Question

Write a system of linear inequalities with the given characteristic:

Answer 1

In quadrant III x-coordinates are less than 0 and y-coordinates are less than 0.

$x$

You can modify it to get other examples:
$x+1$

Answer 2

If all solutions are points in Quadrant-3, then the x-coordinates
and y-coordinates of all those points are all negative.

I believe the only possible system of liner inequalities whose
solutions have those characteristics is . . .

           x < 0
           y < 0 .