Question:What is the solution to the given equation?\(\frac{(\mathrm{x}+4)(\mathrm{x}-1)}{(\mathrm{x}+4)} + 3 = 11\)

1. INFER the domain restriction

• Since we have \((\mathrm{x}+4)\) in the denominator, x cannot equal -4
• This is important to note even though it won't affect our final answer

2. SIMPLIFY by canceling common factors

• Notice that \((\mathrm{x}+4)\) appears in both the numerator and denominator
• We can cancel these: \(\frac{(\mathrm{x}+4)(\mathrm{x}-1)}{(\mathrm{x}+4)} = (\mathrm{x}-1)\) when \(\mathrm{x} \neq -4\)
• Our equation becomes: \((\mathrm{x}-1) + 3 = 11\)

3. SIMPLIFY the linear equation

• Combine like terms on the left side: \(\mathrm{x} - 1 + 3 = \mathrm{x} + 2\)
• So we have: \(\mathrm{x} + 2 = 11\)

4. SIMPLIFY to solve for x

• Subtract 2 from both sides: \(\mathrm{x} = 9\)

5. INFER verification strategy and check

• Substitute back into original equation: \(\frac{(9+4)(9-1)}{(9+4)} + 3\)
• Calculate: \(\frac{(13)(8)}{13} + 3 = 8 + 3 = 11\) ✓

Answer: C (9)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students may try to expand \((\mathrm{x}+4)(\mathrm{x}-1)\) in the numerator instead of recognizing the cancellation opportunity, leading to:

\(\frac{\mathrm{x}^2 + 3\mathrm{x} - 4}{(\mathrm{x}+4)} + 3 = 11\)

This creates a much more complex rational equation that's difficult to solve, causing students to get confused and guess randomly among the choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the cancellation but make arithmetic errors when combining terms, such as:

• Getting \(\mathrm{x} - 1 + 3 = \mathrm{x} + 4\) instead of \(\mathrm{x} + 2\)
• Or making sign errors when solving \(\mathrm{x} + 2 = 11\)

This leads to incorrect values like \(\mathrm{x} = 7\) or \(\mathrm{x} = 13\), causing them to select answer choices that aren't available or to abandon the systematic approach and guess.

The Bottom Line:

This problem tests whether students can recognize and execute the key insight that canceling common factors dramatically simplifies rational equations. The real skill is in SIMPLIFY - seeing the shortcut rather than taking the long algebraic route.