  We investigate the behavior of the maximal violations of the CHSH inequality
and V$\grave{e}$rtesi's inequality under the local filtering operations. An
analytical method has been presented for general two-qubit systems to compute
the maximal violation of the CHSH inequality and the lower bound of the maximal
violation of V$\acute{e}$rtesi's inequality over the local filtering
operations. We show by examples that there exist quantum states whose
non-locality can be revealed after local filtering operation by the
V$\acute{e}$rtesi's inequality instead of the CHSH inequality.
