=-2, x=-1. Check both in the original: |2(5)-4|=|6|=6 ✓ and |2(-1)-4|=|-6|=6 ✓. Both are valid.

Mistake 1: Forgetting the negative case. If you only solve the positive case and get one answer, you've missed half the points. Always remember: there are usually two solutions. Mistake 2: Forgetting to check your solutions. If your check fails, one or both solutions are extraneous (invalid). Mistake 3: Treating absolute value bars like parentheses. |x+3| is not the same as (x+3). The bars mean "distance from zero," which produces two cases. Always expand to both cases before solving.

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.

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: x=-5 (distance of zero is unique). Problem 5: |2x-3|=7. Case 1: 2x-3=7→x=5. Case 2: 2x-3=-7→x=-2. Check both in the original to confirm. Solve all five, writing out both cases and verifying your answers.

Now 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 absolute value problems.

Absolute value equations appear on most ACT Math tests, usually in the medium-difficulty range (questions 30-45). These problems are mechanical once you know the two-case method. Students who master this method pick up 1-2 points because absolute value is predictable and rewards careful application of the method, not deep math knowledge.

Drill the two-case method for one week. Spend 10 minutes daily solving five absolute value problems until both cases become automatic. By test day, you should identify an absolute value equation, expand to two cases, solve both, and check your answers within 90 seconds.