Covering and elimination inequalities are central to combinatorial optimization, yet their role has largely been studied in problem-specific settings or via no-good cuts. This paper introduces a unified perspective that treats these inequalities as primitives for set system approximation in binary integer programs (BIPs). We show that arbitrary set systems admit tight inner and outer monotone approximations, exactly corresponding to covering and elimination inequalities. Building on this, we develop a toolkit that both identifies classical structural correspondences (e.g., paths vs. cuts, spanning trees vs. cycles) and extends polyhedral tools from set covering to general BIPs, including facet conditions and strengthened valid inequalities. We also propose new reformulation techniques for nonlinear and latent monotone systems, such as auxiliary-variable-free bilinear linearization, bimonotone cuts, and interval decompositions. Computational experiments on bilinear supplier–region selection and distributionally robust network site selection show how these tools produce exact reformulations, strengthen root relaxations, and support efficient cut generation. Overall, this unified view clarifies inner/outer approximation criteria, extends classical polyhedral analysis, and provides broadly applicable reformulation strategies for nonlinear BIPs.