1. **State the problem:** Evaluate the expression $$\frac{3}{2-\sqrt{3}} - \frac{2}{3+\sqrt{3}}$$ without using a calculator. 2. **Rationalize the denominators:** To simplify expressions with square roots in the denominator, multiply numerator and denominator by the conjugate of the denominator. 3. For the first term: $$\frac{3}{2-\sqrt{3}} \times \frac{2+\sqrt{3}}{2+\sqrt{3}} = \frac{3(2+\sqrt{3})}{(2)^2 - (\sqrt{3})^2} = \frac{3(2+\sqrt{3})}{4 - 3} = \frac{3(2+\sqrt{3})}{1} = 6 + 3\sqrt{3}$$ 4. For the second term: $$\frac{2}{3+\sqrt{3}} \times \frac{3-\sqrt{3}}{3-\sqrt{3}} = \frac{2(3-\sqrt{3})}{(3)^2 - (\sqrt{3})^2} = \frac{2(3-\sqrt{3})}{9 - 3} = \frac{2(3-\sqrt{3})}{6} = \frac{6 - 2\sqrt{3}}{6} = 1 - \frac{\sqrt{3}}{3}$$ 5. **Combine the two results:** $$\left(6 + 3\sqrt{3}\right) - \left(1 - \frac{\sqrt{3}}{3}\right) = 6 + 3\sqrt{3} - 1 + \frac{\sqrt{3}}{3} = 5 + \left(3\sqrt{3} + \frac{\sqrt{3}}{3}\right)$$ 6. **Simplify the radical terms:** $$3\sqrt{3} + \frac{\sqrt{3}}{3} = \sqrt{3} \left(3 + \frac{1}{3}\right) = \sqrt{3} \times \frac{10}{3} = \frac{10}{3} \sqrt{3}$$ 7. **Final answer:** $$5 + \frac{10}{3} \sqrt{3}$$ This is the simplified form of the expression without a calculator.