The results obtained by measuring the cross sections for the inelastic scattering of very cold neutrons for a number of metals and polymers by the method of a neutron-irradiation analysis are presented. The method is based on simultaneously measuring events of inelastic scattering and neutron capture in the sample under investigation via recording gamma radiation with a semiconductor germanium detector. Neutron capture by a nucleus of the sample is accompanied by the prompt radiation of gamma rays having a known spectrum. Upon inelastic scattering, a neutron acquires thermal energy. Upon leaving the sample, this neutron is absorbed in a special converter that contains the isotope 10B. The capture of the neutron by a 10B nucleus is followed by the emission of a 477-keV gamma ray. The probabilities of capture and inelastic scattering are proportional to the respective neutron-interaction cross sections, and the ratio of the recorded detector counts corresponding to events of the two types does not depend on the spectrum of the incident flux of very cold neutrons or on the trajectory of neutron motion in the sample. The sought inelastic-scattering cross section at a fixed sample temperature is calculated by using this ratio and the known cross section for neutron capture by the sample isotope having a known gamma-radiation spectrum.