Question

which expression is equivalent to the following complex fraction?\\(\frac{\frac{3}{x - 1}-4}{2-\frac{3}{x - 1}}\\)\\(\circ\\) \\(\frac{2(x - 2)}{-4x + 7}\\)\\(\circ\\) \\(\frac{-4x + 7}{2(x - 2)}\\)\\(\circ\\) \\(\frac{-4x + 7}{2(x^2 - 2)}\\)\\(\circ\\) \\(\frac{2(x^2 - 2)}{-4x + 7}\\)

Explanation:

Step1: Simplify numerator

Combine terms over $x-1$:
$\frac{3}{x-1} - 4 = \frac{3 - 4(x-1)}{x-1} = \frac{3 - 4x + 4}{x-1} = \frac{-4x + 7}{x-1}$

Step2: Simplify denominator

Combine terms over $x-1$:
$2 - \frac{2}{x-1} = \frac{2(x-1) - 2}{x-1} = \frac{2x - 2 - 2}{x-1} = \frac{2x - 4}{x-1} = \frac{2(x-2)}{x-1}$

Step3: Divide simplified numerator/denominator

Cancel $x-1$ (where $x
eq 1$):
$\frac{\frac{-4x + 7}{x-1}}{\frac{2(x-2)}{x-1}} = \frac{-4x + 7}{2(x-2)}$

Answer:

$\boldsymbol{\frac{-4x+7}{2(x-2)}}$ (matches the second option: $\boldsymbol{\frac{-4x+7}{2(x-2)}}$)