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]
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]
