1. **State the problem:** Solve the equation $$\frac{2}{3}(x-6)=\frac{1}{2}x+1$$ for $x$. 2. **Write down the equation:** $$\frac{2}{3}(x-6)=\frac{1}{2}x+1$$ 3. **Distribute the fraction on the left side:** $$\frac{2}{3}x - \frac{2}{3} \times 6 = \frac{1}{2}x + 1$$ 4. **Calculate the multiplication:** $$\frac{2}{3}x - 4 = \frac{1}{2}x + 1$$ 5. **Bring all $x$ terms to one side and constants to the other:** $$\frac{2}{3}x - \frac{1}{2}x = 1 + 4$$ 6. **Find common denominator for the $x$ terms:** $$\frac{4}{6}x - \frac{3}{6}x = 5$$ 7. **Simplify the left side:** $$\frac{1}{6}x = 5$$ 8. **Multiply both sides by 6 to solve for $x$:** $$x = 5 \times 6$$ 9. **Calculate the final value:** $$x = 30$$ **Final answer:** $$x=30$$