1. **State the problem:** Solve the equation $$\frac{x - 4}{2} = \frac{x + 3}{1}$$ for $x$. 2. **Formula and rules:** To solve equations with fractions, multiply both sides by the least common denominator (LCD) to eliminate fractions. 3. **Find the LCD:** The denominators are 2 and 1, so the LCD is 2. 4. **Multiply both sides by 2:** $$2 \times \frac{x - 4}{2} = 2 \times \frac{x + 3}{1}$$ 5. **Simplify by canceling denominators:** $$\cancel{2} \times \frac{x - 4}{\cancel{2}} = 2(x + 3)$$ 6. **Simplify left side:** $$x - 4 = 2(x + 3)$$ 7. **Distribute on the right side:** $$x - 4 = 2x + 6$$ 8. **Bring all terms involving $x$ to one side and constants to the other:** $$x - 2x = 6 + 4$$ 9. **Simplify both sides:** $$-x = 10$$ 10. **Divide both sides by -1:** $$\frac{-x}{-1} = \frac{10}{-1}$$ 11. **Simplify:** $$x = -10$$ **Final answer:** $x = -10$