Counterexample or proof of outer measure inequality

Question

Let $X$ be a non empty set and $\mu^{*}:P(X) \longrightarrow [0,\infty]$ an outer measure, that is, $\mu^{*}$ satisfies the following properties:

1. $\mu^{*}(\emptyset)=0$;
2. If $A \subset B$ then $\mu^{*}(A) \leq \mu^{*}(B)$;
3. For any sequence $(A_{n}) \subset P(X)$ we have that $\mu^{*}(\bigcup_{n=1}^{\infty}A_n) \leq \sum_{n=1}^{\infty} \mu^{*}(A_n)$. I want to know weather or not the following inequality holds: $$ \mu^{*}(A \cup B) + \mu^{*}(A \cap B) \leq \mu^{*}(A)+\mu^{*}(B) $$ I wrote $A \cup B$ as $A-(A \cap B) \cup B-(A \cap B) \cup (A \cap B)$ and then tried to use the monotonicity property. Also, I considered that $\mu^{*}(A-B) \geq \mu^{*}(A)-\mu^{*}(B)$. However, even bearing this in mind I haven't been able to prove it nor giving a counterexample.

Any help?

In advance thank you.