We define K-injective and K-projective DG modules in K(A,M) and also K-flat DG modules in K(A). These constitute full triangulated subcategories of K(A,M), and we refer to them as resolving subcategories. The category K(A,M)inj of K-injectives in K(A,M) plays the role of the category J in the abstract of Chapter 8, and the category of K-projectives K(A,M)prj plays the role of the category P there. The K-flat DG modules are resolving for the tensor functor. Furthermore, we prove that the functors Q : K(A,M)inj → D(A,M) and Q : K(A,M)prj → D(A,M) are fully faithful.