In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. 2x−5y≤10, 3x−2y>6

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They can be represented using symbols such as ≤ (less than or equal to) and > (greater than). Understanding how to manipulate and graph inequalities is crucial for solving systems of inequalities, as it allows us to determine the regions of the coordinate plane that satisfy the given conditions.

Graphing Systems of Inequalities

Graphing systems of inequalities involves plotting each inequality on a coordinate plane to visualize the solution set. The area where the shaded regions of the inequalities overlap represents the solutions that satisfy all inequalities in the system. It is important to use dashed lines for strict inequalities (>) and solid lines for inclusive inequalities (≤) to accurately depict the solution set.

Solution Set

The solution set of a system of inequalities is the collection of all points (x, y) that satisfy all inequalities in the system. This set can be represented graphically as a shaded region on the coordinate plane. In some cases, a system may have no solution if the inequalities contradict each other, resulting in no overlapping region.