1. **Solve the equation:** $\frac{1}{2x} = \frac{5}{4x} - \frac{1}{x^2}$. 2. The LCD (Least Common Denominator) for the terms is $4x^2$. 3. Multiply every term by $4x^2$ to clear denominators: $$4x^2 \times \frac{1}{2x} = 4x^2 \times \frac{5}{4x} - 4x^2 \times \frac{1}{x^2}$$ 4. Simplify each term: $$4x^2 \times \frac{1}{2x} = 2x$$ $$4x^2 \times \frac{5}{4x} = 5x$$ $$4x^2 \times \frac{1}{x^2} = 4$$ 5. Substitute back: $$2x = 5x - 4$$ 6. Rearrange to isolate $x$: $$2x - 5x = -4$$ $$\cancel{2x} - 5x = -4$$ $$-3x = -4$$ 7. Divide both sides by $-3$: $$x = \frac{-4}{-3} = \frac{4}{3}$$ **Final answer:** $x = \frac{4}{3}$