Lesson 3.10

Absolute Value Inequalities (Less Than)

"Stay close." That's what a Less Than inequality says. It keeps you on a short leash, trapping the variable between two walls.

Introduction

asks: "Which numbers are less than 5 steps away from zero?" The answer isn't just 1, 2, 3, 4. It's also -1, -2, -3, -4. The absolute value acts like a magnet, pulling everything toward the center.

Past Knowledge

Lesson 3.8 (AND inequalities) and Lesson 2.17 (Intro to Absolute Value).

Today's Goal

Translate into the compound inequality .

Future Success

This visualizes "Tolerance" in engineering (e.g., ).

Key Concepts

The "Less ThAND" Sandwich

Mnemonic: Less ThAND. Less Than problems always become AND problems (sandwiches).

Original Problem

becomes

The Sandwich

You are trapped between the negative and the positive version of the number.

Worked Examples

Example 1: Basic Trap

Basic

Solve .

1

Rewrite

Remove the bars and create the sandwich using -7.

Example 2: Trap then Solve

Intermediate

Solve .

1

Create the Sandwich

Put in the middle of -5 and 5.

2

Solve (Add 2)

Add 2 to the left, middle, AND right.

Example 3: No Solution?

Advanced

Solve .

1

Stop and Think

Absolute value is distance. Distance is always positive. Can a positive number be LESS than -3?

NEVER. No Solution.

Don't try to sandwich this. You'll get nonsense math.

Common Pitfalls

Only Half the Story

Solving as just is incorrect. That includes -1,000,000! You must have the lower boundary ().

Real-Life Applications

Pizza Delivery: "We deliver within 5 miles of the shop." If the shop is at mile marker 20, and you are at mile x, the distance is . The delivery zone is . Solving this () gives the exact street range.

Practice Quiz

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