We study the language inclusion problem L1 ⊆ L2 where L1 is regular or context-free. Our approach relies on abstract interpretation and checks whether an overapproximating abstraction of L1, obtained by successively overapproximating the Kleene iterates of its least fixpoint characterization, is included in L2. We show that a language inclusion problem is decidable whenever this overapproximating abstraction satisfies a completeness condition (i.e. its loss of precision causes no false alarm) and prevents infinite ascending chains (i.e. it guarantees termination of least fixpoint computations). Such overapproximating abstraction function on languages can be defined using quasiorder relations on words where the abstraction gives the language of all words ``greater than or equal to’’ a given input word for that quasiorder. We put forward a range of quasiorders that allow us to systematically design decision procedures for different language inclusion problems such as context-free languages into regular languages and regular languages into trace sets of one-counter nets. We also provide quasiorders for which the induced inclusion checking procedure corresponds to well-known state-of-the-art algorithms like the so-called antichain algorithms. Finally, we provide an equivalent greatest fixpoint language inclusion check which relies on quotients of languages and, to the best of our knowledge, was not previously known