1. **State the problem:** Solve the equation $$\frac{1}{x} = \frac{1}{5x} - \frac{x+3}{x^2}$$ for $x$. 2. **Identify the common denominator:** The denominators are $x$, $5x$, and $x^2$. The least common denominator (LCD) is $$5x^2$$. 3. **Multiply both sides of the equation by the LCD to clear denominators:** $$5x^2 \times \frac{1}{x} = 5x^2 \times \frac{1}{5x} - 5x^2 \times \frac{x+3}{x^2}$$ 4. **Simplify each term:** $$5x^2 \times \frac{1}{x} = 5x$$ $$5x^2 \times \frac{1}{5x} = x$$ $$5x^2 \times \frac{x+3}{x^2} = 5(x+3)$$ So the equation becomes: $$5x = x - 5(x+3)$$ 5. **Expand and simplify the right side:** $$5x = x - 5x - 15$$ 6. **Combine like terms on the right:** $$5x = -4x - 15$$ 7. **Add $4x$ to both sides to collect $x$ terms on the left:** $$5x + 4x = -15$$ $$9x = -15$$ 8. **Divide both sides by 9 to solve for $x$:** $$\cancel{9}x = \frac{-15}{\cancel{9}}$$ $$x = -\frac{15}{9}$$ 9. **Simplify the fraction:** $$x = -\frac{5}{3}$$ **Final answer:** $$x = -\frac{5}{3}$$