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 $|x|=5$ then what does $x$ equal?

## Background Regarding Numbers

*VIDEO: Types of Numbers*

*Video by Irania Vazquez*.

## Absolute Value

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

$|c|=\left\{ \begin{array}{cl}

c, & \text { if } c \text { is positive or zero } \\

-c, & \text { if } c \text { is negative }

\end{array}$

**Example** 1.

- $|-4|=$,
- $|4|=$
- For which real numbers $x$ do you have $|x|=3$?
- Solve for $x:|3 x-4|=5$
- Solve for $x:\quad -2 \cdot|12+3 x|=-18$

*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:

$4<9$ reads as 4 is less than 9

$-3 \leq 2$ reads as -3 is less than or equal to 2

$\frac{7}{6}>1$ reads as $\frac{7}{6}$ is greater than 1

$2 \geq-3$ 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 $x$ :

- $|3 x-5| \geq 11$
- $|12-5 x| \leq 1$
- $|x+7|\leq -3$, $|x+7|\geq -3$

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

*Video by Irania Vazquez*.

#### Exit Question

Solve for $x: \quad|-3-2x| \geq 7$

#### Answer

In interval notation, the solution set is

$S=(-\infty,-5]\cup [2,\infty)$

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.