Which expression is equivalent to the following complex fraction?

[tex]\[ \frac{\frac{3}{x-1} - 4}{2 - \frac{2}{x-1}} \][/tex]

A. \(\frac{2(x-2)}{-4x+7}\)

B. \(\frac{-4x+7}{2(x-2)}\)

C. \(\frac{-4x+7}{2\left(x^2-2\right)}\)

D. [tex]\(\frac{2\left(x^2-2\right)}{-4x+7}\)[/tex]



Answer :

To simplify the given complex fraction and determine which of the provided expressions is equivalent, we can follow these steps:

Given complex fraction:
[tex]\[ \frac{\frac{3}{x-1} - 4}{2 - \frac{2}{x-1}} \][/tex]

### Step 1: Simplify the numerator
The numerator of the complex fraction is:
[tex]\[ \frac{3}{x-1} - 4 \][/tex]

To combine the terms, find a common denominator, which in this case is \( x-1 \):
[tex]\[ \frac{3 - 4(x-1)}{x-1} = \frac{3 - 4x + 4}{x-1} = \frac{7 - 4x}{x-1} \][/tex]

### Step 2: Simplify the denominator
The denominator of the complex fraction is:
[tex]\[ 2 - \frac{2}{x-1} \][/tex]

Similarly, convert to a single fraction with a common denominator:
[tex]\[ 2 - \frac{2}{x-1} = \frac{2(x-1)}{x-1} - \frac{2}{x-1} = \frac{2x - 2 - 2}{x-1} = \frac{2x - 4}{x-1} \][/tex]

### Step 3: Combine numerical and denominal fractions
The complex fraction now is:
[tex]\[ \frac{\frac{7-4x}{x-1}}{\frac{2x-4}{x-1}} \][/tex]

Since both the numerator and the denominator have the same common denominator (x - 1), we can eliminate it:
[tex]\[ \frac{7 - 4x}{2x - 4} \][/tex]

### Step 4: Simplify the resulting fraction
Factor out common terms from the numerator and the denominator if possible:
[tex]\[ \frac{-(4x - 7)}{2(x-2)} = \frac{-1 \cdot (4x - 7)}{2(x-2)} = \frac{-4x + 7}{2(x-2)} \][/tex]

This is one of the provided options.

Thus, the expression equivalent to the given complex fraction is:
[tex]\[ \boxed{\frac{-4x + 7}{2(x-2)}} \][/tex]