Nonlinear Systems of Inequalities

A system of inequalities consists of two or more inequalities, which are statements that one quantity is greater than or less than another. A nonlinear inequality is an inequality that involves a nonlinear expression—a polynomial function of degree 2 or higher. The most common way of solving one inequality with two variables $x$ and $y$ is to shade the region on a graph where the inequality is satisfied. 

Cubic function

This graph of a cubic function is an example of a nonlinear equation.

Every inequality has a boundary line, which is the equation produced by changing the inequality relation to an equals sign. The boundary line is drawn as a dashed line (if $<$ or $>$ is used) or a solid line (if $\leq$ or $\geq$ is used). One side of the boundary will have points that satisfy the inequality, and the other side will have points that falsify it. By testing individual points, the correct region can be shaded. If we have two inequalities, therefore, we shade in the overlap region, where both inequalities are simultaneously satisfied.

Consider, for example, the system including the parabolic nonlinear inequality:

$y>x^2$

and the linear inequality:

$y < x+2$

All points below the line $y=x+2$ satisfy the linear equality, and all points above the parabola $y=x^2$ satisfy the parabolic nonlinear inequality.

Graphing both inequalities reveals one region of overlap: the area where the parabola dips below the line. This area is the solution to the system.

Nonlinear system of inequalities

Any point in the region between the line $y=x+2$ and the parabola $y=x^2$ satisfies the system of inequalities.

The limits of each inequality intersect at $(-1, 1)$ and $(2, 4)$. Note that the area above $y=x^2$ that is also below $ y=x+2$ is closed between those two points. Whereas a solution for a linear system of equations will contain an infinite, unbounded area (lines can only pass one another a maximum of once), in many instances, a solution for a nonlinear system of equations will consist of a finite, bounded area.

This need not be the case with all nonlinear inequalities, but reversing the direction of both inequalities in the previous example would lead to an infinite solution area.