1.40 Linear Inequality System Definitions

Linear Inequality System Definitions establishes the vocabulary for a collection of linear inequalities considered together, describing the boundary lines that separate their individual solution regions, the two-dimensional regions each inequality carves out of the coordinate plane, and the specific overlapping region that satisfies every inequality in the collection at once.

Linear Inequality System

A linear inequality system is a collection of two or more linear inequalities involving the same variables, considered together with the goal of finding all points on the coordinate plane that satisfy every inequality in the collection simultaneously. A linear inequality system in two variables, such as y > x + 1 paired with y < −x + 5, corresponds geometrically to two boundary lines dividing the plane, and solving the system means identifying the region where the conditions from both inequalities hold at once.

Inequality Boundary Line

An inequality boundary line is the straight line obtained by replacing a linear inequality's inequality symbol with an equals sign, marking the exact dividing line between the points that satisfy the inequality and the points that do not. This boundary line is drawn as a dashed line when the original inequality is strict, since points exactly on the line are excluded from the solution, and as a solid line when the original inequality is inclusive, since points exactly on the line are included in the solution.

Solution Half-Plane

A solution half-plane is the entire region of the coordinate plane lying on one side of an inequality boundary line, consisting of every point that satisfies a single linear inequality. Each linear inequality in two variables divides the plane into two half-planes—one where the inequality holds true and one where it does not—and the solution half-plane is specifically the side where the inequality is satisfied, typically shaded to distinguish it visually from the excluded side.

Common Planar Solution

The common planar solution is the region of the coordinate plane formed by the overlap of every solution half-plane from each inequality in a linear inequality system, representing every point that satisfies all of the system's inequalities simultaneously. Graphically, the common planar solution appears as the specific area where the individual shaded half-planes from each inequality intersect, and any point chosen from within this overlapping region will satisfy every inequality in the system when its coordinates are substituted in.

Together, these definitions describe how a linear inequality system is solved geometrically: each inequality's boundary line divides the plane into a solution half-plane and an excluded half-plane, and the common planar solution is the region where every individual solution half-plane overlaps, representing the complete set of points satisfying the entire system at once.