Inequalities in Two Variables

Definition An inequality in two variables compares two expressions in $x$ and $y$ with one of the symbols $<$, $\le$, $>$ or $\ge$. An ordered pair $(a, b)$ is a solution if the inequality is true for $x = a$ and $y = b$. The solution set, also called the graph of the inequality, is the set of all solutions, drawn as a region of the plane. The inequality is linear if it can be written as

with real numbers $A$, $B$, $C$ and with $A$ and $B$ not both zero. Any other inequality is nonlinear, for example $y \ge x^2$, $x^2 + y^2 < 9$ or $xy > 1$.

Definition The boundary curve of an inequality in $x$ and $y$ is the graph of the equation obtained by replacing the inequality symbol with $=$. If both sides are polynomials in $x$ and $y$, the boundary curve splits the rest of the plane into regions, and within each region the inequality is either true at every point or false at every point. A point can switch from true to false only by crossing the boundary. If a side can be undefined, as with a fraction, the points where it is undefined must be treated as part of the boundary too, just as with the sign charts for rational inequalities.

Definition For an inequality with $\le$ or $\ge$, every point of the boundary curve makes both sides equal and so satisfies the inequality. The boundary belongs to the solution set and is drawn as a solid curve. For an inequality with $<$ or $>$, no point of the boundary satisfies it. The boundary is left out of the solution set and is drawn as a dashed curve. Points where a side is undefined are never solutions, so they are always left out.

Definition A line in the plane splits the points not on it into two open half-planes, one on each side. A closed half-plane is an open half-plane together with its boundary line. The solution set of a linear inequality with $<$ or $>$ is an open half-plane, and with $\le$ or $\ge$ it is a closed half-plane. When $B \ne 0$ you can solve for $y$. If $B > 0$, then $Ax + By < C$ means

the half-plane below the line. If $B < 0$, dividing by $B$ reverses the inequality and gives the half-plane above the line. When $B = 0$ the boundary is a vertical line and the half-plane lies to its left or right.

Definition A test point is a point not on the boundary curve that you substitute into the inequality to decide whether its region belongs to the solution set. When both sides are polynomials, the inequality is true in the whole region of a test point that makes it true, and false in the whole region of one that makes it false. One test point per region is enough. A linear inequality has only two regions, but a curved boundary can create more, and then every region needs its own test point.

Definition The intersection $S \cap T$ of two sets of points is the set of points that belong to both $S$ and $T$. A system of inequalities joined by “and” asks for the points that satisfy every inequality at once, so its solution set is the intersection of the individual solution sets.

Definition The union $S \cup T$ of two sets of points is the set of points that belong to $S$ or to $T$ or to both. A compound inequality joined by “or” has as its solution set the union of the individual solution sets.

Definition In a problem with several conditions, called constraints, the feasible region is the set of all points that satisfy every constraint, that is the intersection of their solution sets. A corner point, or vertex, of the region is a point of the region where two nonparallel boundary lines meet. If the feasible region is nonempty and bounded and is described by finitely many linear inequalities with $\le$ or $\ge$, then every linear expression $ax + by$ reaches its largest value on the region at a corner point, and its smallest value at a corner point too.