  In this paper we introduce a special kind of relative (co)resolutions
associated to a pair of classes of objects in an abelian category
$\mathcal{C}.$ We will see that, by studying these relative (co)resolutions, we
get a possible generalization of a part of the Auslander-Buchweitz
approximation theory that is useful for developing $n$-$\mathcal{X}$-tilting
theory in [4]. With this goal, new concepts as $\mathcal{X}$-complete and
$\mathcal{X}$-hereditary pairs are introduced as a generalization of complete
and hereditary cotorsion pairs. These pairs appear in a natural way in the
study of the category of representations of a quiver in an abelian category
[5]. Our main results will include an existence theorem for relative
approximations, among other results related with closure properties of relative
(co)resolution classes and relative homological dimensions which are essential
in the development of $n$-$\mathcal{X}$-tilting theory in [4].
