1. **State the problem:** Solve the equation $$\frac{5x + 1}{2} - \frac{x - 2}{6} = \frac{2x + 4}{3}$$ for $x$. 2. **Identify the formula and rules:** To solve equations with fractions, find a common denominator to eliminate the fractions by multiplying through. 3. **Find the least common denominator (LCD):** The denominators are 2, 6, and 3. The LCD is 6. 4. **Multiply every term by 6 to clear denominators:** $$6 \times \frac{5x + 1}{2} - 6 \times \frac{x - 2}{6} = 6 \times \frac{2x + 4}{3}$$ 5. **Simplify each term:** $$3(5x + 1) - (x - 2) = 2(2x + 4)$$ 6. **Distribute:** $$15x + 3 - x + 2 = 4x + 8$$ 7. **Combine like terms on the left:** $$14x + 5 = 4x + 8$$ 8. **Subtract $4x$ from both sides:** $$14x - 4x + 5 = 8$$ $$10x + 5 = 8$$ 9. **Subtract 5 from both sides:** $$10x = 3$$ 10. **Divide both sides by 10:** $$x = \frac{3}{10}$$ **Final answer:** $$x = \frac{3}{10}$$