## Introduction

How do we use **absolute value** in the real world? Absolute value is useful to calculate temperature variations, differences in height and depth, distances, as well as lots of other applications.

Absolute value, **also known as modulus**, can be defined as the distance of a number from zero, the magnitude of a real number without regard for sign or the numerical value of a number.

To indicate the absolute value of a number, write a **number or variable between two vertical bars**.

In Algebra 1, students learn how to solve, graph, and write solutions for **absolute value equations**, **absolute value inequalities**, and **absolute value functions** for use in the classroom and the real world.

Solutions to absolute value problems may be displayed using **set or interval notation symbols**. Proper notation includes curly braces { }, brackets [ ], parentheses ( ), and other symbols to provide information in a concise format $\mathbb{R}$ $\cap$.

## Absolute Value Equations

**Absolute value equations** can have **more than one solution or no solution** at all.

$\begin{align} |x| & = 2 \\ x & = 2, -2 \\ x & = { -2, 2 } \end{align}$

$\begin{align} |x| & = -2 \\ x & = \text{no solution} \end{align}$

## Absolute Value Inequalities

**Absolute value inequalities** have different formats depending if the inequality is **less than**; **less than or equal to**; **greater than**; or **greater than or equal to**. These are known as **compound inequalities**. The solutions to compound inequalities are written as inequalities separated by “**and**” for less than or less than or equal to inequalities or “**or**” for greater than or greater than or equal to inequalities.

$\begin{align} \left|x\right|\le 2\\ -2\leq x\leq2\\ x\geq -2 \thinspace \text{AND} \thinspace x\leq 2\\ x = [ -2, 2 ] \end{align}$

$\begin{align} \left|x\right|\ge 2 \\ x\leq -2 \quad \text{OR} \quad x\geq 2 \\ x = (-\infty \:,\:-2]\cup \:[2,\:\infty \:) \end{align}$

The **graphs of absolute value inequalities** have a different appearance depending if the graph models less than; less than or equal to; greater than; or greater than or equal to inequalities. The use of **open or closed circles** demonstrates if the end points are included in the solution.

## Absolute Value Functions

**Absolute value functions** have an input and an output, and the graph of absolute functions is in two pieces and has a V-shape. Graphs of functions having more than one piece are known as piecewise functions.

The parent function for absolute value is $|x| = y$.

The vertex for the parent function is located at ( 0 , 0 ).

$| x +1 | = y$

The vertex for this absolute value function is located at ( 0 , -1 ).