Upper bounds for two events of random variable following Poisson distribution

Question

So $X$ follows Poisson with $L>0$ and $\operatorname{E}[X]=\operatorname{Var}(X)=L$. We are dealing with two events: $A=\{X \leq L/2\}$ and $B=\{X \geq 2L\}$. I have calculated that $P(A) \leq 4/L$ and $P(B) \leq 1/L$ and I want to calculate the upper bound for $P(A \cup B)$. Any ideas?

Answer 1

$$A\cap B=\left\{2L\le x\le {L\over 2}\right\}=\emptyset\\\to\\\Pr(A\cup B){=\Pr(A)+\Pr(B)-\Pr(A\cap B)\\=\Pr(A)+\Pr(B)\\\le {5\over L}}$$therefore $$\Pr(A\cup B)\le {5\over L}$$