Nonlinear Inequalities in Two Variables
Big Idea: Just as linear inequalities divide the coordinate plane into two half-planes, nonlinear inequalities in two variables divide the plane into regions bounded by curves — parabolas, circles, ellipses, hyperbolas, and other higher-degree curves.
1. What Is a Nonlinear Inequality in Two Variables?
A nonlinear inequality in two variables is an inequality where at least one variable appears with a degree higher than 1, or the variables are multiplied together. The general idea is:
$$f(x, y) \;<\; 0, \quad f(x, y) \;\leq\; 0, \quad f(x, y) \;>\; 0, \quad f(x, y) \;\geq\; 0$$
where $f(x,y)$ is a polynomial of degree 2 or higher in $x$ and/or $y$.
Common Examples:
| Inequality | Boundary Curve | Type |
|---|---|---|
| $y < x^2$ | Parabola $y = x^2$ | Quadratic |
| $x^2 + y^2 \leq 25$ | Circle $x^2 + y^2 = 25$ | Quadratic (Circle) |
| $y \geq x^3 - 4x$ | Cubic $y = x^3 - 4x$ | Cubic |
| $\frac{x^2}{9} + \frac{y^2}{4} < 1$ | Ellipse | Quadratic (Ellipse) |
| $y > x^2 + 2x - 3$ | Parabola | Quadratic |