It was shown that the Clausius-Clapeyron relation (CCR) determines only the maximum possible slope of the first-order phase transition (PT-1) line in the temperature-pressure coordinates. This conclusion follows from the fact that the real PT-1 it is an irreversible process. An inequality has been obtained that generalizes the CCR to the case of irreversibility at PT-1: the Clausius-Clapeyron inequality (CCI). It is indicated that the CCI passes into equality (i.e., into the CCR) only in the case of reversible PT-1. Analysis of experimental data for various types of PT-1 showed the implementation of CCI in most cases. Herewith, the deviation from the CCR is the more noticeable, the greater the volume change in PT-1. It was found that deviations of experimental data from the CCR correlate with the presence of phase-transition radiation in PT-1. This makes it possible to use the value of the deviation from the CCR for thermodynamic indication of the probability of observing phase-transition radiation at PT-1 in various substances. A method has been proposed for estimating the irreversibility PT-1 value, based on the magnitude of the deviation from the CCR. The obtained CCI has been generalized to the case of the presence of an external homogeneous magnetic (or electric) field acting on the two phases of matter in equilibrium.