Let $X$ be a simplicial complex. Suppose $X=B \cup C$ for subcomplexes $B$ and $C$, and let $A=B \cap C$. Show that the inclusion of $A$ in $B$ induces an isomorphism $H_{*} A \rightarrow H_{*} B$ if and only if the inclusion of $C$ in $X$ induces an isomorphism $H_{*} C \rightarrow H_{*} X$.