1. **State the problem:** Solve for $x$ in the equation $$\frac{x-3}{3} = \frac{4 - x - 2}{2}.$$\n\n2. **Simplify the right side:** Combine like terms inside the numerator on the right side: $$4 - x - 2 = 2 - x.$$ So the equation becomes $$\frac{x-3}{3} = \frac{2 - x}{2}.$$\n\n3. **Use cross multiplication to eliminate denominators:** Multiply both sides by 6 (the least common multiple of 3 and 2) to clear fractions: $$6 \times \frac{x-3}{3} = 6 \times \frac{2 - x}{2}.$$\n\n4. **Simplify both sides:** $$2(x-3) = 3(2 - x).$$\n\n5. **Distribute:** $$2x - 6 = 6 - 3x.$$\n\n6. **Add $3x$ to both sides:** $$2x + 3x - 6 = 6 - \cancel{3x} + 3x \Rightarrow 5x - 6 = 6.$$\n\n7. **Add 6 to both sides:** $$5x - 6 + 6 = 6 + 6 \Rightarrow 5x = 12.$$\n\n8. **Divide both sides by 5:** $$\frac{\cancel{5}x}{\cancel{5}} = \frac{12}{5} \Rightarrow x = \frac{12}{5}.$$\n\n**Final answer:** $$x = \frac{12}{5}.$$