Graphical Solutions of System of Linear Inequalities
Linear programming involves understanding and solving systems of inequalities using graphical methods.

HalfPlane: A halfplane is the area of a graph on one side of a line in the Cartesian coordinate plane. When we graph a linear inequality, it divides the plane into two parts—one of which is the solution set. The line itself can be either included in the solution (for inequalities involving ≤ or ≥) or not included (for inequalities involving < or >).

Graphical Representation: To solve a system of linear inequalities, you first graph each inequality. Each inequality will have a shaded halfplane representing its solution region. The feasible region, which represents the solution to the entire system, is the overlapping section where all inequalities are satisfied.

Vertices (Corner Points): A system of linear inequalities may have boundary lines that intersect at certain points. These intersection points are called vertices or corner points of the feasible region. To find the optimum solution, we often evaluate the objective function at these corner points.