Adding and Subtracting Rational Expressions with Different Denominators

We add or subtract rational expressions in the same way that we add or subtract fractions.

You have already learned how to add and subtract rational expressions with the same denominator. Now we will add and subtract rational expressions with different denominators.

Procedure — To Add (or Subtract) Rational Expressions That Have Different Denominators

Step 1 Find the LCD.

Step 2 Rewrite each rational expression with the LCD as the denominator.

Step 3 Add (or subtract) the numerators. The denominator stays the same.

Step 4 Reduce to lowest terms.

We usually leave the answer in factored form.

Note:

To add (or subtract) fractions with different denominators, say follow these steps:

Step 1 Find the LCD.

24 = 2 · 2 · 2 · 3

15 = 3 · 5

LCD = 2 · 2 · 2 · 3 · 5 = 120

Step 2 Rewrite each fraction with the LCD as the denominator.

Step 4 Reduce to lowest terms.

Example 1

Find:

Solution

Step 1 Find the LCD.
Factor each denominator.
List each factor the greatest number of times it appears in any factorization.	3, 3, w, x
The LCD is 3 · 3 · w · x = 9wx.
Step 2 Rewrite each rational expression with the LCD as the denominator.
Step 3 Add (or subtract) the numerators. The denominator stays the same.
Step 4 Reduce to lowest terms. The numerator and denominator have no common factors other than 1 and -1. Therefore, is in lowest terms.

Thus,

Note:

In , we CANNOT cancel w or x because the operation in the numerator is addition, not multiplication.