1. **State the problem:** Solve the equation $$\frac{x+\frac{1}{2}}{x-\frac{1}{3}} = \frac{3}{2}$$ for $x$. 2. **Rewrite the equation:** To simplify, write the numerator and denominator clearly: $$\frac{x + \frac{1}{2}}{x - \frac{1}{3}} = \frac{3}{2}$$ 3. **Cross-multiply:** Multiply both sides by the denominator to eliminate the fraction: $$\left(x + \frac{1}{2}\right) \times 2 = \left(x - \frac{1}{3}\right) \times 3$$ 4. **Expand both sides:** $$2x + 1 = 3x - 1$$ 5. **Isolate $x$:** Move all $x$ terms to one side and constants to the other: $$2x + 1 = 3x - 1$$ $$2x - 3x = -1 - 1$$ $$\cancel{2x} - 3x = -2$$ $$-x = -2$$ 6. **Solve for $x$:** Multiply both sides by $-1$: $$x = 2$$ 7. **Check for restrictions:** The denominator $x - \frac{1}{3}$ cannot be zero, so $x \neq \frac{1}{3}$. Our solution $x=2$ is valid. **Final answer:** $$x = 2$$