The lever rule applies to any two-component system existing in a state of two-phase equilibrium. Here, several classroom demonstrations are described to help students understand liquid--liquid coexistence and the lever rule using a set of isobutyric acid + water + dye samples ranging in composition from water + dye to isobutyric acid + dye. The critical temperature of the partially miscible isobutyric acid+ water system is just above room temperature, so students can perceive the general shape of the coexistence curve by watching a progression of phase separations take place over time as the set of prewarmed single-phase samples are allowed to cool. Qualitative visualization of the lever rule is demonstrated by showing students the set of samples after equilibration at a temperature well below the critical temperature. An interactive Excel spreadsheet is provided to help students connect their observations of the samples with the phase diagram. Quantitative and intuitive interpretation of the lever rule can be achieved through height measurements of the biphasic sample layers, leading to immediate graphical determination of the compositions of the isothermal equilibrated phases in the classroom. The set of samples may also be used in an extended classroom or laboratory project to map out the coexistence curve, and advanced students can use the coexistence data to estimate the critical composition of the system, as well as the universal critical exponent [beta]. Finally, preparation of a stable hexaphasic immiscible liquid mixture is described.