1. **State the problem:** Solve the equation $$\frac{1}{2}x - \frac{x}{4} + \frac{6}{2} = -\frac{1}{9}x + \frac{3 - x}{3}$$ for $x$. 2. **Write down the equation clearly:** $$\frac{1}{2}x - \frac{x}{4} + 3 = -\frac{1}{9}x + \frac{3 - x}{3}$$ 3. **Find a common denominator to clear fractions:** The denominators are 2, 4, 9, and 3. The least common denominator (LCD) is 36. 4. **Multiply every term by 36 to eliminate fractions:** $$36 \times \left(\frac{1}{2}x\right) - 36 \times \left(\frac{x}{4}\right) + 36 \times 3 = 36 \times \left(-\frac{1}{9}x\right) + 36 \times \left(\frac{3 - x}{3}\right)$$ 5. **Calculate each term:** $$18x - 9x + 108 = -4x + 12(3 - x)$$ 6. **Simplify the right side:** $$18x - 9x + 108 = -4x + 36 - 12x$$ 7. **Combine like terms on the left:** $$9x + 108 = -4x + 36 - 12x$$ 8. **Combine like terms on the right:** $$9x + 108 = -16x + 36$$ 9. **Add $16x$ to both sides:** $$9x + 16x + 108 = 36$$ 10. **Simplify:** $$25x + 108 = 36$$ 11. **Subtract 108 from both sides:** $$25x + \cancel{108} - \cancel{108} = 36 - 108$$ $$25x = -72$$ 12. **Divide both sides by 25:** $$x = \frac{-72}{25}$$ 13. **Final answer:** $$\boxed{x = -\frac{72}{25}}$$