1. Stating the problem: Simplify the expression $$\frac{3\sqrt{42}}{2\sqrt{6}} - \frac{\sqrt{35}}{\sqrt{5}}$$. 2. Simplify each term separately. For the first term: $$\frac{3\sqrt{42}}{2\sqrt{6}} = \frac{3}{2} \cdot \frac{\sqrt{42}}{\sqrt{6}} = \frac{3}{2} \cdot \sqrt{\frac{42}{6}}$$ Since $$\frac{42}{6} = 7$$, we have: $$\frac{3}{2} \cdot \sqrt{7} = \frac{3\sqrt{7}}{2}$$ 3. For the second term: $$\frac{\sqrt{35}}{\sqrt{5}} = \sqrt{\frac{35}{5}}$$ Since $$\frac{35}{5} = 7$$, we get: $$\sqrt{7}$$ 4. Now the expression is: $$\frac{3\sqrt{7}}{2} - \sqrt{7}$$ Rewrite $$\sqrt{7}$$ as $$\frac{2\sqrt{7}}{2}$$ to have a common denominator: $$\frac{3\sqrt{7}}{2} - \frac{2\sqrt{7}}{2} = \frac{3\sqrt{7} - 2\sqrt{7}}{2}$$ 5. Combine like terms: $$\frac{(3 - 2)\sqrt{7}}{2} = \frac{\sqrt{7}}{2}$$ Final answer: $$\boxed{\frac{\sqrt{7}}{2}}$$