Lesson 2.19

Splitting the Equation

Once the absolute value is isolated, the "box" opens. But be careful—what comes out is not one equation, but two. The positive case and the negative case.

Introduction

If I tell you my number is 5 units from zero (), my number could be 5 OR it could be -5. To solve , you must write two separate equations: one for and one for .

Past Knowledge

Isolating the absolute value (Lesson 2.18).

Today's Goal

Split an absolute value equation into two cases and solve both.

Future Success

Crucial for solving inequalities like (Unit 3).

Key Concepts

The Split Maneuver

Once (and ONLY once) the bars are isolated, split into two worlds.

Isolated

Split!

Case 1 (Positive)

Case 2 (Negative)

IMPORTANT: The bars DISAPPEAR when you split.

Worked Examples

Example 1: Basic Split

Basic

Solve .

1

Check Isolation

Bars are alone on the left. We can split immediately.

2

Split & Solve

Case 1

Case 2

Example 2: Isolate Then Split

Intermediate

Solve .

1

Isolate First

2

Split & Solve

Example 3: No Solution

Advanced

Solve .

1

Stop & Think

Can a distance be negative? No.

No Solution

Absolute value cannot equal a negative number.

Common Pitfalls

Only One Answer

Solving and getting only is only 50% correct. Always check for the negative case.

Changing the Inside

When splitting , do NOT change it to . The inside stays exactly the same; only the answer side changes sign ( and ).

Real-Life Applications

Quality Control: If a chip must be , that means the acceptable range is defined by . Solving this split gives us the minimum () and maximum () size allowances.

Practice Quiz

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