Which expression is equivalent to the following complex fraction? \( \frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}} \)

To simplify the complex fraction \( \frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}} \), we first need a common denominator for both the numerator and the denominator. In the numerator \( \frac{3}{x-1} - 4 \), the common denominator is \( x-1 \): \[ = \frac{3 - 4(x-1)}{x-1} = \frac{3 - 4x + 4}{x-1} = \frac{7 - 4x}{x-1}. \] In the denominator \( 2 - \frac{2}{x-1} \), the common denominator is again \( x-1 \): \[ = \frac{2(x-1) - 2}{x-1} = \frac{2x - 2 - 2}{x-1} = \frac{2x - 4}{x-1}. \] Now we substitute these simplified forms back into the complex fraction: \[ \frac{\frac{7 - 4x}{x-1}}{\frac{2x - 4}{x-1}}. \] We can simplify by multiplying by the reciprocal of the denominator: \[ = \frac{7 - 4x}{2x - 4}. \] Next, notice that \( 2x - 4 \) can be factored as \( 2(x - 2) \): \[ = \frac{7 - 4x}{2(x - 2)}. \] The expression \( 7 - 4x \) can be rearranged as \( -4x + 7 \) or \( -(4x - 7) \): \[ = \frac{-(4x - 7)}{2(x - 2)}. \] So, the final simplified expression for the complex fraction is: \[ \frac{4x - 7}{2(2 - x)} \quad \text{or} \quad -\frac{4x - 7}{2(x - 2)}. \] Either form conveys the same equivalent expression!