Step by Step
1
Isolate the absolute value expression completely
Before splitting into two cases, get the absolute value bars completely alone on one side of the equation — nothing multiplied, added, or subtracted around them yet.
2
Split into two equations
Once isolated, |expression| = k splits into: expression = k, OR expression = −k.
3
Solve both equations separately
Solve each resulting equation using standard algebra steps — you'll typically get two different solutions.
4
Check both answers
Plug both solutions back into the original equation to confirm they work — occasionally one solution turns out to be extraneous.
Applied Walkthrough
1
Solve 3|x + 1| − 2 = 10. You cannot split into two cases yet — the absolute value isn't isolated; there's a 3 multiplying it and a −2 outside it.
2
Isolate first: add 2 to both sides (3|x+1| = 12), then divide by 3 (|x+1| = 4).
3
Now split: x + 1 = 4, OR x + 1 = −4. Solving each: x = 3, or x = −5.
4
Check both in the original equation — both satisfy 3|x+1| − 2 = 10, so both are valid solutions.
Exam Application
Exams test whether you isolate the absolute value expression completely before splitting into two cases — a step many students skip — and whether you check both resulting solutions.
⚠ Common Trap
The most common trap is splitting into two cases before the absolute value is fully isolated. If there's still a coefficient or constant outside the bars, isolate it first, then split.
✓ Quick Self-Check
1. What does |x| = k split into?
x = k OR x = −k.
Tap to reveal / hide
2. What must happen before you split an absolute value equation into two cases?
The absolute value expression must be completely isolated on one side of the equation.
Tap to reveal / hide
3. Solve 3|x+1| − 2 = 10.
x = 3 or x = −5.
Tap to reveal / hide
4. Why do you need to check both solutions at the end?
Because occasionally one of the two solutions turns out to be extraneous and doesn't actually satisfy the original equation.
Tap to reveal / hide
5. What does the mnemonic |x| = k represent conceptually?
That distance from zero is always positive, so a value k away from zero could be either +k or −k.
Tap to reveal / hide