Triangle Inequality of Tensor Products

Question

If $$\|A - x\|_1 \le \epsilon$$ and $$\|B - y\|_1 \le \epsilon$$ where $A, B, x, y \in Herm(H_A)$, where $Herm(H_A)$ are the set of Hermitian matrices in a Hilbert space $H_A$, then can we say, by Triangle Inequality that, $$\|A \otimes B - x \otimes y\|_1 \le 2\epsilon$$ How do I derive it?