1. **State the problem:** Solve the equation $$\frac{x + 1}{2} = 2 - \frac{x + 2}{7}$$. 2. **Write down the equation:** $$\frac{x + 1}{2} = 2 - \frac{x + 2}{7}$$. 3. **Goal:** Find the value of $x$ that satisfies this equation. 4. **Eliminate the fractions by finding the least common denominator (LCD):** The denominators are 2 and 7, so the LCD is 14. 5. **Multiply both sides of the equation by 14:** $$14 \times \frac{x + 1}{2} = 14 \times \left(2 - \frac{x + 2}{7}\right)$$ 6. **Simplify each term:** $$7(x + 1) = 14 \times 2 - 2(x + 2)$$ 7. **Distribute:** $$7x + 7 = 28 - 2x - 4$$ 8. **Simplify the right side:** $$7x + 7 = 24 - 2x$$ 9. **Add $2x$ to both sides to get all $x$ terms on one side:** $$7x + 2x + 7 = 24 - \cancel{2x} + 2x$$ $$9x + 7 = 24$$ 10. **Subtract 7 from both sides:** $$9x + 7 - 7 = 24 - 7$$ $$9x = 17$$ 11. **Divide both sides by 9 to solve for $x$:** $$\frac{\cancel{9}x}{\cancel{9}} = \frac{17}{9}$$ $$x = \frac{17}{9}$$ **Final answer:** $$x = \frac{17}{9}$$