To find which expression is equivalent to [tex]\(\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}}\)[/tex], we need to simplify the given expression step by step.
Given expression:
[tex]\[
\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}}
\][/tex]
### Step 1: Simplify the Numerator
First, we will simplify the numerator [tex]\(\frac{3}{x-2} - 5\)[/tex].
Rewrite [tex]\( -5 \)[/tex] as a fraction:
[tex]\[
-5 = \frac{-5(x-2)}{x-2} = \frac{-5x + 10}{x-2}
\][/tex]
Now, combine the fractions:
[tex]\[
\frac{3}{x-2} - 5 = \frac{3}{x-2} + \frac{-5x + 10}{x-2} = \frac{3 - 5x + 10}{x-2} = \frac{13 - 5x}{x-2}
\][/tex]
### Step 2: Simplify the Denominator
Next, we need to simplify the denominator [tex]\( 2 - \frac{4}{x-2} \)[/tex].
Rewrite [tex]\( 2 \)[/tex] as a fraction:
[tex]\[
2 = \frac{2(x-2)}{x-2} = \frac{2x - 4}{x-2}
\][/tex]
Now, combine the fractions:
[tex]\[
2 - \frac{4}{x-2} = \frac{2x - 4}{x-2} - \frac{4}{x-2} = \frac{2x - 4 - 4}{x-2} = \frac{2x - 8}{x-2}
\][/tex]
### Step 3: Divide the Numerator by the Denominator
Now that we have simplified both the numerator and the denominator, we can divide one by the other:
[tex]\[
\frac{\frac{13 - 5x}{x-2}}{\frac{2x - 8}{x-2}}
\][/tex]
Since both fractions have the same denominator [tex]\((x-2)\)[/tex], they cancel out:
[tex]\[
\frac{13 - 5x}{2x - 8}
\][/tex]
### Simplified Expression
The equivalent simplified expression is:
[tex]\[
\frac{13 - 5x}{2x - 8}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{13-5x}{2x-8}}
\][/tex]
