$X=X_1\cup X_2$, $X_i, i=1,2$ are closed and disjoint.
$i_{2_*}: H_k(X_2)\rightarrow H_k(X),$ induced by the inclusion $i_2: X_2\rightarrow X$ and $j_{1_*}: H_k(X)\rightarrow H_k(X,X_1)$ induced by the inclusion $j: X\rightarrow (X,X_1)$, we prove easily that $j_{1_*}\circ i_{2_*}:H_k(X_2)\rightarrow H_k(X,X_1)$ is an isomorphism , and $i_{2_*}$ is injective, so i have that $H_k(X_2)\simeq \text{Im}~i_{2_*}$ then i can say that $j_{1_*}: H_k(X)\rightarrow \text{Im}~ i_{2_*},$
My question is how to prove that $H_k(X)=\ker j_{1_*}\oplus \text{Im}~i_{2_*}$
I proved that $\ker j_{1_*}\cap \text{Im}~ i_{2_*}=\lbrace 0\rbrace$ but I can't find $[x]\in H_k(x)$ such that $[x]=[y]+[z]$ where $[y]\in \ker j_{1_*}, [z]\in \text{Im}~i_{2_*}.$