Ion passage through ion-exchange membranes is vital in electrodialysis desalination, batteries and fuel cells, and water splitting. Simplified models of ion transport through such membranes frequently assume complete exclusion of co-ions (ions with the same sign of charge as the fixed charge in the membrane) from the membrane. However, a second assumption of constant counterion electrochemical potentials across the membrane leads to simple analytical expressions for ion fluxes and transmembrane potentials. Moreover, linear corrections to account for a small membrane electrical resistance yield analytical expressions with a wider applicability. For bi-ionic potential measurements and current-induced concentration polarization at low salt concentrations, these analytical solutions match the fluxes and potentials obtained numerically without the limiting assumptions. This gives confidence in both the limiting assumptions (under appropriate conditions) and the numerical solutions. At low ion concentrations, the analytical solutions may enable rapid characterization of membrane coatings or boundary layers in solution, and such boundary layers are important in many applications of ion-exchange membranes. In fact, the assumption of complete co-ion exclusion is sometimes more limiting than the constraint of constant electrochemical potentials of counterions across the membrane. Remarkably, this limiting case readily yields the ion accumulation and depletion regions above "ion-exchange patches" that reside beneath a solution with an applied electric field. Such regions are important for sample preconcentration in microfluidic devices.