1. **State the problem:** Evaluate the definite integral $$\int_1^9 \frac{x - 1}{\sqrt{x}} \, dx$$ which is rewritten as $$\int_1^9 \left(x^{\frac{1}{2}} - x^{-\frac{1}{2}}\right) \, dx$$. 2. **Recall the integral formula:** For any real number $n \neq -1$, $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$. 3. **Apply the formula to each term:** $$\int_1^9 x^{\frac{1}{2}} \, dx = \left[ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right]_1^9 = \left[ \frac{2}{3} x^{\frac{3}{2}} \right]_1^9$$ $$\int_1^9 x^{-\frac{1}{2}} \, dx = \left[ \frac{x^{\frac{1}{2}}}{\frac{1}{2}} \right]_1^9 = \left[ 2 x^{\frac{1}{2}} \right]_1^9$$ 4. **Evaluate each definite integral:** $$\left[ \frac{2}{3} x^{\frac{3}{2}} \right]_1^9 = \frac{2}{3} (9^{\frac{3}{2}} - 1^{\frac{3}{2}}) = \frac{2}{3} (27 - 1) = \frac{2}{3} \times 26 = \frac{52}{3}$$ $$\left[ 2 x^{\frac{1}{2}} \right]_1^9 = 2 (9^{\frac{1}{2}} - 1^{\frac{1}{2}}) = 2 (3 - 1) = 2 \times 2 = 4$$ 5. **Combine the results:** $$\int_1^9 \left(x^{\frac{1}{2}} - x^{-\frac{1}{2}}\right) \, dx = \frac{52}{3} - 4 = \frac{52}{3} - \frac{12}{3} = \frac{40}{3}$$ **Final answer:** $$\boxed{\frac{40}{3}}$$