inequality regarding the gamma function

Question

I am curious to know if the following is true:

Let $\alpha >0$. Then \begin{equation} \int_0^K x^{\alpha -1} e^{-x} \,dx \leq \Gamma (\alpha), \quad \forall K \in \mathbb{R}. \end{equation}

This is trivial if $K \geq 0$. However, I am not sure if this holds when $K<0$. Using the substitution $u =-x$ gives us a term $e^{u}$ which doesn't really help.

Answer 1

Set $K = -100$. Consider $\alpha = 2$. You get $$\int_0^{-100} x e^{-x} \,dx.$$Now compare with $\Gamma (2) = 1! =1.$