To subtract two fractions, they need to have a common denominator.
The common denominator of 3 and 2 is 6, so we can write:
\begin{align*}
\frac{2x-1}{3}-\frac{x+3}{2} &= \frac{2(2x-1)}{2 \cdot 3}-\frac{3(x+3)}{3 \cdot 2} \\
&= \frac{4x-2}{6}-\frac{3x+9}{6} \\
&= \frac{4x-3x-2-9}{6} \\
&= \frac{x-11}{6}
\end{align*}
Therefore, the answer is \(\frac{x-11}{6}\).