1. **State the problem:** Solve the equation $$\frac{1}{2} \left(x + \frac{2}{3}\right) = \frac{13}{6}$$ for $x$. 2. **Use the distributive property:** Multiply both sides by 2 to eliminate the fraction on the left side. $$2 \times \frac{1}{2} \left(x + \frac{2}{3}\right) = 2 \times \frac{13}{6}$$ 3. **Simplify:** $$\cancel{2} \times \frac{1}{\cancel{2}} \left(x + \frac{2}{3}\right) = \frac{2 \times 13}{6}$$ $$x + \frac{2}{3} = \frac{26}{6}$$ 4. **Simplify the right side fraction:** $$\frac{26}{6} = \frac{13}{3}$$ So the equation is now: $$x + \frac{2}{3} = \frac{13}{3}$$ 5. **Isolate $x$ by subtracting $\frac{2}{3}$ from both sides:** $$x + \frac{2}{3} - \frac{2}{3} = \frac{13}{3} - \frac{2}{3}$$ $$x = \frac{13}{3} - \frac{2}{3}$$ 6. **Subtract the fractions:** $$x = \frac{13 - 2}{3} = \frac{11}{3}$$ **Final answer:** $$x = \frac{11}{3}$$