Systems of Inequalities in 2D

Definition

A system of inequalities consists of two or more inequalities. The solution is the set of all ordered pairs $(x, y)$ that satisfy all inequalities in the system simultaneously, usually represented as a shaded region in the Cartesian plane.

• How to read: “Ordered pairs (x, y).”
• Meaning: The solution is every $(x, y)$ satisfying all inequalities simultaneously—a shaded region, not a single point.

Why It Matters

This is the mathematical foundation of optimization under constraints. Whether it’s maximizing profit or minimizing fuel consumption, systems of inequalities define the ‘feasible region’ where all requirements are met, identifying the boundaries of the possible.

Core Concepts

• Boundary Line: The graph of the equation $f(x, y) = c$.

• How to read: “f of x comma y equals c.”
• Meaning: The boundary line (or curve) separates the plane into regions where the inequality is true or false.
  • Solid Line: Includes the boundary ($\leq, \geq$).
  • Dashed Line: Excludes the boundary ($<, >$).

• Test Point Method: Choosing a point (often $(0,0)$) not on the boundary to determine which side of the line satisfies the inequality.
• Feasible Region: The intersection (overlap) of all individual solution sets.

Connected Concepts

• Calculus
• Systems Thinking