Lesson 1.10

Absolute Value

Absolute value strips away the sign and answers one question: "How far is this number from zero?" It's the distance function of the number line.

Introduction

If you're 5 steps to the left of zero OR 5 steps to the right, you're still 5 units away. That's what measures — pure distance, always positive (or zero).

Past Knowledge

You know positive and negative numbers and can locate them on a number line.

Today's Goal

Evaluate absolute value expressions and simplify expressions that combine absolute value with operations.

Future Success

Absolute value equations and inequalities appear throughout algebra and calculus.

Key Concepts

1. Definition

In plain English: if the number is already positive (or zero), keep it. If it's negative, flip the sign.

2. Key Facts

Always ≥ zero

Zero is the only number with

Opposites have equal absolute values

3. Absolute Value as Grouping

Treat absolute value bars like parentheses: simplify inside first, then take the absolute value, then continue with operations outside.

Worked Examples

Example 1: Simple Absolute Value

Basic

Evaluate and .

is negative → flip to positive:

is positive → stays:

Example 2: Expression Inside

Intermediate

Evaluate .

Simplify Inside

Apply Absolute Value

Add

Example 3: Negation Outside

Advanced

Evaluate .

Each Absolute Value

Apply the Outer Negative & Add

Common Pitfalls

Absolute Value "Makes Everything Positive"

, not . The negative sign outside the bars is not affected.

Splitting the Bars

. You must simplify inside first, then take the absolute value of the result.

Real-Life Applications

GPS calculates the distance between two points. Distance is always positive — you can't drive miles. Under the hood, GPS uses absolute values (and their higher-dimensional cousin, the norm) to ensure every measurement makes physical sense.