Lesson 3.7

Adding & Subtracting (Unlike Denominators)

When the denominators are different, you must first build equivalent fractions with the LCD — then combine as usual.

Introduction

This is the real challenge. You need to combine Lesson 3.5 (finding the LCD) with Lesson 3.6 (combining numerators). The process: find the LCD, build equivalent fractions, then add or subtract.

Past Knowledge

Finding the LCD (Lesson 3.5) and adding with like denominators (Lesson 3.6).

Today's Goal

Rewrite each fraction with the LCD, combine numerators, and simplify.

Future Success

This skill is critical for solving rational equations (Chapter 10) and simplifying complex fractions (Lesson 3.8).

Key Concepts

The Full Process

1

Factor all denominators

2

Find the LCD

3

Multiply each fraction by the "missing" factor(s)

4

Combine the new numerators over the LCD

5

Simplify: combine like terms, factor, cancel

Numeric Reminder

LCD = 12. Multiply by and by .

Worked Examples

Example 1: Distinct Linear Denominators

Basic

Add .

1

LCD =

2

Build equivalent fractions

3

Combine & simplify

Example 2: One Denominator Factors

Intermediate

Subtract .

1

Factor:

LCD =

2

Build equivalent fractions

3

Combine

Example 3: Result Simplifies

Advanced

Add .

1

LCD =

2

Build & combine

3

Simplify numerator

Common Pitfalls

Multiplying Only the Numerator

When building equivalent fractions, multiply both numerator and denominator by the same factor. You're multiplying by .

Distribution Errors

When expanding , don't forget: that gives , not .

Real-Life Applications

Electrical engineers regularly combine impedances in parallel circuits using the formula , which requires adding rational expressions with unlike denominators to find the total resistance.

Practice Quiz

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