Lesson 3.8

Solving "AND" Inequalities

The sandwich returns. When limits come from both sides, whatever you do to the middle, you must do to the left AND the right.

Introduction

We usually solve inequalities like . But what if there are two boundaries? . This creates three sections, and we must balance all of them simultaneously.

Past Knowledge

Lesson 3.7 (Intro to AND/OR) and basic solving skills.

Today's Goal

Solve compound "AND" inequalities by isolating the variable in the middle.

Future Success

This "sandwich" logic is the basis for Absolute Value Less Than problems!

Key Concepts

The 3-Sided Balance

Think of it as having two equals signs. The goal is to get alone in the center.

Standard Equation

-5     -5

Two sides.

Tri-Inequality

-5     -5     -5

THREE SIDES!

Worked Examples

Example 1: Simple Subtraction

Basic

Solve .

1

Subtract from All Sides

Subtract 3 from left, middle, AND right.

Example 2: Two-Step Solving

Intermediate

Solve .

1

Add 3 Everywhere

2

Divide by 2 Everywhere

Example 3: Standard Intersection

Advanced

Solve .

Sometimes it's written as two separate parts.

1

Merge Logic

If x is bigger than 3 AND smaller than 8, it must be between them.

Common Pitfalls

Forgetting a Side

Students often add 3 to the middle and the right, but forget the left. The inequality becomes unbalanced and WRONG.

Flipping Chaos

If you divide by a negative, you must flip BOTH inequality signs. .

Real-Life Applications

Manufacturing: A machine must cut a pipe to be 10 meters long, with an error margin of 0.1m. This means the length must satisfy . Quality control depends on solving these ranges.

Practice Quiz

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