Rationalize the denominator and simplify. \( \frac{2 \sqrt{x y}-3 \sqrt{y}}{\sqrt{x}}, x>0, y \geq 0 \) \( -\sqrt{x y} \) \( \frac{2 x \sqrt{y}-3 \sqrt{x y}}{x} \) \( \frac{-\sqrt{x}}{x} \) \( \frac{2 x-\sqrt{y}}{x} \)

To rationalize the denominator of \( \frac{2 \sqrt{x y}-3 \sqrt{y}}{\sqrt{x}} \), you can multiply both the numerator and the denominator by \( \sqrt{x} \). This results in: \[ \frac{(2 \sqrt{x y}-3 \sqrt{y}) \sqrt{x}}{\sqrt{x} \cdot \sqrt{x}} = \frac{2 \sqrt{x^2 y} - 3 \sqrt{x y}}{x} \] This simplifies to: \[ \frac{2 x \sqrt{y} - 3 \sqrt{x y}}{x} \] Now we can express the fraction in simplified form: \[ \frac{2 x \sqrt{y} - 3 \sqrt{x y}}{x} \] This matches the earlier option of \( \frac{2 x \sqrt{y} - 3 \sqrt{x y}}{x} \). So the next step would be to confirm that all the calculations remain concise, remembering to avoid knotted mistakes which often occur during fractions and square roots. Simplifying rational expressions might seem trivial, but careful step-by-step handling will save a lot of time!