用代数方法求解非线性不等式：代入

Background

We just saw how to solve systems of nonlinear inequalities using graphing, and we even talked about what to do when there is no solution. Similar to what we've seen previously with solving linear systems with substitution, we will use substitution to solve systems on nonlinear inequalities.

Substitution

To start, let's define what a system of nonlinear inequalities is. A system of nonlinear inequalities is a system of $2+$ inequalities, where at least one inequality is nonlinear. Our graph will have multiple inequalities with multiple shaded regions. The overlapping shaded region is your solution! In contrast, in a system of equations, the solutions to the system are the points of intersection.

When we are using substitution in systems of nonlinear inequalities, we substitute one inequality into the other.