Existance of solution of $Ax < b$

Question

How to check if the inequality $Ax < b$ admits at least one solution. Entries of $A$, $x$ and $b$ are taken in $\mathbb{Z}$

Answer 1

First assume $A , b > 0$; in this case by taking any integer $x<0$ you are done.

Now assume $A < 0$ and $b > 0$; in this case by taking any integer $x>0$ you are done again.

Now assume $A > 0$ and $b < 0$; in this case by taking $x = b$ you are done.

Finally, assume $A < 0$ and $b < 0$; in this case by taking $x = -b$ you are done.