1. **State the problem:** Solve the equation $$x - \frac{2 - x}{3} = \frac{3}{2} - \frac{x + 1}{3}$$. 2. **Identify the goal:** We want to find the value of $x$ that makes the equation true. 3. **Clear the denominators:** Multiply every term by 6 (the least common multiple of 2 and 3) to eliminate fractions: $$6 \times \left(x - \frac{2 - x}{3}\right) = 6 \times \left(\frac{3}{2} - \frac{x + 1}{3}\right)$$ 4. **Distribute multiplication:** $$6x - 6 \times \frac{2 - x}{3} = 6 \times \frac{3}{2} - 6 \times \frac{x + 1}{3}$$ 5. **Simplify each term:** $$6x - 2(2 - x) = 3 \times 3 - 2(x + 1)$$ 6. **Expand the parentheses:** $$6x - 4 + 2x = 9 - 2x - 2$$ 7. **Combine like terms:** $$6x + 2x - 4 = 9 - 2 - 2x$$ $$8x - 4 = 7 - 2x$$ 8. **Add $2x$ to both sides:** $$8x + 2x - 4 = 7 - 2x + 2x$$ $$10x - 4 = 7$$ 9. **Add 4 to both sides:** $$10x - 4 + 4 = 7 + 4$$ $$10x = 11$$ 10. **Divide both sides by 10:** $$\frac{\cancel{10}x}{\cancel{10}} = \frac{11}{10}$$ $$x = \frac{11}{10}$$ **Final answer:** $$x = \frac{11}{10}$$ or 1.1.