The influence of small changes in solute-solvent interactions on the solvent density augmentation under supercritical conditions is examined by integral equation calculations. It is shown here, through the use of a Yukawa model for the solute-solvent interaction in a Lennard-Jones solvent, that variations in the solute size or interaction strength are not so relevant. Rather, small differences in the range of the solute-solvent interaction can lead to dramatical changes in the increase of solvation that occurs in a supercritical solvent around the critical density. It is speculated that such features may serve as an explanation to large supercritical solubility differences between structurally similar molecules such as xanthines in supercritical carbon dioxide and methanol mixtures, for example. Two temperatures emerge from the present analysis that can be used to generally characterize supercritical fluids. One is the Boyle temperature T-B that is shown here to be a rigorous upper bound to the supercritical temperatures. The second, T-x, that is, in fact equivalent to the Boyle temperature for the solute-solvent interaction, is shown to characterize the attractiveness or repulsiveness of the infinitely dilute solute for the near-critical solvent. The magnitude of the attractiveness (repulsiveness) is defined by a parameter X introduced herein, and its pertinence is equally analyzed by comparison with infinite-dilution partial molar volume calculations. The ability of various integral equations to describe the supercritical fluid state is critically examined by exploring both the subcritical and the supercritical region, particularly for the case of a Lennard-Jones solvent. It appears that, bearing few differences that have their rationale in the subcritical region, all these approximate methods are essentially qualitatively equivalent in the supercritical region. This is, however, no longer the case when a solute is inserted into the supercritical solvent, and important differences can be found between different integral equations.