1. **State the problem:** Solve the equation $$\frac{x}{x+2} + \frac{1}{x} = 1$$ and determine the nature of its solutions (valid or extraneous). 2. **Identify the domain restrictions:** The denominators cannot be zero, so: $$x+2 \neq 0 \Rightarrow x \neq -2$$ $$x \neq 0$$ 3. **Find a common denominator and combine terms:** The common denominator is $$x(x+2)$$. Multiply both sides by $$x(x+2)$$ to clear denominators: $$x \cdot x + 1 \cdot (x+2) = 1 \cdot x(x+2)$$ which simplifies to: $$x^2 + x + 2 = x^2 + 2x$$ 4. **Simplify the equation:** Subtract $$x^2$$ from both sides: $$\cancel{x^2} + x + 2 = \cancel{x^2} + 2x$$ which gives: $$x + 2 = 2x$$ 5. **Solve for $$x$$:** $$x + 2 = 2x$$ Subtract $$x$$ from both sides: $$\cancel{x} + 2 = \cancel{x} + x$$ which simplifies to: $$2 = x$$ 6. **Check for extraneous solutions:** The solution $$x=2$$ is not excluded by the domain restrictions ($$x \neq 0$$ and $$x \neq -2$$), so it is valid. 7. **Conclusion:** The equation has one valid solution ($$x=2$$) and no extraneous solutions. **Final answer:** Option C. The equation has one valid solution and no extraneous solutions.