Table of Contents

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 1: The Absolute Value

**Topic**. This lesson covers Session 1: The Absolute Value.

**Learning Outcomes.**

- Solve Absolute Value equations
- Understand and use interval and inequality notation
- Solve Absolute Value Inequalities

**WeBWorK**. There are two WeBWorK assignments on today’s material:

- Interval Notation
- Absolute Value Inequalities

**Additional Video Resources.**

**Question of the Day: **If then what does equal?

## Background Regarding Numbers

*VIDEO: Types of Numbers*

*Video by Irania Vazquez*.

## Absolute Value

**Definition**. The **absolute value** of a real number , denoted by , is the non-negative number which is equal in magnitude (or size) to , i.e., is the number resulting from disregarding the sign:

**Example** 1.

- ,
- For which real numbers do you have ?
- Solve for
- Solve for

*VIDEO: Example 1 – Absolute Value Equations*

*Video by Irania Vazquez*.

## Inequalities and Interval Notation

In this section we need to discuss two ideas:

First, the notion of *ordering* or *inequalit*y. For example:

reads as 4 is less than 9

reads as -3 is less than or equal to 2

reads as is greater than 1

reads as 2 is greater than or equal to -3

The second is the called *intervals – *an interval is a set of numbers on the number line lying (for example) between two endpoints. We can give an interval in three ways: using inequalities, using interval notation, or by drawing a number line. It’s important to be able to describe the same interval in different ways. The following table shows how these three methods are connected:

## Absolute Value Inequalities

To solve an inequality containing an absolute value, we use the following steps:

Step 1: Solve the corresponding equality. The solution of the equality divides the real number line into several subintervals.

Step 2: Using step 1, check the inequality for a number in each of the subintervals. This check determines the intervals of the solution set.

Step 3: Check the endpoints of the intervals.

**Example 2. **Solve for :

- ,

*VIDEO: Example 2 – Absolute Value Inequalities and Interval Notation*

*Video by Irania Vazquez*.

#### Exit Question

Solve for

#### Answer

In interval notation, the solution set is

Good job! You are now ready to practice on your own – give the WeBWorK assignment a try. If you get stuck, try using the “Ask for Help” button to ask a question on the Q&A site.