Show that if $(X,A)$ is a CW pair of dimension $n$ (so all cells of $X-A$ have dimension at most $n$) then the map $H_n(A)\to H_n(X)$ induced by the inclusion $A\to X$ is injection with image a direct summand of $H_n(X)$.
Finding a retraction would be impossible but here's one thing: $(X,A)$ is a good pair so $X\times\{0\}\cup A\times I$ is a (deformation) retract of $X\times I$. Anyway, any idea how to prove this?