1. Addition and Subtraction with Unlike Denominators
6.4
Adding and Subtracting with LCDs
When Factors Are Opposites

2. To Add or Subtract Rational Expressions Having Different Denominators
1. Find the LCD.
2. Multiply each rational expression by a form of 1 made up of the factors of the LCD missing from that expression’s denominator.
3. Add or subtract the numerators, as indicated. Write the sum or difference over the LCD.
4. Simplify, if possible.

3. 2. Multiply each expression by the appropriate number to get the LCD.
Add:
Solution
1. First, we find the LCD:
9 = 3  3
12 = 2  2  3
2. Multiply each expression by the appropriate number to get the LCD.
LCD = 2  2  3  3 = 36

4. 3. Next we add the numerators: 4
3. Next we add the numerators: 4. Since 16x2 + 15x and 36 have no common factor, cannot be simplified any further. Subtraction is performed in much the same way.

5. The denominator 9x must be multiplied by 4x to obtain the LCD.
Subtract:
Solution We follow the four steps as shown in the previous example. First, we find the LCD.
9x = 3  3  x
12x2 = 2  2  3  x  x
The denominator 9x must be multiplied by 4x to obtain the LCD.
The denominator 12x2 must be multiplied by 3 to obtain the LCD.
LCD = 2  2  3  3  x  x = 36x2

6. Multiply to obtain the LCD and then we subtract and, if possible, simplify.
Caution! Do not simplify these rational expressions or you will lose the LCD.
This cannot be simplified, so we are done.

7. Solution First, we find the LCD: a2  4 = (a  2)(a + 2)
Add:
Solution First, we find the LCD:
a2  4 = (a  2)(a + 2)
a2  2a = a(a  2)
We multiply by a form of 1 to get the LCD in each expression:
LCD = a(a  2)(a + 2)

8. 3a2 + 2a + 4 will not factor so we are done.

9. Subtract:
Solution First, we find the LCD. It is just the product of the denominators: LCD = (x + 4)(x + 6).
We multiply by a form of 1 to get the LCD in each expression. Then we subtract and try to simplify.
Multiplying out numerators.

10. When subtracting a numerator with more than one term, parentheses are important.
Removing parentheses and subtracting every term.

11. Add:
Solution
Adding numerators

12. When Factors are Opposites
When one denominator is the opposite of the other, we can first multiply either expression by 1 using 1/1.

13. Add: Solution Multiplying by 1 using 1/1
The denominators are now the same.

14. Add: Solution
 3 + x = x + (3) = x  3

15. Add:
Solution