We consider different variants of Variable Consistency Dominance-based Rough Set Approach (VC-DRSA). These variants produce more general (extended) lower approximations than those computed by Dominance-based Rough Set Approach (DRSA), (i.e., lower approximations that are supersets of those computed by DRSA). They define lower approximations that contain objects characterized by a strong but not necessarily certain relation with approximated sets. This is achieved by introduction of parameters that control consistency of objects included in lower approximations. We show that lower approximations generalized in this way enable us to observe dependencies that remain undiscovered by DRSA. Extended lower approximations are also a better basis for rule generation. In the paper, we focus our considerations on different definitions of generalized lower approximations. We also show definitions of VC-DRSA decision rules, as well as their application to classification/sorting and ranking/choice problems.