Lesson 3.1

Graphing Inequalities

An equation like is a single point. An inequality like is an infinite range. We draw pictures to handle infinity.

Introduction

"You must be at least 48 inches tall to ride." This loop-the-loop doesn't just clear exactly 48-inch people. It clears 49, 50, 72... anyone greater than or equal to 48.

Past Knowledge

Comparing numbers () from Lesson 1.8.

Today's Goal

Graph solutions on a number line using open and closed circles.

Future Success

This visual language is used for Domain and Range in Unit 4.

Key Concepts

The Circle Type Matters

Open Circle

<>

The boundary is NOT included.
"Less than" or "Greater than".

Closed Circle

≤≥

The boundary IS included.
"Or equal to".

Worked Examples

Example 1: Greater Than

Basic

Graph .

1

Choose Circle

The symbol is (no bar). Use an Open Circle at 2.

2

Shade

Numbers greater than 2 are to the right (3, 4, 100...).

Example 2: Less Than or Equal To

Intermediate

Graph .

1

Choose Circle

The symbol is (solid bar). Use a Closed Circle at -1.

2

Shade

Numbers less than -1 are to the left (-2, -10, -50...).

Example 3: Rewriting to Graph

Advanced

Graph .

1

Flip It

It is hard to read "3 is greater than x". It is easier to read "x is less than 3".

Note: When you swap sides, you MUST swap the symbol.

2

Graph

Closed circle at 3, shade left.

Common Pitfalls

Shading the Wrong Way

If the variable is on the left (), the arrow points the way you shade. points right, points left. If variable is on the right (), this trick fails!

Wrong Circle

Forgetting that "or equal to" (the little bar) makes the circle solid. No bar = clear hole (open).

Real-Life Applications

Safety Systems: An elevator has a maximum capacity: lbs. A bridge has a clearance height: ft. Using the wrong inequality (like ) could be catastrophic.

Practice Quiz

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