Solution region of a system of linear inequalities in two variables

What it shows

A linear inequality ax + by ≤ c in two variables is satisfied by a half-plane bounded by the line ax + by = c, and a test point off the line shows which side. The solution set of a system is the intersection of these half-planes, often a convex polygon; the boundary is included for ≤ and ≥ but not for < and >. In linear programming, the level lines of F = ax + by are parallel, so on a bounded polygonal region F reaches its maximum and minimum at vertices.

How to use

Enter up to three inequalities in Inequality 1, Inequality 2 and Inequality 3, type an expression for Objective function F(x; y) = and set Axis range (up to). Students drag point P to test it against each inequality and watch the dashed level line F = F(P) move with it. Dragging a boundary line changes its constant and reshapes the region. Finally, Show vertices & F at vertices lists F at each corner and marks the optimum.

Parameters you can change

• Inequality 1 (e.g. x + y <= 4)
• Inequality 2 (leave blank if unused)
• Inequality 3 (leave blank if unused)
• Objective function F(x; y) =
• Axis range (up to) 4–20

Questions to explore

1. How does a test point tell you which side of a boundary line belongs to the solution region?
2. Does the boundary line belong to the solution region of a strict inequality such as x + y < 4?
3. Why can the maximum of F not occur inside the region, judging by the level line through P?