For subspaces $A,B\subset X$ such that $X=A\cup B$, the Mayer-Vietoris sequence is: $$\longrightarrow\cdots H_n(A\cap B)\overset{\Phi}{\longrightarrow} H_n(A)\oplus H_n(B)\overset{\Psi}{\longrightarrow} H_n(X)\overset{\partial}{\longrightarrow} H_{n-1}(A\cap B)\to\dots\to H_0(X)\to 0.$$ Then the Mayer-Vietoris sequence is the long exact sequence of homology groups given by the short exact sequence of chain complexes formed by short exact sequences $$0\to C_n(A\cap B)\overset{\phi}{\longrightarrow} C_n(A)\oplus C_n(B)\overset{\psi} {\longrightarrow}C_n(A+B)\to 0.$$
Is it then true that the long exact Mayer Vietoris sequence can be decomposed into short exact sequences of the form $0\to H_n(A\cap B)\overset{\Phi}{\longrightarrow} H_n(A)\oplus H_n(B)\overset{\Psi}{\longrightarrow} H_n(X)\to 0$?