We present measurements of partial branching fractions of B→K+X, B→K0X, and B→π+X, where X denotes any accessible final state above the endpoint for B decays to charmed mesons, specifically for momenta of the candidate hadron greater than 2.34 (2.36) GeV for kaons (pions) in the B rest frame. These measurements are sensitive to potential new-physics particles which could enter the b→s(d) loop transitions. The analysis is performed on a data sample consisting of 383×106BB̅ pairs collected with the BABAR detector at the PEP-II e+e- asymmetric energy collider. We observe the inclusive B→π+X process, and we set upper limits for B→K+X and B→K0X. Our results for these inclusive branching fractions are consistent with those of known exclusive modes, and exclude large enhancements due to sources of new physics.