1. **State the problem:** Solve the equation $$\frac{2x-5}{3} - \frac{x}{2} = \frac{4-x}{6}$$ for $x$. 2. **Identify the formula and rules:** To solve equations with fractions, find a common denominator to clear the fractions by multiplying both sides. 3. **Find the least common denominator (LCD):** The denominators are 3, 2, and 6. The LCD is 6. 4. **Multiply both sides of the equation by 6 to clear denominators:** $$6 \times \left(\frac{2x-5}{3} - \frac{x}{2}\right) = 6 \times \frac{4-x}{6}$$ 5. **Distribute multiplication:** $$6 \times \frac{2x-5}{3} - 6 \times \frac{x}{2} = 4 - x$$ 6. **Simplify each term:** $$2 \times (2x - 5) - 3 \times x = 4 - x$$ 7. **Expand terms:** $$4x - 10 - 3x = 4 - x$$ 8. **Combine like terms on the left:** $$x - 10 = 4 - x$$ 9. **Add $x$ to both sides:** $$x + x - 10 = 4 - x + x$$ $$2x - 10 = 4$$ 10. **Add 10 to both sides:** $$2x - 10 + 10 = 4 + 10$$ $$2x = 14$$ 11. **Divide both sides by 2:** $$\frac{\cancel{2}x}{\cancel{2}} = \frac{14}{2}$$ $$x = 7$$ **Final answer:** $$x = 7$$