1. **Solve the equation:** $$4 - \frac{x + 1}{2} = \frac{2 - x}{7}$$ 2. **State the problem:** We want to find the value of $x$ that satisfies the equation. 3. **Use the formula:** To solve equations with fractions, multiply both sides by the least common denominator (LCD) to clear fractions. 4. **Find the LCD:** The denominators are 2 and 7, so $$\text{LCD} = 14$$. 5. **Multiply both sides by 14:** $$14 \times \left(4 - \frac{x + 1}{2}\right) = 14 \times \frac{2 - x}{7}$$ 6. **Distribute:** $$14 \times 4 - 14 \times \frac{x + 1}{2} = 14 \times \frac{2 - x}{7}$$ $$56 - 7(x + 1) = 2(2 - x)$$ 7. **Expand terms:** $$56 - 7x - 7 = 4 - 2x$$ 8. **Simplify left side:** $$49 - 7x = 4 - 2x$$ 9. **Bring variables to one side and constants to the other:** $$49 - 7x + 2x = 4$$ $$49 - 5x = 4$$ 10. **Subtract 49 from both sides:** $$\cancel{49} - 5x = 4 - \cancel{49}$$ $$-5x = -45$$ 11. **Divide both sides by -5:** $$\frac{-5x}{\cancel{-5}} = \frac{-45}{\cancel{-5}}$$ $$x = 9$$ **Final answer:** $$x = 9$$