Question

Do all systems of linear inequalities have solutions? If not, write a system of inequalities that has no solution. What would the graph of a system of linear inequalities with no solution look like?

Answer 1

Answer: There are systems with no solutions, and the graphs may show two regions with no intersections (as you know, the solution set is in the intersection of the sets of solutions for each inequality)

Explanation:

Ok, suppose that our system is:

y > x

and

y < x.

This system obviously does not have any solution, because y can not be larger and smaller than x at the same time.

The graph of y > x is where we shade all the region above the line y = x (the line is not included)

and the graph of y < x is where we sade all the region under the line y = x (the line is not included)

So we will look at a graph where we never have a region with the two shades overlapping (so we do not have a intersection in the sets of solutions), meaning that we have no solutions.