We study TE-wave propagation in a hollow waveguide with a graded transition from a lossy right-handed material (RHM) filling the left-hand half of the waveguide to the impedance-matched lossy left-handed material (LHM) filling the right-hand half of the waveguide. The transition between the two media is graded along the direction perpendicular to the boundary between the two materials (chosen to be the z-direction), and the permittivity epsilon(omega, z) and permeability mu(omega, z) are chosen to vary according to hyperbolic tangent functions along the z-direction. We obtain exact analytical solutions to Maxwell's equations for lossy media, and the solutions for the field components confirm the expected properties of RHM-LHM structures. Thereafter, a numerical study of the wave propagation over an impedance-matched graded RHM-LHM interface is performed, using COMSOL software. The numerical study shows an excellent agreement between the numerical simulations and analytical results. Compared to other solution methods, the present approach has the advantage of being able to model more realistic smooth transitions between different materials. However, in the limiting case, it includes correct results for abrupt transitions as well. In the present approach we also introduce interface width as an additional degree of freedom that can be used in the design of practical RHM-LHM interfaces.