The inelastic scattering of neutrons by nuclei has been treated historically as a stepchild of elastic scattering. Few analytical studies have been performed which focus attention on inelastic scattering as a primary energy transfer mechanism. In this paper, we consider neutrons slowing down in the presence of inelastic scatterers. We take the host nuclei to be very heavy, so that in an inelastic collision a precise amount of energy is lost in the laboratory system. The slowing-down equation we obtain in the steady state has the form of a differential difference equation. We study its solutions in a variety of cases (cross-section models) and compare them with those obtained from conventional approaches. The techniques and results presented may be useful in evaluating complicated algorithms for the machine solution of problems in fast-reactor physics.