Lesson 3.4

Solving Inequalities (×/÷)

This is the only difference between equations and inequalities. When you multiply or divide by a negative, the world turns upside down.

Introduction

We know . But if we multiply both sides by -1, we get -2 and -5. Which is bigger now? . Negative numbers flip the order.

Past Knowledge

Multiplying fractions and negative numbers (Unit 1).

Today's Goal

Identify when to FLIP the inequality symbol and when to leave it alone.

Future Success

This rule is automatic for experts but the #1 cause of lost points for beginners.

Key Concepts

The Golden Rule of Flipping

If you Multiply or Divide by a NEGATIVE number, you MUST FLIP the symbol.

Multiplying by Positive (Do Nothing)

Multiply by 2

Still True!

Multiplying by Negative (FLIP!)

Multiply by -2

Must Flip to be True!

Worked Examples

Example 1: Positive Divider

Basic

Solve .

1

Divide

Divide by 2. Since 2 is positive, keep the symbol the same.

Example 2: Negative Divider

Intermediate

Solve .

1

Divide & Flip

Divide by -3. Since -3 is negative, FLIP to .

Example 3: Multiplying by Negative

Advanced

Solve .

1

Multiply & Flip

Multiply both sides by -4. FLIP to .

Check it: Pick a number greater than 8, like 12. Is ? Yes (). Correct.

Common Pitfalls

The "Result" Trap

Students see a negative answer and want to flip. . Divide by 2. The result is -5, but we divided by POSITIVE 2. DO NOT FLIP. Only flip if the number you move (the divisor/multiplier) is negative.

Flipping for Addition

Subtracting a negative () looks like negative math, but subtraction never flips. Only multiplication and division do.

Real-Life Applications

Science: In chemistry, reaction rates often depend on inverse relationships. If you multiply a concentration by a fraction (decreasing it), the reaction time might increase (flip direction).

Practice Quiz

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