show that if $G/K, G/N$ are solvable then $G/(K \cap N)$ is also solvable

Question

Let $G$ be a group and let $N,K \lhd G$ be her normal subgroups. It is given that $G/N$ and $G/K$ are solvable.

I need to prove that $G/(K \cap N)$ is also solvable.

I thought about proving somehow that $(G/K) \times (G/N) \cong (G/(K \cap N))$, using the theorem which says that: if $H,M \lhd G$ and $HM = G$ and $H \cap M = {e}$, then $G \cong H \times M$

Then it will follow, since I already proved once that if $H,M$ are solvable, then $H \times M$ is also solvable.

However, I didn't manage to prove the isomorphism part.

Help would be appreciated.

Answer 1

Hint: construct an injective homomorphism from $G/(K \cap N)$ to $G/K \times G/N$ and use that products and subgroups of solvable groups are again solvable.