1. **State the problem:** Solve the equation $$\frac{2}{3}x - 2 - \frac{x}{6} + \frac{x}{2} = \frac{x+2}{3}$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to isolate zero: $$\frac{2}{3}x - \frac{x}{6} + \frac{x}{2} - \frac{x+2}{3} = 2$$ 3. **Find a common denominator:** The denominators are 3, 6, 2, and 3. The least common denominator (LCD) is 6. 4. **Rewrite each term with denominator 6:** $$\frac{2}{3}x = \frac{4}{6}x, \quad -\frac{x}{6} = -\frac{x}{6}, \quad \frac{x}{2} = \frac{3}{6}x, \quad -\frac{x+2}{3} = -\frac{2(x+2)}{6}$$ 5. **Substitute back:** $$\frac{4}{6}x - \frac{x}{6} + \frac{3}{6}x - \frac{2(x+2)}{6} = 2$$ 6. **Combine the fractions on the left:** $$\frac{4x - x + 3x - 2(x+2)}{6} = 2$$ 7. **Simplify numerator:** $$4x - x + 3x - 2x - 4 = (4x - x + 3x - 2x) - 4 = (4x - x + 3x - 2x) - 4 = 4x$$ Calculate stepwise: $$4x - x = 3x$$ $$3x + 3x = 6x$$ $$6x - 2x = 4x$$ So numerator is $$4x - 4$$ 8. **Rewrite equation:** $$\frac{4x - 4}{6} = 2$$ 9. **Multiply both sides by 6 to clear denominator:** $$\cancel{6} \times \frac{4x - 4}{\cancel{6}} = 2 \times 6$$ $$4x - 4 = 12$$ 10. **Add 4 to both sides:** $$4x - 4 + 4 = 12 + 4$$ $$4x = 16$$ 11. **Divide both sides by 4:** $$\frac{\cancel{4}x}{\cancel{4}} = \frac{16}{4}$$ $$x = 4$$ **Final answer:** $$x = 4$$