\(\mathrm{(p + 3) + 8 = 10}\) What value of p is the solution to the given equation?...

1. INFER the solving strategy

• Given equation: \(\mathrm{(p + 3) + 8 = 10}\)
• Goal: Find the value of p that makes this equation true
• Strategy: Isolate p by undoing the operations step by step

2. SIMPLIFY by combining like terms on the left side

• \(\mathrm{(p + 3) + 8 = 10}\)
• \(\mathrm{p + 3 + 8 = 10}\)
• \(\mathrm{p + 11 = 10}\)

3. SIMPLIFY by subtracting 11 from both sides

• \(\mathrm{p + 11 = 10}\)
• \(\mathrm{p + 11 - 11 = 10 - 11}\)
• \(\mathrm{p = -1}\)

4. Verify the solution

• Check: \(\mathrm{(-1 + 3) + 8 = 2 + 8 = 10}\) ✓

Answer: A. -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not understand the systematic approach to "undoing" operations to isolate the variable. They might try to guess and check with the answer choices instead of using algebraic manipulation, or they might perform operations incorrectly (like adding instead of subtracting).

This leads to confusion and random answer selection rather than systematic solving.

Second Most Common Error:

Poor SIMPLIFY execution: Students might make sign errors when dealing with negative numbers, particularly when subtracting to get \(\mathrm{p = -1}\). They may struggle with the concept that subtracting a larger number from a smaller number yields a negative result.

This may lead them to select an incorrect positive answer choice or abandon the systematic approach.

The Bottom Line:

This problem tests fundamental equation-solving skills. Students need to understand both the conceptual approach (isolate the variable) and execute it correctly with proper attention to arithmetic signs. The negative answer can be particularly tricky for students who expect all solutions to be positive.