1. **State the problem:** Solve the equation $$\frac{1-\sqrt{2}}{2-\sqrt{2}} = \frac{\sqrt{3}x}{3\sqrt{3}}$$ and express the answer in surd form. 2. **Simplify the right side:** Since $$\frac{\sqrt{3}x}{3\sqrt{3}} = \frac{x}{3}$$ (because $$\frac{\sqrt{3}}{3\sqrt{3}} = \frac{1}{3}$$), the equation becomes: $$\frac{1-\sqrt{2}}{2-\sqrt{2}} = \frac{x}{3}$$ 3. **Rationalize the denominator on the left side:** Multiply numerator and denominator by the conjugate of the denominator $$2 + \sqrt{2}$$: $$\frac{(1-\sqrt{2})(2 + \sqrt{2})}{(2-\sqrt{2})(2+\sqrt{2})} = \frac{x}{3}$$ 4. **Calculate the denominator:** $$(2-\sqrt{2})(2+\sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2$$ 5. **Calculate the numerator:** $$(1-\sqrt{2})(2 + \sqrt{2}) = 1 \times 2 + 1 \times \sqrt{2} - \sqrt{2} \times 2 - \sqrt{2} \times \sqrt{2} = 2 + \sqrt{2} - 2\sqrt{2} - 2 = (2 - 2) + (\sqrt{2} - 2\sqrt{2}) = 0 - \sqrt{2} = -\sqrt{2}$$ 6. **Substitute back:** $$\frac{-\sqrt{2}}{2} = \frac{x}{3}$$ 7. **Solve for $$x$$:** Multiply both sides by 3: $$x = 3 \times \left(-\frac{\sqrt{2}}{2}\right) = -\frac{3\sqrt{2}}{2}$$ **Final answer:** $$x = -\frac{3\sqrt{2}}{2}$$