If (x-5)/7 = (x-5)/9, the value of x - 5 is between which of the following pairs of values?

1. INFER the key insight about the equation structure

• Given equation: \(\frac{\mathrm{x-5}}{7} = \frac{\mathrm{x-5}}{9}\)
• Key insight: Both fractions have the same numerator \(\mathrm{(x-5)}\)
• Strategy: Use cross multiplication to solve

2. SIMPLIFY using cross multiplication

• Cross multiply: \(\mathrm{9(x-5) = 7(x-5)}\)
• Expand both sides: \(\mathrm{9x - 45 = 7x - 35}\)
• Collect like terms: \(\mathrm{9x - 7x = -35 + 45}\)
• Simplify: \(\mathrm{2x = 10}\)
• Solve for x: \(\mathrm{x = 5}\)

3. INFER the final answer

• We need \(\mathrm{x - 5}\), not just x
• Calculate: \(\mathrm{x - 5 = 5 - 5 = 0}\)
• Check the ranges: 0 falls between -3 and 3

Answer: B. -3 and 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize the most efficient approach and might try complex algebraic manipulations instead of seeing that when \(\frac{\mathrm{x-5}}{7} = \frac{\mathrm{x-5}}{9}\), cross multiplication leads directly to the solution.

Some students might get overwhelmed by the fraction setup and attempt to clear denominators in inefficient ways, leading to more complex algebra than necessary. This can cause calculation errors and confusion about what they're solving for.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors during the cross multiplication and simplification steps, such as sign errors when collecting terms or mistakes in basic arithmetic.

For example, when simplifying \(\mathrm{2x = 10}\), they might incorrectly get \(\mathrm{x = 4}\), leading to \(\mathrm{x - 5 = -1}\), which would make them select Choice B for the wrong reason or become confused about which range contains -1.

The Bottom Line:

This problem tests whether students can recognize an efficient solution path for rational equations and execute basic algebraic manipulations accurately. The key insight is that the equation structure immediately suggests cross multiplication as the cleanest approach.