1. **State the problem:** Solve the equation $$\frac{2X+5}{2} = \frac{3X}{5} + \frac{1}{4}$$ for $X$. 2. **Identify the formula and rules:** To solve for $X$, we need to eliminate the fractions by finding a common denominator and then isolate $X$. 3. **Find the least common denominator (LCD):** The denominators are 2, 5, and 4. The LCD is 20. 4. **Multiply both sides of the equation by 20 to clear denominators:** $$20 \times \frac{2X+5}{2} = 20 \times \frac{3X}{5} + 20 \times \frac{1}{4}$$ 5. **Simplify each term:** $$10(2X+5) = 4(3X) + 5(1)$$ 6. **Distribute:** $$20X + 50 = 12X + 5$$ 7. **Bring like terms together:** $$20X - 12X = 5 - 50$$ 8. **Simplify:** $$8X = -45$$ 9. **Solve for $X$:** $$X = \frac{-45}{8}$$ **Final answer:** $$X = -\frac{45}{8}$$