Simplify the given value of x

Express $x = 3 + 2\sqrt{2}$ as a perfect square.

$x = (\sqrt{2})^2 + 1^2 + 2(\sqrt{2})(1) = (\sqrt{2} + 1)^2$

Calculate the square root of x

Taking the square root on both sides, we get the value of $\sqrt{x}$.

$\sqrt{x} = \sqrt{(\sqrt{2} + 1)^2} = \sqrt{2} + 1$

Substitute into the expression

Substitute $x = 3 + 2\sqrt{2}$ and $\sqrt{x} = \sqrt{2} + 1$ into $\frac{x-1}{\sqrt{x}}$.

$\frac{3 + 2\sqrt{2} - 1}{\sqrt{2} + 1} = \frac{2 + 2\sqrt{2}}{\sqrt{2} + 1}$

Final simplification

Factor out 2 from the numerator to simplify.

$\frac{2(1 + \sqrt{2})}{\sqrt{2} + 1} = 2$