1. **Problem:** Simplify the expression $$\frac{1}{3 + \frac{2}{x}}$$. 2. **Formula and rules:** When simplifying complex fractions, combine terms by finding a common denominator or rewrite the expression to eliminate the complex fraction. 3. **Step-by-step simplification:** 1. Start with $$\frac{1}{3 + \frac{2}{x}}$$. 2. Rewrite the denominator as a single fraction: $$3 + \frac{2}{x} = \frac{3x}{x} + \frac{2}{x} = \frac{3x + 2}{x}$$. 3. Substitute back: $$\frac{1}{\frac{3x + 2}{x}}$$. 4. Dividing by a fraction is the same as multiplying by its reciprocal: $$1 \times \frac{x}{3x + 2} = \frac{x}{3x + 2}$$. 4. **Final answer:** $$\boxed{\frac{x}{3x + 2}}$$