1. **State the problem:** Solve the equation $$\frac{4x - 3}{4} - \frac{2x - 1}{2} - \frac{31}{12} = \frac{2(4x - 2)}{6} + \frac{1}{3}x - \frac{1}{2}$$ 2. **Identify the least common denominator (LCD):** The denominators are 4, 2, 12, 6, 3, and 2. The LCD is 12. 3. **Multiply both sides of the equation by 12 to clear denominators:** $$12 \times \left(\frac{4x - 3}{4} - \frac{2x - 1}{2} - \frac{31}{12}\right) = 12 \times \left(\frac{2(4x - 2)}{6} + \frac{1}{3}x - \frac{1}{2}\right)$$ 4. **Simplify each term:** $$12 \times \frac{4x - 3}{4} = 3(4x - 3) = 12x - 9$$ $$12 \times \frac{2x - 1}{2} = 6(2x - 1) = 12x - 6$$ $$12 \times \frac{31}{12} = 31$$ $$12 \times \frac{2(4x - 2)}{6} = 2 \times 2(4x - 2) = 4(4x - 2) = 16x - 8$$ $$12 \times \frac{1}{3}x = 4x$$ $$12 \times \frac{1}{2} = 6$$ 5. **Rewrite the equation with these simplifications:** $$ (12x - 9) - (12x - 6) - 31 = (16x - 8) + 4x - 6 $$ 6. **Simplify left side:** $$12x - 9 - 12x + 6 - 31 = -34$$ 7. **Simplify right side:** $$16x - 8 + 4x - 6 = 20x - 14$$ 8. **Set the simplified sides equal:** $$-34 = 20x - 14$$ 9. **Add 14 to both sides:** $$-34 + 14 = 20x - 14 + 14$$ $$-20 = 20x$$ 10. **Divide both sides by 20:** $$\frac{\cancel{-20}}{\cancel{20}} = \frac{20x}{20}$$ $$-1 = x$$ **Final answer:** $$x = -1$$