1. **State the problem:** Simplify the expression $$\frac{3}{x-2} - \frac{2}{x-3}$$. 2. **Formula and rules:** To subtract fractions, find a common denominator and combine the numerators. 3. **Find the common denominator:** The denominators are $x-2$ and $x-3$, so the common denominator is $$(x-2)(x-3)$$. 4. **Rewrite each fraction with the common denominator:** $$\frac{3}{x-2} = \frac{3(x-3)}{(x-2)(x-3)}$$ $$\frac{2}{x-3} = \frac{2(x-2)}{(x-2)(x-3)}$$ 5. **Subtract the numerators:** $$\frac{3(x-3)}{(x-2)(x-3)} - \frac{2(x-2)}{(x-2)(x-3)} = \frac{3(x-3) - 2(x-2)}{(x-2)(x-3)}$$ 6. **Expand the numerators:** $$3(x-3) = 3x - 9$$ $$2(x-2) = 2x - 4$$ 7. **Substitute back and simplify:** $$\frac{3x - 9 - (2x - 4)}{(x-2)(x-3)} = \frac{3x - 9 - 2x + 4}{(x-2)(x-3)} = \frac{(3x - 2x) + (-9 + 4)}{(x-2)(x-3)} = \frac{x - 5}{(x-2)(x-3)}$$ 8. **Final answer:** $$\boxed{\frac{x - 5}{(x-2)(x-3)}}$$ This matches option B.