jmc

The inverse of $5(\mathrm{mod}7)$ is 3, since $5\cdot 3\equiv 1(\mathrm{mod}7)$. Also, inverse of $2(\mathrm{mod}7)$ is 4, since $2\cdot 4\equiv 1(\mathrm{mod}7)$. Finally, the inverse of $3(\mathrm{mod}7)$ is 5 (again because $5\cdot 3\equiv 1(\mathrm{mod}7)$). So the residue of $2^{-1}+3^{-1}$ is the residue of $4+5(\mathrm{mod}7)$, which is $2$. Thus $L-R=3-2=\boxed{1}$. Since the left-hand side $L$ and the right-hand side $R$ of the equation $$(a+b{)}^{-1}\stackrel{?}{=}{a}^{-1}+b^{-1}(\mathrm{mod}m)$$are not equal, we may conclude that the equation is not true in general.