Definition of the gamma function for non-integer negative values The gamma function is defined as 
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt$$
for $x>0$.
Through integration by parts, it can be shown that for $x>0$,
$$\Gamma(x)=\frac{1}{x}\Gamma(x+1).$$
Now, my textbook says we can use this definition to define $\Gamma(x)$ for non-integer negative values. I don't understand why. The latter definition was derived by assuming $x>0$. So shouldn't the whole definition not be valid for any $x$ value less than zero?
P.S. I have read other mathematical sources and most of them explain things in mathematical terms that are beyond my level. It would be appreciated if things could be kept in relatively simple terms.

To add on the other answers :

This is one of the 1st example of analytic continuation. It is clear that $$\Gamma(z) = \frac{\Gamma(z+1)}{z}=\frac{\Gamma(z+2)}{z(z+1)}=\frac{\Gamma(z+n+1)}{z(z+1)\ldots(z+n)}$$ makes $\Gamma(z)$ well-defined  for $z \in \mathbb{C}, -z \not \in \mathbb{N}$. 
But it is not so obvious (without a lot of theorems in complex analysis) that this continuation is the only one being analytic, in the same way that $\frac{1}{1-z}$ is the only one analytic continuation of $\sum_{n=0}^\infty z^n$ beyond $|z|< 1$
