1. **State the problem:** Solve the equation $$x - \frac{x+2}{3} + 3(x-3) = 2 + \frac{2x+1}{3}$$ for $x$. 2. **Write down the equation and identify the goal:** We want to isolate $x$ by simplifying and solving the equation. 3. **Eliminate the fractions by multiplying both sides by 3:** $$3 \times \left(x - \frac{x+2}{3} + 3(x-3)\right) = 3 \times \left(2 + \frac{2x+1}{3}\right)$$ 4. **Distribute multiplication:** $$3x - \cancel{3} \times \frac{x+2}{\cancel{3}} + 9(x-3) = 6 + \cancel{3} \times \frac{2x+1}{\cancel{3}}$$ 5. **Simplify the canceled terms:** $$3x - (x+2) + 9(x-3) = 6 + (2x+1)$$ 6. **Expand the terms:** $$3x - x - 2 + 9x - 27 = 6 + 2x + 1$$ 7. **Combine like terms on the left:** $$ (3x - x + 9x) - 2 - 27 = 6 + 2x + 1$$ $$11x - 29 = 7 + 2x$$ 8. **Bring all $x$ terms to one side and constants to the other:** $$11x - 2x = 7 + 29$$ 9. **Simplify both sides:** $$9x = 36$$ 10. **Divide both sides by 9 to solve for $x$:** $$\frac{\cancel{9}x}{\cancel{9}} = \frac{36}{9}$$ 11. **Simplify the division:** $$x = 4$$ **Final answer:** $$x = 4$$