We propose an algorithm based on infeasible irreducible subsystems (IIS) to solve general binary chance-constrained problems. By leveraging on the problem structure we are able to generate good quality upper bounds to the optimal value early in the algorithm, and the discrete domain is used to guide us efficiently in the search of solutions. We apply our methodology to individual and joint binary chance-constrained problems, demonstrating the ability of our approach to solve those problems. Extensive numerical experiments show that, in some cases, the number of nodes explored by our algorithm is drastically reduced when compared to a commercial solver. Keywords: Chance-constrained programming; Infeasible irreducible subsystems; Integer programming.