Velocity-resolved reverberation mapping suggests that the broad-line regions (BLRs) of AGNs can have significant net inflow. We use the STOKES radiative transfer code to show that electron and Rayleigh scattering off the BLR and torus naturally explains the blueshifted profiles of high-ionization lines and the ionization dependence of the blueshifts. This result is insensitive to the geometry of the scattering region. If correct, this model resolves the long-standing conflict between the absence of outflow implied by velocity-resolved reverberation mapping and the need for outflow if the blueshifting is the result of obscuration. The accretion rate implied by the inflow is sufficient to power the AGN. We suggest that the BLR is part of the outer accretion disk and that similar MHD processes are operating. In the scattering model the blueshifting is proportional to the accretion rate so high-accretion-rate AGNs will show greater high-ionization line blueshifts as is observed. Scattering can lead to systematically too high black hole mass estimates from the C IV line. We note many similarities between narrow-line region (NLR) and BLR blueshiftings, and suggest that NLR blueshiftings have a similar explanation. Our model explains the higher blueshifts of broad absorption line QSOs if they are more highly inclined. Rayleigh scattering from the BLR and torus could be more important in the UV than electron scattering for predominantly neutral material around AGNs. The importance of Rayleigh scattering versus electron scattering can be assessed by comparing line profiles at different wavelengths arising from the same emission-line region. ; Comment: 10 pages, 7 figures, Astrophysical Journal in press. The only changes from the previous version are to include some additional discussion of the plausibility of supersonic inflow velocities (see section 5.2) and some additional references