Lesson 3.7

Compound Intro: AND vs OR

Sometimes one rule isn't enough. "You must be older than 12 AND younger than 18." This lesson combines inequalities into complex logic.

Introduction

A compound inequality is just two inequalities glued together. The glue matters: AND means intersection (overlap), while OR means union (everything).

Past Knowledge

Graphing simple inequalities on number lines (Lesson 3.1).

Today's Goal

Visualize and graph logical combinations of inequalities.

Future Success

Absolute Value inequalities break down into AND/OR cases in Lessons 3.10 and 3.11.

Key Concepts

Visualizing Boundaries

AND (Intersection)

Both conditions must be true. It's the overlap (Sandwich).

OR (Union)

Either condition can be true. It goes outward (Wings).

Worked Examples

Example 1: The Sandwich (AND)

Basic

Graph .

Identify Endpoints

Left: -3 (Closed circle because )
Right: 2 (Open circle because )

Shade Between

Since x is larger than -3 AND smaller than 2, shade the middle.

Example 2: The Wings (OR)

Intermediate

Graph .

Graph Left Wing

Open circle at -1, shade left.

Graph Right Wing

Closed circle at 3, shade right.

Example 3: Word Problem Translation

Advanced

A water park requires riders to be at least 48 inches tall but under 78 inches. Write and graph the compound inequality for rider height .

Translate Each Phrase

"At least 48" means (closed circle — 48 is included).
"Under 78" means (open circle — 78 is NOT included).

Choose AND or OR

The rider must satisfy both conditions at the same time — tall enough AND not too tall. This is AND (intersection).

Write the Sandwich

Graph

Closed circle at 48, open circle at 78, shade between.

Common Pitfalls

Writing "OR" as a Sandwich

You CANNOT write . This implies x is bigger than 3 AND smaller than 1. Impossible! OR graphs must remain two separate inequalities.

Endpoint Confusion

Ensure you check the "or equal to" bar for EACH endpoint separately. One side can be open while the other is closed.

Real-Life Applications

Safety Ranges: A car engine operates safely when its RPM is between 1000 and 6000 (). If it drops below 1000 OR goes above 6000 (), the check engine light turns on.