Certainly! Let’s simplify the given complex fraction step-by-step. The complex fraction is:
[tex]\[
\frac{\frac{3}{5-x} + \frac{4}{x-5}}{\frac{2}{x} + \frac{3}{x-5}}
\][/tex]
### Step 1: Simplify the Numerator
The numerator of the complex fraction is:
[tex]\[
\frac{3}{5-x} + \frac{4}{x-5}
\][/tex]
Notice that [tex]\( \frac{4}{x-5} \)[/tex] can be rewritten because [tex]\( x-5 = -(5-x) \)[/tex]. Thus,
[tex]\[
\frac{4}{x-5} = \frac{4}{-(5-x)} = -\frac{4}{5-x}
\][/tex]
Now the numerator becomes:
[tex]\[
\frac{3}{5-x} - \frac{4}{5-x}
\][/tex]
Combine the fractions since they have the same denominator:
[tex]\[
\frac{3 - 4}{5-x} = \frac{-1}{5-x}
\][/tex]
### Step 2: Simplify the Denominator
The denominator of the complex fraction is:
[tex]\[
\frac{2}{x} + \frac{3}{x-5}
\][/tex]
We need a common denominator to combine these fractions. The common denominator is [tex]\( x(x-5) \)[/tex]. Rewrite each fraction with this common denominator:
[tex]\[
\frac{2}{x} = \frac{2(x-5)}{x(x-5)} = \frac{2x - 10}{x(x-5)}
\][/tex]
[tex]\[
\frac{3}{x-5} = \frac{3x}{x(x-5)}
\][/tex]
Putting them together:
[tex]\[
\frac{2x - 10}{x(x-5)} + \frac{3x}{x(x-5)}
\][/tex]
Combine the fractions:
[tex]\[
\frac{(2x - 10) + 3x}{x(x-5)} = \frac{5x - 10}{x(x-5)}
\][/tex]
Factorize the numerator:
[tex]\[
\frac{5(x - 2)}{x(x-5)}
\][/tex]
### Step 3: Combine the Simplified Numerator and Denominator
Now we have:
[tex]\[
\frac{\frac{-1}{5-x}}{\frac{5(x - 2)}{x(x-5)}}
\][/tex]
To divide by a fraction, multiply by its reciprocal:
[tex]\[
\frac{-1}{5-x} \times \frac{x(x-5)}{5(x - 2)}
\][/tex]
Since [tex]\( 5-x = -(x-5) \)[/tex], we can rewrite the fraction:
[tex]\[
\frac{-1}{-(x-5)} \times \frac{x(x-5)}{5(x - 2)}
\][/tex]
This can be further simplified to:
[tex]\[
\frac{1}{x-5} \times \frac{x(x-5)}{5(x - 2)}
\][/tex]
Now the [tex]\( x-5 \)[/tex] terms cancel each other out:
[tex]\[
\frac{1}{1} \times \frac{x}{5(x - 2)} = \frac{x}{5(x - 2)}
\][/tex]
Therefore, the final simplified form of the complex fraction is:
[tex]\[
\boxed{\frac{x}{5(x-2)}}
\][/tex]
