Lesson 3.11

Absolute Value Inequalities (Greater Than)

If "Less Than" keeps you close, "Greater Than" pushes you away. This is the math of social distancing and outer limits.

Introduction

asks: "Which numbers are more than 5 steps away from zero?" The answers are far out: numbers like 6, 7, 8 (right side) OR numbers like -6, -7, -8 (left side).

Past Knowledge

Lesson 3.9 ("OR" inequalities) and Lesson 3.10 ("Less Than").

Today's Goal

Translate into disjoint "OR" inequalities.

Future Success

This completes Unit 3! You now have the full toolkit for linear inequalities.

Key Concepts

The "GreatOR" Split

Mnemonic: GreatOR. Greater Than problems always become OR problems (wings).

Original Problem

becomes

The Split

One case stays exactly the same. The other flips the symbol AND the sign.

Worked Examples

Example 1: The Basic Split

Basic

Solve .

1

Apply GreatOR Rule

Case 1 (Pos)
Case 2 (Neg)

Example 2: Isolate First

Intermediate

Solve .

1

Isolate Absolute Value

Subtract 4 first.

2

Split into OR

Case 1 (Left Wing)

Case 2 (Right Wing)

Example 3: All Real Numbers?

Advanced

Solve .

1

Stop and Think

Absolute value is always positive (or zero). Is a positive number bigger than -2? ALWAYS.

All Real Numbers.

Common Pitfalls

Writing "OR" as a Sandwich

Never write . This says numbers are bigger than 5 AND smaller than -5. Impossible. Keep them separate.

Real-Life Applications

Safety Zones: "Keep at least 10 feet away from the heavy machinery." If the machine is at position 0, your position x must be . You must be far to the left OR far to the right.

Practice Quiz

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