Lesson 3.6

Inequalities with Variables on Both Sides

When is everywhere, the first goal is to get it to one place. Strategy matters here: move the smaller pile to avoid negatives.

Introduction

In , we have two "x-terms". Just like a balance scale, we can remove the same weight from both sides. Removing from both sides simplifies the problem instantly.

Past Knowledge

Lesson 2.9 (Variables on Both Sides for equations).

Today's Goal

Consolidate variables to one side and interpret special solutions (All Real Numbers vs No Solution).

Future Success

This skill is critical for comparing linear functions in Unit 5 (e.g., when is Graph A higher than Graph B?).

Key Concepts

The "Move Smaller" Strategy

You have a choice: move the left x or the right x. Always move the smaller coefficient. This keeps your x-term positive, so you don't have to worry about flipping the symbol!

Smart Move

Subtract 3x (smaller).

(Positive x. Easy.)

Hard Move

Subtract 5x (larger).

(Now you have negative x. Danger zone.)

Special Solutions

All Real Numbers

If variables cancel out and leave a TRUE statement.

Any number works.

No Solution

If variables cancel out and leave a FALSE statement.

Impossible.

Worked Examples

Example 1: Basic Strategy

Basic

Solve .

1

Move Unknowns

2x is smaller than 7x. Subtract 2x from both sides.

2

Solve Two-Step

Subtract 3, then divide by 5.

Example 2: All Real Numbers

Intermediate

Solve .

1

Distribute

2

Subtract Variables

Subtract 2x from both sides. They cancel completely.

3

Interpret

Is 8 greater than 1? ALWAYS. The answer is All Real Numbers ().

Example 3: No Solution

Advanced

Solve .

1

Distribute

2

Subtract Variables

Subtract 4x from both sides.

3

Interpret

Is 10 less than or equal to 4? NEVER. No Solution ().

Common Pitfalls

The "Zero" Error

If variables cancel and you get , students write "x=0". No! The variables are gone. It's either All Reals or No Solution.

Thinking False = 0

"No Solution" is not the same as . is a valid answer (the point zero). "No Solution" means no number exists.

Real-Life Applications

Cell Phone Plans: Plan A is . Plan B is . When is Plan A cheaper? . Solving this tells you exactly how many minutes you need to use for one plan to beat the other.

Practice Quiz

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