Question

which expression is equivalent to \\(\frac{\frac{3}{x - 2} - 5}{2 - \frac{4}{x - 2}}\\)? \\(\bigcirc\\) \\(\frac{2x - 8}{13 - 5x}\\) \\(\bigcirc\\) \\(\frac{-5x - 8}{2x - 8}\\) \\(\bigcirc\\) \\(\frac{-5x - 4}{x - 2}\\) \\(\bigcirc\\) \\(\frac{13 - 5x}{2x - 8}\\)

Explanation:

Step1: Simplify numerator expression

Combine terms in $\frac{3}{x-2}-5$ by getting common denominator $x-2$:
$$\frac{3}{x-2} - 5 = \frac{3 - 5(x-2)}{x-2} = \frac{3 -5x +10}{x-2} = \frac{13-5x}{x-2}$$

Step2: Simplify denominator expression

Combine terms in $2-\frac{4}{x-2}$ by getting common denominator $x-2$:
$$2 - \frac{4}{x-2} = \frac{2(x-2) -4}{x-2} = \frac{2x -4 -4}{x-2} = \frac{2x-8}{x-2}$$

Step3: Divide simplified numerator/denominator

Divide the two results, cancel $x-2$ (where $x
eq2$):
$$\frac{\frac{13-5x}{x-2}}{\frac{2x-8}{x-2}} = \frac{13-5x}{2x-8}$$

Answer:

$\boldsymbol{\frac{13-5x}{2x-8}}$ (corresponding to the last option)