Graphing Inequalities Makes Linear Programming Click

Linear inequalities show up everywhere in optimization problems. If you have a budget constraint, a time constraint, and a resource constraint, each one is an inequality. The feasible region -- where all constraints are satisfied simultaneously -- is the shaded area where all inequalities overlap. Finding the optimal solution within that region is what linear programming does. But understanding why the optimal solution sits at a vertex of the feasible region, or why adding a constraint can dramatically change the answer, is much easier when you can see the inequalities graphed. How inequality graphing works A linear inequality like y >= 2x + 1 divides the plane into two half-planes. The boundary line y = 2x + 1 separates them. Points above the line satisfy the inequality; points below it do not. To graph an inequality: Graph the boundary line (solid for >= or <=, dashed for > or <) Shade the side that satisfies the inequality Test a point (typically the origin) to verify which side to