It is an exercise in Hatcher's Algebraic Topology text to derive the long exact sequence of a pair $(X,A)$ from the Mayer-Vietoris sequence applied to $X \cup CA$. We may use the isomorphism $H_n(X,A) \cong \tilde{H}_n(X \cup CA)$ for all $n$.
But, the Mayer-Vietoris sequence is applied under the assumptions that we have a space $Y$ such that $Y = \text{int}(C) \cup \text{int}(D)$ for subspaces $C$ and $D$ of $Y$. (And, if we add the assumption that $C \cap D \neq \emptyset$, we get the Mayer-Vietoris sequence for reduced homology.)
In our situation, does the problem aim to have $X \cup CA$ play the role of $Y$? If so, how are we to write $X \cup CA$ as a union of interiors of subspaces of $X \cup CA$? I don't see why it would be true that $X \cup CA = \text{int}(X) \cup \text{int}(CA)$, so is the goal to apply the Mayer-Vietoris sequence to $Y = \text{int}(X) \cup \text{int}(CA)$?
The wording of the problem confused me and prevented me from getting started.
Thank you!