Systems of Linear Inequalities
Intersect half-planes to find the feasible region.
The idea
A system of linear inequalities picks out the region of the plane where several linear conditions hold at once.
One condition first. A linear equation in $x$ and $y$ draws a line; the corresponding inequality, with $\le$ or $\ge$ in place of $=$, holds at every point on one side of that line, a region called a half-plane. A strict
lt;$ or gt;$ leaves the boundary line out of the solution set, while $\le$ or $\ge$ keeps it in. To decide which side, test a point not on the boundary — the origin is usually easiest: if the point satisfies the inequality, its side is the solution side; if not, the other side is.Definition (Feasible region).
The solution set of a system of linear inequalities, also called its feasible region, is the set of points that satisfy every inequality in the system. It is the overlap of the individual half-planes.
One failed inequality rules a point out, however many of the others it passes. And because every boundary is straight, the feasible region is a polygon whose corners are the points where boundary lines cross.
Ways to work on it
- Walkthrough. Test a point, identify the feasible region, and check membership.
- Practice. Decide whether a point lies in a system's feasible region.
- Hardest. Find a corner of the feasible region as the intersection of two boundaries.
Not sure where to start? Take the ten-question placement test.