An inertia term is introduced in the AUSM +-up scheme. The resulting scheme, called AUSM-IT (IT for Inertia Term), is designed as an extension of the AUSM +-up scheme allowing for full Mach number range calculations of unsteady flows including acoustic features. In line with the continuous asymptotic analysis, the AUSM-IT scheme satisfies the conservation of the discrete linear acoustic energy at first order in the low Mach number limit. Its capability to properly handle low Mach number unsteady flows, that may include acoustic waves or discontinuities, is numerically illustrated. The approach for building the AUSM-IT scheme from the AUSM +-up scheme is applicable to any other Godunov-type scheme.