Lesson 2.13

Clearing Fractions

Fractions are messy. Instead of struggling with common denominators at every step, we can use a "nuclear option" to wipe them all out in a single move.

Introduction

Solving the old way requires finding common denominators for every addition. But since it's an equation, we have a superpower: we can multiply the ENTIRE equation by a number that cancels every denominator at once.

Past Knowledge

You need to know how to find the Least Common Multiple (LCM) of numbers like 3 and 4.

Today's Goal

Eliminate all fractions in the first step by multiplying by the LCD.

Future Success

This acts as a "cheat code" for rational equations in Intermediate Algebra.

Key Concepts

The LCD Strategy

If you have denominators of 3, 4, and 6, the Least Common Denominator (LCD) is 12. Multiplying EVERY term by 12 turns all fractions into integers.

Before

Multiply ALL by 12 →

After (No Fractions!)

Worked Examples

Example 1: Single Denominator

Basic

Solve .

1

Identify LCD

Only denominator is 5. So LCD = 5.

2

Multiply EVERY Term by 5

3

Solve

Add 15 to both sides.

Example 2: Multiple Fractions

Intermediate

Solve .

1

Find LCD of 3, 4, 6

Multiples of 6: 6, 12... 4 goes into 12. 3 goes into 12. LCD = 12.

2

Multiply Everything by 12

3

Solve

Example 3: Binomial in Numerator

Advanced

Solve .

1

Multiply by LCD (6)

Be careful! The LCD multiplies the entire numerator group.

2

Distribute and Solve

Common Pitfalls

Forgetting the Whole Numbers

In , students multiply the by 2 but forget to multiply the 3 and the 5. You must multiply everything.

Bad Distribution

When you have and multiply by 3, you get . Don't accidentally multiply the top by 3 too.

Real-Life Applications

Construction Framework: Carpenters deal with fractions of an inch constantly (). When scaling up a design or calculating total material needed, converting everything to "eighths" (LCD) makes the math much faster and safer than working with mixed fractions.

Practice Quiz

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