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Which expression is equivalent to [tex]\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}}[/tex]?

A. [tex]\frac{2 x-8}{13-5 x}[/tex]
B. [tex]\frac{-5 x-8}{2 x-8}[/tex]
C. [tex]\frac{-5 x-4}{x-2}[/tex]
D. [tex]\frac{13-5 x}{2 x-8}[/tex]

Sagot :

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]