1. **State the problem:** Solve the equation $$3 - \left( \frac{2}{3}x + \frac{1}{9} \right) = x - \left( \frac{x}{3} - \frac{2}{3} \right)$$ for $x$. 2. **Rewrite the equation and remove parentheses:** $$3 - \frac{2}{3}x - \frac{1}{9} = x - \frac{x}{3} + \frac{2}{3}$$ 3. **Combine like terms on the left side:** $$3 - \frac{1}{9} = \frac{27}{9} - \frac{1}{9} = \frac{26}{9}$$ So the equation becomes: $$\frac{26}{9} - \frac{2}{3}x = x - \frac{x}{3} + \frac{2}{3}$$ 4. **Combine like terms on the right side:** $$x - \frac{x}{3} = \frac{3}{3}x - \frac{1}{3}x = \frac{2}{3}x$$ So the right side is: $$\frac{2}{3}x + \frac{2}{3}$$ 5. **Rewrite the equation:** $$\frac{26}{9} - \frac{2}{3}x = \frac{2}{3}x + \frac{2}{3}$$ 6. **Add $\frac{2}{3}x$ to both sides to get all $x$ terms on the right:** $$\frac{26}{9} = \frac{2}{3}x + \frac{2}{3}x + \frac{2}{3} = \frac{4}{3}x + \frac{2}{3}$$ 7. **Subtract $\frac{2}{3}$ from both sides:** $$\frac{26}{9} - \frac{2}{3} = \frac{4}{3}x$$ Convert $\frac{2}{3}$ to ninths: $$\frac{2}{3} = \frac{6}{9}$$ So: $$\frac{26}{9} - \frac{6}{9} = \frac{20}{9}$$ Thus: $$\frac{20}{9} = \frac{4}{3}x$$ 8. **Solve for $x$ by dividing both sides by $\frac{4}{3}$:** $$x = \frac{\frac{20}{9}}{\frac{4}{3}} = \frac{20}{9} \times \frac{3}{4}$$ 9. **Simplify:** $$x = \frac{20 \times 3}{9 \times 4} = \frac{60}{36}$$ Cancel common factor 12: $$x = \frac{\cancel{60}^{5} \times 12}{\cancel{36}^{3} \times 12} = \frac{5}{3}$$ **Final answer:** $$x = \frac{5}{3}$$