Question

Martha solved the absolute value equation below. Her final answer is x = 3
|2x + 4| - 2 = 8
Step 1: Isolate the absolute value by adding 2 to each side and simplify.

Step 2: Remove absolute value symbols and set up equation.
Step 3: Subtract 4 from both sides and simplify.
Step 4: Divide by 2 from both sides and simplify.
Explain what Martha did incorrectly and/or is missing. Then, solve the problem correctly and completely, showing and/or explaining your work.

Answer 1

Martha only kept the positive case when she removed the absolute value. When |A| = B (with B ≥ 0) you must use A = B OR A = −B. She used A = B but omitted A = −B.

Solve correctly:

1. Isolate the absolute value:
|2x + 4| − 2 = 8 ⇒ |2x + 4| = 10.

2. Split into two cases:
Case 1: 2x + 4 = 10
2x = 6
x = 3

Case 2: 2x + 4 = −10
2x = −14
x = −7

3. Check both solutions:
x = 3: |2(3)+4| − 2 = |10| − 2 = 10 − 2 = 8 ✓
x = −7: |2(−7)+4| − 2 = |−10| − 2 = 10 − 2 = 8 ✓

Final answer: x = 3 or x = −7.