Topic: Linear Equations and Inequalities

Subtopic: Solving Linear Inequalities

**Overview**

A linear inequality is like a linear equation except it contains an inequality symbol (<, ≤, >, ≥) rather than an equals sign (=). While linear equations have a single solution (eg. x=-4), linear inequalities can have an infinite number of solutions (eg. x≥1).

The process of solving a linear inequality is identical to solve a linear inequality with one big exception. When solving an inequality, watch the direction of the inequality sign. If you multiply or divide both sides of an inequality by a *negative* number you must reverse the direction of the inequality. Don't forget this important step!

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 2.5.1 Solve linear inequalities algebraically
- 2.5.2 When solving an inequality, know when the direction of the inequality sign must reverse and be able to perform this step correctly
- 2.5.3 Understand some basic applications of linear inequalities
- 2.5.4 Recognize when a linear inequality has "no solution" or "all solutions"

**Terminology**

Define: linear inequality in one variable

**Text Notes**

When studying your textbook for this lesson here are a few things to watch.

- When you text introduces solving linear inequalities they often mention an addition and a multiplication property. I recommend that you put the "additional property of inequality" in your own words instead of making this so complicated. For instance, "It is OK to add (or subtract) a number to both sides of an inequality." Similarly, the "multiplication property of inequality" could be stated as, "It is OK to multiply (or divide) both sides of an inequality by a
*positive*number, but if you multiply (or divide) both sides by a*negative*number you must reverse the direction of the inequality." Remembering to reverse the direction of the inequality sign whenever you multiply or divide by a*negative*number is very important!

- Pay attention to the "no solution" and "all solutions" special cases when solving linear inequalities. Watch the class discussion boards examples.