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. |2x +4| = 10

Step 2: Remove absolute value symbols and set up equation. 2x + 4 = 10
Step 3: Subtract 4 from both sides and simplify. 2x = 6
Step 4: Divide by 2 from both sides and simplify. x = 3
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 forgot the negative case when removing the absolute value. If |A| = 10, then A can be 10 or −10. She only solved A = 10.

Correct solution:

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

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

Case 2: 2x + 4 = −10
2x = −14
x = −7

3. Check both answers in the original equation:
For x = 3: |2(3)+4| − 2 = |10| − 2 = 10 − 2 = 8 ✓
For x = −7: |2(−7)+4| − 2 = |−10| − 2 = 10 − 2 = 8 ✓

Final solutions: x = 3 or x = −7.