Could a system of linear inequalities ever have no solutions? What would this look like on a graph? Explain.
Yes, a system of linear inequalities can indeed have no solutions. This occurs when the set of feasible solutions of each inequality does not overlap with the other inequalities, resulting in an empty set of solutions that fulfill all the given inequalities simultaneously.
When graphing a system of linear inequalities, we plot the individual graphs of each inequality and identify the overlapping regions that satisfy all the inequalities. However, if there is no overlapping region, it means that there are no common feasible solutions that satisfy all the inequalities simultaneously.
On a graph, this would appear as separate and non-overlapping shaded regions for each inequality. The resulting empty region between or outside the individual shaded regions would represent the absence of feasible solutions for the system of linear inequalities.