Complex fraction made of sums involving rational expressions: roodiem... Simplify. \( \frac{6}{\frac{6}{x-5}+2} \) \( \frac{27}{x-5}+9 \) \( \square \)

To simplify the complex fraction: \[ \frac{6}{\frac{6}{x-5} + 2} \] Let's follow these steps: 1. **Simplify the Denominator:** Start by simplifying the denominator of the complex fraction: \[ \frac{6}{x-5} + 2 \] To combine these terms, find a common denominator: \[ \frac{6}{x-5} + 2 = \frac{6 + 2(x - 5)}{x - 5} = \frac{6 + 2x - 10}{x - 5} = \frac{2x - 4}{x - 5} \] 2. **Rewrite the Original Fraction:** Substitute the simplified denominator back into the original complex fraction: \[ \frac{6}{\frac{2x - 4}{x - 5}} \] Dividing by a fraction is the same as multiplying by its reciprocal: \[ 6 \times \frac{x - 5}{2x - 4} \] 3. **Simplify the Expression:** Factor out common terms in the numerator and denominator: \[ 2x - 4 = 2(x - 2) \] So the expression becomes: \[ 6 \times \frac{x - 5}{2(x - 2)} = \frac{6}{2} \times \frac{x - 5}{x - 2} = 3 \times \frac{x - 5}{x - 2} \] Therefore, the simplified form is: \[ \frac{3(x - 5)}{x - 2} \] **Final Answer:** \[ \boxed{\dfrac{3(x - 5)}{\,x - 2\,}} \]