We investigated the effects of asymmetry of the potential and noises on the mean first-passage time (MFPT) in two opposite directions in an asymmetric bistable system subjected to multiplicative colored and additive white noise with a correlated noise. The expression of the stationary probability distribution is obtained by means of an extended unified colored-noise approximation and the expressions of the MFPTs are derived through the steepest-descent approach. Numerical results show that the influence of the asymmetry coefficient r on T± is different and because of the correlated noise，even for r=0, T+(xs1→xs2,r) is not equal to T-(xs2→xs1,r). The curves of　lnT± versus r in both cases have mono-valleys but the mono-valleys move to the opposite directions with increasing correlation intensity λ.