1. **Stating the problem:** Solve the equation $$\frac{1}{x} + a = \frac{1}{b}$$ for $x$. 2. **Formula and rules:** We want to isolate $x$. The equation involves fractions and addition. To solve for $x$, we will first isolate $\frac{1}{x}$ and then invert it. 3. **Step-by-step solution:** Start with the equation: $$\frac{1}{x} + a = \frac{1}{b}$$ Subtract $a$ from both sides: $$\frac{1}{x} = \frac{1}{b} - a$$ Rewrite the right side with a common denominator $b$: $$\frac{1}{x} = \frac{1}{b} - \frac{a b}{b} = \frac{1 - a b}{b}$$ Now, take the reciprocal of both sides to solve for $x$: $$x = \frac{b}{1 - a b}$$ Show the cancellation step explicitly (if any): No common factors to cancel here. 4. **Final answer:** $$\boxed{x = \frac{b}{1 - a b}}$$ This means $x$ is equal to $b$ divided by $1$ minus $a$ times $b$. This completes the solution.