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

Absolute value |x| measures distance from zero, so |x|=a has two solutions: x=a or x=-a. To solve any absolute value equation, (1) isolate the absolute value on one side. (2) Split into two cases: one where the inside is positive, one where it is negative. (3) Solve both cases separately. (4) Check both solutions in the original equation. This framework is foolproof because absolute value algebraically has two branches; solving both captures all answers.

Example: |2x-3|=7. Case 1 (positive): 2x-3=7, so 2x=10, x=5. Case 2 (negative): 2x-3=-7, so 2x=-4, x=-2. Check: |2(5)-3|=|10-3|=|7|=7 ✓ and |2(-2)-3|=|-4-3|=|-7|=7 ✓. Both solutions are valid. Many students forget to solve Case 2, missing half the answer and losing points.

Mistake 1: Solving only the positive case and ignoring the negative case. This loses one solution. Mistake 2: Forgetting to isolate the absolute value first. |3x-5|+2=10 becomes |3x-5|=8 before you split cases. Mistake 3: Not checking solutions. Sometimes one case yields an extraneous solution that does not satisfy the original equation. Mistake 4: Confusing absolute value inequalities with equations. |x|<3 means -33 means x<-3 or x>3 (OR). Inequalities are the trickiest variant; spend extra time on these if you struggle.

Practice: |x|<5 means -55 means x<-5 or x>5. The inside must be both less than 5 and greater than -5 for the inequality version; this is why AND is used.

Equation 1: |x|=4. Equation 2: |x-2|=6. Equation 3: |3x+1|=10. Equation 4: |2x-5|=3. Equation 5: |x+4|=0. Equation 6: |x-1|=-5. Equation 7: |-2x+3|=7. Equation 8: |x|/2=3. For each, (1) isolate the absolute value, (2) write both cases, (3) solve both, (4) check your solutions. Equations 5 and 6 test edge cases: Equation 5 has one solution (x=-4), and Equation 6 has no solution because absolute value is always non-negative.

Answers: 1) x=±4. 2) x=8 or x=-4. 3) x=3 or x=-11/3. 4) x=4 or x=1. 5) x=-4 (only one). 6) No solution. 7) x=2 or x=-5. 8) x=±6.

Absolute value appears on every ACT Math section and becomes more important in intermediate algebra, where it leads into inequalities and distance problems. Solid absolute value skills let you approach piecewise functions and complex inequalities without hesitation, unlocking 1-2 additional points on harder test questions.

Commit to the eight-problem drill this week. Do it twice, correcting errors until the two-case method feels automatic. By test day, you will never lose a point to forgotten cases or extraneous solutions.