Liquid-phase activity coefficient models (or excess Gibbs energy models) with three or fewer adjustable parameters are often sufficient to model the liquid solutions present for the vapor-liquid equilibria (VLE) or the vapor-liquid-liquid equilibria (VLLE) calculations of phase equilibria. Our purpose here is to show the flexibility of three such equations in the correlation and prediction of (1) extrema and inflection points on the popular ln gamma(i) vs. x(1) diagram for binary solutions, (2) liquid-liquid phase splitting, and (3) homogeneous azeotropes. The empirical equations investigated here are (a) the two-constant Margules model, (b) the van Laar model, and (c) the three-constant NRTL model. New mathematical procedures lead to new graphs and equations, which allow the reader to identify the presence of any of the above three phenomena by inspection without performing detailed calculations such as a Gibbs energy minimization for liquid phase splitting. (C) 2004 American Institute of Chemical Engineers.