1. **State the problem:** Solve the linear equation $$\frac{1}{3}(x - 1) - \frac{1}{2}(1 - x + 2) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{6}$$. 2. **Rewrite the equation clearly:** $$\frac{1}{3}(x - 1) - \frac{1}{2}(3 - x) = \frac{1}{6}$$ 3. **Distribute the fractions:** $$\frac{1}{3}x - \frac{1}{3} - \frac{1}{2} \times 3 + \frac{1}{2}x = \frac{1}{6}$$ 4. **Simplify the terms:** $$\frac{1}{3}x - \frac{1}{3} - \frac{3}{2} + \frac{1}{2}x = \frac{1}{6}$$ 5. **Combine like terms on the left:** $$\left(\frac{1}{3}x + \frac{1}{2}x\right) - \left(\frac{1}{3} + \frac{3}{2}\right) = \frac{1}{6}$$ 6. **Find common denominators and add:** $$\frac{2}{6}x + \frac{3}{6}x - \left(\frac{2}{6} + \frac{9}{6}\right) = \frac{1}{6}$$ $$\frac{5}{6}x - \frac{11}{6} = \frac{1}{6}$$ 7. **Add $\frac{11}{6}$ to both sides:** $$\frac{5}{6}x - \frac{11}{6} + \frac{11}{6} = \frac{1}{6} + \frac{11}{6}$$ $$\frac{5}{6}x = \frac{12}{6}$$ 8. **Simplify right side:** $$\frac{5}{6}x = 2$$ 9. **Divide both sides by $\frac{5}{6}$:** $$x = 2 \div \frac{5}{6} = 2 \times \frac{6}{5}$$ 10. **Simplify multiplication:** $$x = \frac{12}{5}$$ **Final answer:** $$x = \frac{12}{5}$$ or 2.4.