1. **State the problem:** Solve the equation $$\frac{1}{1 - \sqrt{x}} = 1 - \frac{\sqrt{x}}{\sqrt{x} - 1}$$ for $x$. 2. **Rewrite the equation:** Notice that $\sqrt{x} - 1 = -(1 - \sqrt{x})$. This will help simplify the right side. 3. **Simplify the right side:** $$1 - \frac{\sqrt{x}}{\sqrt{x} - 1} = 1 - \frac{\sqrt{x}}{-(1 - \sqrt{x})} = 1 + \frac{\sqrt{x}}{1 - \sqrt{x}}$$ 4. **Rewrite the equation with this simplification:** $$\frac{1}{1 - \sqrt{x}} = 1 + \frac{\sqrt{x}}{1 - \sqrt{x}}$$ 5. **Bring all terms to a common denominator:** $$1 = \frac{1 - \sqrt{x}}{1 - \sqrt{x}}$$, so $$1 + \frac{\sqrt{x}}{1 - \sqrt{x}} = \frac{1 - \sqrt{x}}{1 - \sqrt{x}} + \frac{\sqrt{x}}{1 - \sqrt{x}} = \frac{1 - \sqrt{x} + \sqrt{x}}{1 - \sqrt{x}} = \frac{1}{1 - \sqrt{x}}$$ 6. **So the right side simplifies to:** $$\frac{1}{1 - \sqrt{x}}$$ 7. **Therefore, the equation becomes:** $$\frac{1}{1 - \sqrt{x}} = \frac{1}{1 - \sqrt{x}}$$ 8. **This is an identity for all $x$ where the expressions are defined.** 9. **Domain restrictions:** - Denominator $1 - \sqrt{x} \neq 0 \Rightarrow \sqrt{x} \neq 1 \Rightarrow x \neq 1$ - Denominator $\sqrt{x} - 1 \neq 0 \Rightarrow x \neq 1$ 10. **Final solution:** All real $x \geq 0$ except $x = 1$ satisfy the equation. **Answer:** $$\boxed{\{x \in \mathbb{R} : x \geq 0, x \neq 1\}}$$