|p| + 61 = 65 Which value is a solution to the given equation?...

1. SIMPLIFY to isolate the absolute value

• Given: \(|\mathrm{p}| + 61 = 65\)
• Subtract 61 from both sides:
  \(|\mathrm{p}| = 65 - 61\)
  \(|\mathrm{p}| = 4\)

2. INFER what \(|\mathrm{p}| = 4\) means using absolute value definition

• The definition of absolute value tells us that \(|\mathrm{p}| = 4\) means:
  • \(\mathrm{p} = 4\), or
  • \(\mathrm{p} = -4\)
• Both values make the original equation true because both have an absolute value of 4

3. CONSIDER ALL CASES but select from available choices

• We found \(\mathrm{p} = 4\) or \(\mathrm{p} = -4\)
• Looking at the answer choices: A. 65/61, B. 4, C. 126, D. 130
• Only \(\mathrm{p} = 4\) appears as an option

Answer: B. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students may struggle with the algebraic step of isolating \(|\mathrm{p}|\), either forgetting to subtract 61 from both sides or making arithmetic errors in calculating \(65 - 61\).

This leads to confusion about what the absolute value equals, causing them to get stuck and guess randomly.

Second Most Common Error:

Missing conceptual knowledge about absolute value definition: Students might remember that absolute value makes things positive but not recall that \(|\mathrm{x}| = \mathrm{a}\) has two solutions. They may only consider \(\mathrm{p} = 4\) and miss that \(\mathrm{p} = -4\) is also mathematically valid.

However, since -4 doesn't appear in the choices, this conceptual gap fortunately doesn't lead to selecting an incorrect answer in this particular problem.

The Bottom Line:

This problem tests whether students can systematically work with absolute value equations by first isolating the absolute value term, then correctly applying the definition to find all solutions. The key insight is recognizing that while absolute value equations often have two solutions, you only need to identify which solutions match the given answer choices.