Math101 · Independent Ontario learning library
Math101Linear Inequalities in Two Variables
A linear inequality in two variables describes a half-plane of solutions separated by a boundary line. Points on the boundary are included for $\le$ or $\ge$ and excluded for $<$ or $>$.
Half-planes model feasible combinations in budgeting, optimization, design, and systems of constraints. Graphing turns infinitely many solutions into one region.
Intuition and core definition
A linear inequality in two variables describes a half-plane of solutions separated by a boundary line. Points on the boundary are included for $\le$ or $\ge$ and excluded for $<$ or $>$. Every shaded point, not just lattice points, satisfies the inequality.
Notation, language, and conditions
$Ax+By<C$ has boundary $Ax+By=C$. A solid line marks inclusion; a dashed line marks exclusion. In slope-intercept form $y>mx+b$, shading is above the line and $y<mx+b$ shades below, but a test point is safer for general forms.
Why this idea matters
A two-variable linear inequality models an entire feasible half-plane, with its boundary style showing whether equality is permitted.
A dependable method
- Replace the inequality symbol with equality and graph the boundary line.
- Choose solid or dashed style based on endpoint inclusion.
- Select a test point not on the boundary, often $(0,0)$.
- Substitute it into the original inequality.
- Shade the side containing the test point if true, otherwise the opposite side, and verify another point.
