Abstract Inconsistent-tolerant semantics, like the IAR and ICAR semantics, have been proposed as means to compute meaningful query answers over inconsistent Description Logic (DL) ontologies. In the current paper we present a framework for scalable query answering under both the IAR and ICAR semantics, which is based on highly efficient data saturation systems. Our approach is sound and complete for ontologies expressed in the lightweight DL DL-Lite, but for more expressive DLs the problem is known to be intractable, hence our algorithm only computes upper approximations. Nevertheless, its structure motivates a new type of ICARlike semantics which can be computed in polynomial time for a very large family of DLs. We have implemented our techniques and conducted an experimental evaluation obtaining encouraging results as both our IARand ICAR-answering approaches are far more efficient than existing available IAR-based answering systems.