Mathematics and Statistics · Ch 17 — Linear Inequations
Systems of Linear Inequations and the Feasible Region
Systems of Linear Inequations and the Feasible Region
In real problems several conditions must hold at the same time — a factory's labour hours and its material and the non-negativity of quantities. A system of linear inequations is a collection of two or more inequations that must all be satisfied together.
The feasible region
Feasible region
The feasible region of a system of linear inequations is the set of points that satisfy every inequation of the system simultaneously — i.e. the common (overlapping) region of all the individual half-planes.
Geometrically, you shade the half-plane of each inequation on the same axes; the feasible region is where all the shadings overlap. It is always a region bounded by straight-line pieces (a polygon or an unbounded polygonal region).
Non-negativity constraints
and restrict you to the first quadrant
In commercial problems the variables usually count real quantities (units produced, hours used), which cannot be negative. The constraints and therefore appear constantly; together they confine the feasible region to the first quadrant (including its bounding axes).
Corner (vertex) points
The feasible region meets its boundary lines at corner points (vertices). These are found by solving the relevant boundary lines two at a time as simultaneous equations. They matter enormously in Std XII: in Linear Programming the optimum of a linear objective always occurs at a corner point of the feasible region. Learning to locate the region and its corners now is the direct foundation for that work. …
Two or more linear inequations in the same variables that must all hold s …
The common region satisfying every inequation of the system — the overlap of all the half-planes. Bounded by straight-line segments; confined to the first qu …
A point where two boundary lines of the feasible region meet, found by solving those lines as simultaneous equations. Corner points are decis …