Graphical Solution of Inequalities: Explained with Examples

A system of inequalities is solved graphically by drawing every inequality's region on the same axes and keeping only the overlap. That overlapping patch — the feasible region — contains every point satisfying all the inequalities at once.

This is the visual heart of linear programming: businesses use it to find the best production mix under limited resources, and a famous result says the optimal answer always sits at a corner of this region.

Key Concepts

• System of inequalities: several inequalities that must all hold simultaneously.
• Feasible region: the intersection (overlap), not the union, of the individual half-planes.
• Corner (vertex) points: where boundary lines meet; these are the candidates for optimisation.
• Bounded vs unbounded: a feasible region may be a closed polygon or stretch to infinity.
• First-quadrant restriction: x ≥ 0 and y ≥ 0 confine the region to non-negative values, typical for quantities like items produced.

How It Works

1. Graph each inequality as its own half-plane with the correct solid or dashed boundary.
2. Find the overlap of all shaded regions — that is the feasible region.
3. Mark the corner points where pairs of boundary lines intersect.
4. Verify corners by substituting each into every original inequality.

Worked Example

Solve the system x + y ≤ 4, x ≥ 0, y ≥ 0. The line x + y = 4 is solid and passes through (4, 0) and (0, 4); the test point (0, 0) satisfies 0 ≤ 4, so shade below the line. The conditions x ≥ 0 and y ≥ 0 keep only the first quadrant. The feasible region is the triangle with corners (0, 0), (4, 0), and (0, 4). Check each corner against all three inequalities: (0, 0) gives 0 ≤ 4; (4, 0) gives 4 ≤ 4; (0, 4) gives 4 ≤ 4 — all pass. An interior point like (1, 1) gives 2 ≤ 4 and satisfies x ≥ 0, y ≥ 0 as well.

Common Misconceptions

• Taking the union instead of the intersection. A point must satisfy every inequality, so only the common overlap counts.
• Including boundary lines of strict inequalities — corners on a dashed boundary are excluded from the solution.
• Assuming a feasible region always exists. Contradictory constraints like x ≥ 5 together with x ≤ 3 leave an empty region and no solution.

Real-World Applications

• Linear programming: maximising profit or minimising cost under resource constraints.
• Factory scheduling with limits on machine time, labour, and materials.
• Portfolio constraints in finance, such as caps on risky assets.
• Transport and logistics planning with capacity limits on routes.

Graphical solution of inequalities means drawing each half-plane and reading the overlap as the feasible region. Its corners are the key points for optimisation, and verifying them against every constraint keeps the solution honest.