A non-electrolyte equation-of-state model was used to describe the phase behavior of binary systems containing alkyl-methyimidazolium bis(trifluoromethyl-sulfonyl)imide ionic liquids. A methodology is suggested for modeling this phase behavior by using the Non-Random Hydrogen-Bonding (NRHB) model. According to this methodology, the scaling constants of the ionic liquid are calculated using limited available experimental data on liquid densities and Hansen's solubility parameters, while all electrostatic interactions (polar, hydrogen bonding and ionic) are treated as strong specific interactions. Using the aforementioned methodology, the model is applied to describe the vapor-liquid and the liquid-liquid equilibria in mixtures of ionic liquids with various polar or quadrupolar solvents at low and high pressures. In all cases, one temperature-independent binary interaction parameter was used. Accurate correlations were obtained for the majority of the systems, both, for vapor-liquid and liquid-liquid equilibria.