用代数方法求解非线性不等式：消去

Background

Past sections covered how to use elimination to solve linear systems of equations. We did this with addition, subtraction, and multiplication. Then, we learned methods for solving nonlinear inequalities. These included graphing and using the quadratic formula.

This lesson covers elimination when dealing with nonlinear inequalities!

Review: Elimination

But before we dive in, let's do a quick review of elimination. Elimination is when we add or subtract terms within two equations in order to eliminate variables.

Let's say we were trying to solve the system of equations for:

$$\begin{align} 2x + 3y &= 21 \\ -2x + 9y &= 51 \\ \end{align}$$

Elimination is a good option for this system of equations because the $x$-terms would cancel out.

$$\begin{align} \cancel{2x}+3y&=21\\ +\ \ \ \cancel{-2x}+9y&=51\\ \hline 0x+12y&=72\\ 12y&=72\\ y&=\boxed{6} \end{align}$$

Elimination offered us a simple way to solve for $y$. We use this idea when using elimination to solve a nonlinear system of inequalities. However, there are a couple of small changes.

Solving a System of Nonlinear Inequalities: Using Elimination

When using elimination to solve systems of inequalities, there are a few points to keep in mind:

• Elimination only works if the inequality signs are in the same direction. That means that you cannot set up a $>$ inequality with a $<$ inequality.

• Systems of inequalities have a special rule. We can only eliminate inequalities with addition. Subtracting inequalities indicates a sign change, which can cause incorrect solutions.

• Stack the inequalities with the like terms on top of each other. We can do this because of the Addition Property of Equality.

Now that we understand those basics, let's walk through an example problem.

Example

Use elimination method to solve the following system of nonlinear inequalities:

$$\begin{align} y > x^2 + 2x + 1 \\ y > x^2 -2x +3 \\ \end{align}$$

Solution:

First, we notice that the inequalities are facing the same direction. This means we can use elimination to solve this system. Notice the second terms of each inequality. They are each other's additive inverse, so they can be eliminated.

$$\begin{align} y > x^2 + 2x + 1 \\ y > x^2 -2x +3 \\ \hline y > x^2 \cancel{+ 2x} + 1 \\ y > x^2 \cancel{-2x} +3 \\ \hline 2y > 2x^2 + 4 \\ y > x^2 +2 \\ \end{align}$$

Systems of Nonlinear Inequalities: Elimination

In this lesson, we will discuss using elimination to solve systems of nonlinear inequalities.

The goal of solving systems of nonlinear inequalities is to find the solution region.

With systems of equations, the solution is the point where the graphs cross. That point must work for every equation.

With systems of inequalities, the solution is different. Instead of one point, the solution is the overlapping shaded area on the graph.

This means you are not finding just one answer. You are finding a range of values that make all the inequalities true at the same time.

One of the easiest ways to solve a system of nonlinear inequalities is to graph them. This helps you to see the possible solution sets. But in this lesson, we will focus on elimination.

Elimination

We have seen the method of elimination before, with nonlinear equations. The idea is basically the same, but there are three rules that you should always follow. These rules are illustrated in the image above.

1. The inequality signs must face the same direction.

2. Only addition can be used during elimination. Subtraction can create sign changes that may lead to incorrect solutions.

3. Stack "like terms" on top of each other.

(Video) Elimination in a System of Nonlinear Inequalities

by Anusha Rahman

In this video by Anusha Rahman, you will learn how to use elimination to solve a system of nonlinear inequalities.

Summary

The goal of elimination in a system of nonlinear inequalities is to remove one variable. This leaves you with an inequality that has only one variable, which is easier to work with.

Before you can use elimination, there are three important rules you must follow:

• The inequality signs must point in the same direction. This makes sure the comparison stays valid when you combine the inequalities.

• Use addition to eliminate a variable. Adding the inequalities helps cancel out one variable safely.

• Line up like terms. Make sure matching variables are stacked on top of each other before you combine the inequalities.

Once all three of these conditions are met, you are ready to move forward with elimination.