Abstract : Several inconsistency-tolerant semantics have been introduced for querying inconsistent description logic knowledge bases. This paper addresses the problem of explaining why a tuple is a (non-)answer to a query under such semantics. We define explanations for positive and negative answers under the brave, AR and IAR semantics. We then study the computational properties of explanations in the lightweight description logic DL-LiteR. For each type of explanation, we analyze the data complexity of recognizing (preferred) explanations and deciding if a given assertion is relevant or necessary. We establish tight connections between intractable explanation problems and variants of propositional satisfiabil-ity (SAT), enabling us to generate explanations by exploiting solvers for Boolean satisfaction and optimization problems. Finally, we empirically study the efficiency of our explanation framework using the well-established LUBM benchmark.