= Solution
Take a <purification of a density operator> $|\Psi\rangle_{ABCD}$ of $\rho_{ABC}$. Apply <Strong subadditivity of Von Neumann entropy> to the reduced state on $ACD$, with $A$ the conditioning system:
$$
S(AC)+S(AD)\geq S(A)+S(ACD).
$$
Complementary subsystems of a <pure state> have the same <Von Neumann entropy>, by the <Schmidt decomposition>. Therefore $S(AD)=S(BC)$ and $S(ACD)=S(B)$. Substitution gives
$$
\boxed{S(A)+S(B)\leq S(AC)+S(BC).}
$$
This is <weak monotonicity of quantum entropy>. The <purification of a density operator> is only a proof device; no purity assumption is imposed on $\rho_{ABC}$.
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