If $\frac{\sqrt{x}}{\sqrt{x-1}}=\frac32$, solve for $x$. Express your answer in simplest fractional form.
We can start by cross-multiplying: \begin{align*} 3\sqrt{x-1}&=2\sqrt{x}
\\\Rightarrow \qquad (3\sqrt{x-1})^2 &=(2\sqrt{x})^2
\\\Rightarrow \qquad 9(x-1)& =4(x)
\\\Rightarrow \qquad 9x-9& =4x
\\ \Rightarrow \qquad5x&=9
\\ \Rightarrow \qquad x&=\boxed{\frac9{5}}.
\end{align*}Checking, we see that this value of $x$ does indeed work, so it is not an extraneous solution.