1. **State the problem:** Solve the equation $$\frac{2}{3}X - 2 - \frac{X}{6} + \frac{X}{2} = X + \frac{2}{3}$$ for $X$. 2. **Combine like terms on the left side:** The terms with $X$ are $$\frac{2}{3}X - \frac{X}{6} + \frac{X}{2}$$. 3. **Find a common denominator for the $X$ terms:** The denominators are 3, 6, and 2. The least common denominator is 6. 4. **Rewrite each term with denominator 6:** $$\frac{2}{3}X = \frac{4}{6}X, \quad -\frac{X}{6} = -\frac{1}{6}X, \quad \frac{X}{2} = \frac{3}{6}X$$ 5. **Sum the $X$ terms:** $$\frac{4}{6}X - \frac{1}{6}X + \frac{3}{6}X = \frac{4 - 1 + 3}{6}X = \frac{6}{6}X = X$$ 6. **Rewrite the equation:** $$X - 2 = X + \frac{2}{3}$$ 7. **Subtract $X$ from both sides:** $$\cancel{X} - 2 = \cancel{X} + \frac{2}{3} \implies -2 = \frac{2}{3}$$ 8. **Analyze the result:** The statement $$-2 = \frac{2}{3}$$ is false, so there is no solution. **Final answer:** There is no solution to the equation because it leads to a contradiction.