Lesson 7.13

Linear Inequalities

An equation is a tightrope walk—you must be exactly ON the line. An inequality is a swimming pool—you just need to be on the correct side of the line.

Introduction

We used to graph . Now we graph . The process is almost identical, but with "Shading".

Past Knowledge

Lesson 4.1 (Inequalities on a Number Line). Remember open circles vs. closed circles? This is the same, but with lines.

Today's Goal

Graph linear inequalities by choosing the line type (Dashed/Solid) and shading region (Above/Below).

Future Success

This visual foundation is needed for Linear Programming in Finite Math and Business Calculus.

Key Concepts

The Two New Decision

Decision 1: Line Type

StrictDASHED Line
InclusiveSOLID Line

Decision 2: Shading

GreaterShade ABOVE
LessShade BELOW

Note: Shading rules only work if Y is isolated on the left side!

Worked Examples

Example 1: Greater Than (Dashed)

Basic

Graph:

Analysis

  • Slope: 2, Y-Int: -3
  • Symbol: (Strict) → Dashed Line
  • Direction: "Y is bigger" → Shade Above

Notice how every point in the shaded region makes the inequality true.

Example 2: Less Than or Equal (Solid)

Intermediate

Graph:

Analysis

  • Slope: -1/2, Y-Int: 2
  • Symbol: (Inclusive) → Solid Line
  • Direction: "Y is smaller" → Shade Below

Example 3: Standard Form (Danger Zone)

Advanced

Graph:

Step 1: Isolate Y

ALARM: Division by Negative!

When dividing by -3, the sign FLIPS.

Analysis

It started as but became .

So Shade ABOVE.

Common Pitfalls

Shading from Standard Form

Students see and think "Greater than = Right". No! You must isolate Y. means "Above".

Test Points

If you are unsure where to shade, pick a test point like . Plug it in. Is ? False. So shade the OTHER side.

Real-Life Applications

Budget Constraints:

  • "You can spend AT MOST $500" translates to .
  • Graphing this shows the "Feasible Region"—every possible combination of items you can afford.

Practice Quiz

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