Which expression is equivalent to \(\frac{8x(x-7)-3(x-7)}{(2x-14)}\), where x gt 7?

1. TRANSLATE the problem information

• Given expression: \(\mathrm{8x(x-7)-3(x-7)/(2x-14)}\) where \(\mathrm{x \gt 7}\)
• Need to find: An equivalent expression from the answer choices
• Key insight: This should be read as \(\mathrm{[8x(x-7) - 3(x-7)]/(2x-14)}\)

2. INFER the simplification strategy

• Notice that both terms in the numerator contain the factor \(\mathrm{(x-7)}\)
• The denominator \(\mathrm{2x-14}\) can also be factored
• Since we have common factors, we can likely simplify by canceling

3. SIMPLIFY the denominator

• Factor out 2 from the denominator: \(\mathrm{2x - 14 = 2(x - 7)}\)
• Expression becomes: \(\mathrm{[8x(x-7) - 3(x-7)]/[2(x - 7)]}\)

4. SIMPLIFY the numerator

• Factor out \(\mathrm{(x-7)}\): \(\mathrm{8x(x-7) - 3(x-7) = (x-7)[8x - 3]}\)
• Expression becomes: \(\mathrm{[(x-7)(8x - 3)]/[2(x - 7)]}\)

5. SIMPLIFY by canceling common factors

• Since \(\mathrm{x \gt 7}\), we know \(\mathrm{(x-7) ≠ 0}\), so we can safely cancel \(\mathrm{(x-7)}\)
• Result: \(\mathrm{(8x - 3)/2}\)

Answer: B. \(\mathrm{(8x-3)/2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misread the expression structure, interpreting it as \(\mathrm{8x(x-7) - [3(x-7)/(2x-14)]}\) instead of \(\mathrm{[8x(x-7) - 3(x-7)]/(2x-14)}\).

This leads them to try simplifying \(\mathrm{3(x-7)/(2x-14) = 3/2}\) first, then subtract this from \(\mathrm{8x(x-7)}\), creating a complex expression that doesn't match any answer choice. This causes confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students might expand everything instead of looking for common factors, calculating \(\mathrm{8x^2 - 56x - 3x + 21}\) in the numerator and missing the elegant factoring approach.

This may lead them to select Choice C \(\mathrm{(8x^2-3x-14)/(2x-14)}\) or Choice D \(\mathrm{(8x^2-3x-77)/(2x-14)}\) by making arithmetic errors during expansion.

The Bottom Line:

Success depends on correctly interpreting the expression structure and recognizing the common factor pattern rather than expanding and making the algebra more complicated.