Unstable waves on the equitorial beta-plane are studied when the mean flow U(y) is a function of latitude only, which permits separation of variables. The additional restrictions of (i) a linear shear U(y)=Sy where S is a constant and y is a one-dimensional latitude and (ii) small zonal wavenumber k permit simple approximations that allow identification of two distinct modes of instability. The ''mixed'' Kelvin--inertial'' mode is an unstable Kelvin wave for ‖S‖2. In contrast to normal hydrodynamics problems, there is no minimum shear for instability but instead Im(c) is proportional to exp[-5.34/‖S‖> for small S. Although the inertial instability was known previously, the two branches of unstable Kelvin waves and the correction between them and the neutral and inertially unstable gravity waves are original discoveries. For weak shear, only the Kelvin waves is unstable.