What is the situation when two inequalities have no solution?
This simply means that the parameters marked out by the inequality cannot be satisfied.
An example of this would be:
{ x|x belongs to real numbers (R) ; x<0 }
A real number would have to be equal to or greater than zero,but this inequality requires x<0.Hence this has no solution.

What are the conditions when the solution of a system of inequalities will be in the first quadrant?
For a system of inequalities to lie in the first quadrant, the straight-forward condition would be:
0 < theta < 90,
where theta is an angle under consideration. This puts it in the first quadrant.

If you need any more help,or if you have any questions,do ask.
Hope this helps,
-Prabhat