jmc

If $1<2x<2$, then, dividing all the expressions in these inequalities by $2$, we have $\frac{1}{2}<x<1$.

If $1<3x<2$, then, dividing all the expressions by $3$, we have $\frac{1}{3}<x<\frac{2}{3}$.

Given that $x$ satisfies both inequalities, we must have $\frac{1}{2}<x<\frac{2}{3}$. In interval notation, the set of common solutions is $\boxed{\left(\frac{1}{2},\frac{2}{3}\right)}$.