I want to find a simple way to determine, if the following system of inequalities has a non-trivial solution :
$a_{1,1}x_1+a_{1,2}x_2+\ldots+a_{1,n}x_n \le 0$
$a_{2,1}x_1+a_{2,2}x_2+\ldots+a_{2,n}x_n \le 0$
$\vdots$
$a_{p,1}x_1+a_{p,2}x_2+\ldots+a_{p,n}x_n \le 0$
$b_{1,1}x_1+b_{1,2}x_2+\ldots+b_{1,n}x_n < 0$
$b_{2,1}x_1+b_{2,2}x_2+\ldots+b_{2,n}x_n < 0$
$\vdots$
$b_{q,1}x_1+b_{q,2}x_2+\ldots+b_{q,n}x_n < 0,$
including the case, when $p=0$ or $q=0$ (i.e. all inequalities are strict or non-strict). Non-trivial solution means, that there exists $i$ for which $x_i\not = 0$.
It is assumed, that $(x_1, x_2, \ldots, x_n)$ is to be from real euclidean space.