\(\frac{2(\mathrm{x} + 1)}{(\mathrm{x} + 5)} = 1 - \frac{1}{(\mathrm{x} + 5)}\) What is the solution to the equation above?...

1. INFER the strategic approach

• The right side has both a whole number (1) and a fraction \(\frac{1}{\mathrm{x}+5}\)
• To work with equations involving fractions, we need common denominators
• Key insight: We can rewrite 1 as \(\frac{\mathrm{x}+5}{\mathrm{x}+5}\) to match the denominator

2. SIMPLIFY the right side by combining fractions

• \(1 - \frac{1}{\mathrm{x} + 5} = \frac{\mathrm{x} + 5}{\mathrm{x} + 5} - \frac{1}{\mathrm{x} + 5}\)
• Combine: \(\frac{\mathrm{x} + 5 - 1}{\mathrm{x} + 5} = \frac{\mathrm{x} + 4}{\mathrm{x} + 5}\)
• Now our equation becomes: \(\frac{2(\mathrm{x} + 1)}{\mathrm{x} + 5} = \frac{\mathrm{x} + 4}{\mathrm{x} + 5}\)

3. INFER that we can eliminate the common denominator

• Since both sides have the same denominator \((\mathrm{x} + 5)\), we can equate the numerators
• This gives us: \(2(\mathrm{x} + 1) = \mathrm{x} + 4\)

4. SIMPLIFY the linear equation

• Apply distributive property: \(2(\mathrm{x} + 1) = 2\mathrm{x} + 2\)
• Equation becomes: \(2\mathrm{x} + 2 = \mathrm{x} + 4\)
• Subtract x from both sides: \(\mathrm{x} + 2 = 4\)
• Subtract 2 from both sides: \(\mathrm{x} = 2\)

Answer: B. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing how to handle the mixed terms on the right side \(\left(1 - \frac{1}{\mathrm{x}+5}\right)\)

Students often try to work with these terms separately or attempt to "cross multiply" incorrectly, leading to algebraic confusion. Without the key insight to rewrite 1 as \(\frac{\mathrm{x}+5}{\mathrm{x}+5}\), they can't proceed systematically.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors during the distributive step

Even when students correctly set up \(2(\mathrm{x} + 1) = \mathrm{x} + 4\), they might distribute incorrectly (getting \(2\mathrm{x} + 1\) instead of \(2\mathrm{x} + 2\)) or make sign errors when moving terms. These small mistakes propagate through the solution.

This may lead them to select Choice A (0) or Choice C (3) depending on the specific arithmetic error.

The Bottom Line:

This problem tests whether students can strategically rewrite expressions to create workable forms - a skill that separates systematic problem-solvers from those who get stuck on mixed algebraic expressions.