1. **State the problem:** Solve the equation $$\frac{4}{3} - x + \frac{7}{6} - \frac{3x + 1}{2} = -\frac{7}{9}x + \frac{3 - x}{3}$$ for $x$. 2. **Combine like terms and find a common denominator:** The denominators are 3, 6, 2, 9, and 3. The least common denominator (LCD) is 18. 3. **Multiply every term by 18 to clear denominators:** $$18 \times \left(\frac{4}{3} - x + \frac{7}{6} - \frac{3x + 1}{2}\right) = 18 \times \left(-\frac{7}{9}x + \frac{3 - x}{3}\right)$$ 4. **Calculate each term:** $$18 \times \frac{4}{3} = 24$$ $$18 \times (-x) = -18x$$ $$18 \times \frac{7}{6} = 21$$ $$18 \times \frac{-(3x + 1)}{2} = -9(3x + 1) = -27x - 9$$ $$18 \times -\frac{7}{9}x = -14x$$ $$18 \times \frac{3 - x}{3} = 6(3 - x) = 18 - 6x$$ 5. **Rewrite the equation:** $$24 - 18x + 21 - 27x - 9 = -14x + 18 - 6x$$ 6. **Simplify both sides:** Left side: $$24 + 21 - 9 - 18x - 27x = 36 - 45x$$ Right side: $$-14x - 6x + 18 = -20x + 18$$ 7. **Set the equation:** $$36 - 45x = -20x + 18$$ 8. **Add $45x$ to both sides:** $$36 = 25x + 18$$ 9. **Subtract 18 from both sides:** $$36 - 18 = 25x$$ $$18 = 25x$$ 10. **Divide both sides by 25:** $$x = \frac{18}{25}$$ 11. **Check for simplification:** 18 and 25 have no common factors other than 1, so the fraction is in simplest form. **Final answer:** $$x = \frac{18}{25}$$