Mathematics.

equations

Systems of Inequalities

Algebra I20 minDifficulty3 out of 10

You should know: linear inequalities two variables

Overview

A system of inequalities is a set of two or more inequalities considered together. Its solution is the set of all points satisfying EVERY inequality simultaneously — graphically, the overlapping region where all the individual shaded half-planes intersect. Systems of inequalities are the foundation of linear programming, where a feasible region defined by constraints is searched for an optimal value.

Formal Definition

Definition

A system of two linear inequalities in two variables:

{A1x+B1y □1 C1A2x+B2y □2 C2\begin{cases} A_1x+B_1y \ \Box_1\ C_1 \\ A_2x+B_2y \ \Box_2\ C_2 \end{cases}

Each □ represents <, ≤, >, or ≥

General system

Properties

Solution region

The intersection of all individual half-plane solutions\text{The intersection of all individual half-plane solutions}

Feasible region

The overlapping shaded region, often bounded by a polygon in optimization problems\text{The overlapping shaded region, often bounded by a polygon in optimization problems}

Worked Examples

  1. Graph y = x - 2 as a dashed line, shading above it (for y > x - 2).

  2. Graph y = -x + 4 as a solid line, shading below/on it (for y ≤ -x + 4).

  3. The solution is the overlapping region satisfying both shadings.

Answer: The region above the dashed line y = x - 2 AND on/below the solid line y = -x + 4

Practice Problems

Difficulty 3/10

A system requires x ≥ 0, y ≥ 0, and x + y ≤ 6. Describe the feasible region.

Common Mistakes

Common Mistake

Shading the union of the individual solution regions rather than their intersection.

A system of inequalities requires ALL inequalities to be satisfied simultaneously — the solution is the INTERSECTION (overlap) of the individual shaded regions, not their union.

Summary

  • A system of inequalities is solved where all individual solution regions overlap.
  • Graph each inequality's boundary and shading, then identify the common overlapping region.
  • Systems of inequalities define the feasible region used in linear programming optimization.

References

  1. WebsiteWikipedia — Linear programming