1. **State the problem:** Simplify the complex fraction $$\frac{\frac{x}{81} + \frac{1}{x}}{\frac{5}{81} - \frac{5}{9x}}$$. 2. **Rewrite the numerator and denominator:** Numerator: $$\frac{x}{81} + \frac{1}{x}$$ Denominator: $$\frac{5}{81} - \frac{5}{9x}$$ 3. **Find common denominators inside numerator and denominator:** - Numerator common denominator is $$81x$$: $$\frac{x}{81} = \frac{x \cdot x}{81 \cdot x} = \frac{x^2}{81x}$$ $$\frac{1}{x} = \frac{1 \cdot 81}{x \cdot 81} = \frac{81}{81x}$$ So numerator becomes: $$\frac{x^2}{81x} + \frac{81}{81x} = \frac{x^2 + 81}{81x}$$ - Denominator common denominator is $$81x$$: $$\frac{5}{81} = \frac{5 \cdot x}{81 \cdot x} = \frac{5x}{81x}$$ $$\frac{5}{9x} = \frac{5 \cdot 9}{9x \cdot 9} = \frac{45}{81x}$$ So denominator becomes: $$\frac{5x}{81x} - \frac{45}{81x} = \frac{5x - 45}{81x}$$ 4. **Rewrite the complex fraction:** $$\frac{\frac{x^2 + 81}{81x}}{\frac{5x - 45}{81x}}$$ 5. **Divide numerator by denominator:** $$= \frac{x^2 + 81}{81x} \times \frac{81x}{5x - 45}$$ 6. **Cancel common factors:** $$= \frac{x^2 + 81}{\cancel{81x}} \times \frac{\cancel{81x}}{5x - 45} = \frac{x^2 + 81}{5x - 45}$$ 7. **Factor denominator:** $$5x - 45 = 5(x - 9)$$ 8. **Final simplified expression:** $$\boxed{\frac{x^2 + 81}{5(x - 9)}}$$