  In this paper we introduce a special kind of relative resolutions and
coresolutions associated to a pair of classes of objects in an abelian
category. We will see that by studying this resolutions and coresolutions we
get a generalization of the Auslander-Buchweitz approximation theory. With this
goal, new concepts as $\mathcal{X}$-complete pairs and $\mathcal{X}$-hereditary
pairs are introduced as a generalization of complete and hereditary cotorsion
pairs. Our main results will include a generalization of Salce's Lemma and
Garcia Rozas's Lemma, among other classical results from Auslander-Buchweitz
and Auslander-Reiten theory.
