1. **State the problem:** Simplify the expression $$\frac{1 + \frac{1}{x}}{x - \frac{1}{x}}$$. 2. **Rewrite the expression:** To simplify, write the numerator and denominator with a common denominator: $$\frac{1 + \frac{1}{x}}{x - \frac{1}{x}} = \frac{\frac{x}{x} + \frac{1}{x}}{\frac{x^2}{x} - \frac{1}{x}} = \frac{\frac{x+1}{x}}{\frac{x^2 - 1}{x}}$$ 3. **Divide the fractions:** Dividing by a fraction is the same as multiplying by its reciprocal: $$= \frac{x+1}{x} \times \frac{x}{x^2 - 1}$$ 4. **Cancel common factors:** The $x$ in numerator and denominator cancels out: $$= \frac{x+1}{\cancel{x}} \times \frac{\cancel{x}}{x^2 - 1} = \frac{x+1}{x^2 - 1}$$ 5. **Factor the denominator:** Recognize $x^2 - 1$ as a difference of squares: $$x^2 - 1 = (x-1)(x+1)$$ 6. **Simplify the fraction:** $$\frac{x+1}{(x-1)(x+1)} = \frac{\cancel{x+1}}{x-1 \times \cancel{x+1}} = \frac{1}{x-1}$$ **Final answer:** $$\frac{1}{x-1}$$