1. **State the problem:** Simplify the complex fraction $$\frac{3 - \frac{1}{2}}{1 + \frac{\frac{1}{4}}{-1 + \frac{1}{3}}}$$ 2. **Simplify the numerator:** Calculate $3 - \frac{1}{2}$: $$3 - \frac{1}{2} = \frac{6}{2} - \frac{1}{2} = \frac{5}{2}$$ 3. **Simplify the denominator's inner denominator:** Calculate $-1 + \frac{1}{3}$: $$-1 + \frac{1}{3} = -\frac{3}{3} + \frac{1}{3} = -\frac{2}{3}$$ 4. **Simplify the fraction inside the denominator:** Calculate $\frac{\frac{1}{4}}{-\frac{2}{3}}$: $$\frac{1}{4} \div -\frac{2}{3} = \frac{1}{4} \times -\frac{3}{2} = -\frac{3}{8}$$ 5. **Simplify the entire denominator:** Calculate $1 + \left(-\frac{3}{8}\right)$: $$1 - \frac{3}{8} = \frac{8}{8} - \frac{3}{8} = \frac{5}{8}$$ 6. **Combine numerator and denominator:** Calculate $$\frac{\frac{5}{2}}{\frac{5}{8}} = \frac{5}{2} \times \frac{8}{5} = \frac{5 \times 8}{2 \times 5} = \frac{8}{2} = 4$$ **Final answer:** $$4$$