Lesson 3.5

Multi-Step Inequalities

Just like equations, we peel the onion layer by layer. The only catch? Watching out for that negative multiplier at the very end.

Introduction

Solving uses the exact same steps directly from Unit 2: Inverse Operations. Undo addition/subtraction first, then multiplication/division.

Past Knowledge

Solving Two-Step Equations (Lesson 2.4) and The Flip Rule (Lesson 3.4).

Today's Goal

Solve complex inequalities involving distribution and combining like terms.

Future Success

Essential for "Compound Inequalities" where you solve two of these at once.

Key Concepts

The Priority Checklist

  1. Distribute any parentheses.
  2. Combine Like Terms on each side.
  3. Add/Subtract constants away from the variable.
  4. Multiply/Divide the coefficient. FLIP IF NEGATIVE!

Worked Examples

Example 1: Basic Two-Step

Basic

Solve .

1

Add 7

2

Divide by 3

Positive divisor → No flip.

Example 2: The Negative Flip

Intermediate

Solve .

1

Subtract 4

2

Divide by -2

Negative divisor → FLIP SYMBOL!

Example 3: Distribution

Advanced

Solve .

1

Distribute

2

Add 6

3

Divide by -3

Negative divisor → FLIP SYMBOL!

Common Pitfalls

Forgetting the Flip

It usually happens at the very end of a long problem. You focus so hard on the addition/subtraction that you autopilot the division and forget to flip.

Distributing Errors

In , remembering that is key. Sign errors ruin the whole problem.

Real-Life Applications

Party Planning: "I have $100. The room costs $40 and pizza is $5 per person. How many people can I invite?"

. Solving this gives the real constraint on your guest list.

Practice Quiz

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