1. Stating the problem: Solve the equation $$\frac{\sqrt{8}}{\sqrt{x-1}} - \sqrt{2} = \sqrt{2}$$ for $x$. 2. Simplify and isolate terms: Note that $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$. The equation becomes: $$\frac{2\sqrt{2}}{\sqrt{x-1}} - \sqrt{2} = \sqrt{2}$$ 3. Add $\sqrt{2}$ to both sides to isolate the fraction: $$\frac{2\sqrt{2}}{\sqrt{x-1}} = \sqrt{2} + \sqrt{2} = 2\sqrt{2}$$ 4. Multiply both sides by $\sqrt{x-1}$ to get rid of the denominator: $$2\sqrt{2} = 2\sqrt{2} \cdot \sqrt{x-1}$$ 5. Divide both sides by $2\sqrt{2}$: $$1 = \sqrt{x-1}$$ 6. Square both sides to eliminate the square root: $$1^2 = (\sqrt{x-1})^2 \implies 1 = x - 1$$ 7. Solve for $x$: $$x = 1 + 1 = 2$$ 8. Check the solution by substituting back into the original equation to verify no extraneous solutions. Final answer: $$\boxed{2}$$