1. **State the problem:** Given $x = 3 + \sqrt{8}$, find the value of $x - \frac{1}{x}$. 2. **Recall the formula and rules:** To find $x - \frac{1}{x}$, we need to calculate $\frac{1}{x}$ and then subtract it from $x$. 3. **Calculate $\frac{1}{x}$:** $$\frac{1}{x} = \frac{1}{3 + \sqrt{8}}$$ To rationalize the denominator, multiply numerator and denominator by the conjugate $3 - \sqrt{8}$: $$\frac{1}{3 + \sqrt{8}} \times \frac{3 - \sqrt{8}}{3 - \sqrt{8}} = \frac{3 - \sqrt{8}}{(3)^2 - (\sqrt{8})^2} = \frac{3 - \sqrt{8}}{9 - 8} = 3 - \sqrt{8}$$ 4. **Substitute back:** $$x - \frac{1}{x} = (3 + \sqrt{8}) - (3 - \sqrt{8})$$ 5. **Simplify:** $$= 3 + \sqrt{8} - 3 + \sqrt{8} = 2\sqrt{8}$$ 6. **Simplify the radical:** $$2\sqrt{8} = 2 \times 2\sqrt{2} = 4\sqrt{2}$$ **Final answer:** $$x - \frac{1}{x} = 4\sqrt{2}$$