Between or Beyond? Understanding Compound Inequalities

Lesson Plan - Get It!

Audio:

What Is a Compound Inequality?

A compound inequality combines two inequalities into one statement.

Instead of solving one rule, you’re solving two at once.

There are two types.

• “And” statements

• “Or” statements

Each one works differently, so knowing which one you’re dealing with is key

“And” Inequalities: Stay Between

An “and” inequality means:

Both conditions must be true at the same time.

Example:

x > 5 and x < 9

This means:

x must be greater than 5 AND less than 9

So the solution is everything between 5 and 9.

You can write it as:

5 < x < 9

Visualizing “And”

Think of it like a safe zone.

Only the overlap counts.

If the two parts don’t overlap, there is no solution

Example:

x > 3 and x < 1

There are no numbers that can do both.

So:

No solution

Solving “And” Inequalities (Two Ways)

Method 1: Split and Solve

Example:

4 < x + 2 < 10

Split it.

4 < x + 2 and x + 2 < 10

Solve each part.

4 < x + 2 ? 2 < x

x + 2 < 10 ? x < 8

Combine:

2 < x < 8

Method 2: Solve All Three Parts Together

Instead of splitting, you can do the same step to all three parts.

Example:

-6 < 2x + 4 < 10

Step 1: Subtract 4 from all parts.

-10 < 2x < 6

Step 2: Divide all parts by 2.

-5 < x < 3

Same answer—faster method.

Watch Out: Negatives Change Things

If you multiply or divide all parts by a negative number:

• Flip both inequality signs.

• Or switch the outside numbers to keep the order correct.

Example:

-8 < -2x < 6

Divide all parts by -2.

4 > x > -3

Rewrite.

-3 < x < 4

“Or” Inequalities: Go Beyond

An “or” inequality means:

Only one condition needs to be true.

Example:

x < 2 or x > 6

This means:

• Values less than 2 work.

• Values greater than 6 work.

You’re outside the middle.

Visualizing “Or”

Instead of overlap, you get two separate regions.

Solving “Or” Inequalities

Example:

x - 3 > 5 or x - 3 < -2

Solve each side.

x > 8

x < 1

Final answer:

x > 8 or x < 1

You do NOT combine these into one statement.

Quick Comparison

“And” = Between (overlap)

“Or” = Outside (either side)

What You Can Do Now

You can:

• Tell the difference between “and” and “or."

• Solve compound inequalities step by step.

• Recognize when solutions overlap—or don’t.

• Write answers in combined form or separate parts.

Get Ready to Practice

Now it’s your turn to decide—are you staying in the middle or heading beyond?