1. **State the problem:** Solve the equation $$\frac{x - 2}{4} + \frac{x + 1}{2} = 1$$. 2. **Formula and rules:** To solve equations with fractions, find a common denominator to combine terms or clear fractions by multiplying both sides by the least common denominator (LCD). 3. **Find the LCD:** The denominators are 4 and 2. The LCD is 4. 4. **Multiply both sides by the LCD to clear fractions:** $$4 \times \left(\frac{x - 2}{4} + \frac{x + 1}{2}\right) = 4 \times 1$$ 5. **Distribute multiplication:** $$4 \times \frac{x - 2}{4} + 4 \times \frac{x + 1}{2} = 4$$ 6. **Simplify each term:** $$\cancel{4} \times \frac{x - 2}{\cancel{4}} + 2 \times (x + 1) = 4$$ 7. **Rewrite:** $$x - 2 + 2(x + 1) = 4$$ 8. **Distribute 2:** $$x - 2 + 2x + 2 = 4$$ 9. **Combine like terms:** $$3x + 0 = 4$$ 10. **Simplify:** $$3x = 4$$ 11. **Divide both sides by 3:** $$\frac{3x}{\cancel{3}} = \frac{4}{\cancel{3}}$$ 12. **Final solution:** $$x = \frac{4}{3}$$