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:

1. Interval Notation
2. Absolute Value Inequalities

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.

1. ,
2. 3. For which real numbers do you have ?
4. Solve for 5. 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 inequality. 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 intervalsan 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  