1. 9.5 Adding and Subtracting Rational Expressions
5/3/2013

2. Same Denominator:
When adding or subtracting fractions with same denominators, add/subtract the numerators and keep the same denominator

3. Perform the indicated operation.
Example 1
Add and Subtract with Like Denominators
Perform the indicated operation. a. 2x 3 5 + x 3 + b. 2 –
Add numerators and simplify. a. 2x 3 5 + = 8 x 4
Subtract numerators. x 3 + b. 2 – = 3

4. Perform the indicated operation.
Checkpoint
Add and Subtract with Like Denominators
Perform the indicated operation. 1. 3x 2 4 +
ANSWER x 2 2 2. x –
ANSWER – 1 x 2x 4 + 3. – 3
ANSWER x 4 + 3

5. Different Denominators:
If the denominators are different, find the least common denominator (LCD).

6. LCD: 2•3 (x-2)(x+3) = 6(x-2)(x+3)
Example 2
Find the LCD of Rational Expressions
Find the least common denominator. a. 6x 1
8x 3 1 b. 1
2x 4 – 1
3x 9 + , , a.
6x = 2 •3•x
8x3 =2•2•2•x•x•x
LCD:2•2•2•3•x•x•x = 24x3 b.
2x - 4 = 2(x – 2)
3x + 9 = 3(x + 3)
LCD: 2•3 (x-2)(x+3) = 6(x-2)(x+3) 6

7. Find the least common denominator.
Checkpoint
Find the LCD of Rational Expressions
Find the least common denominator. 4. 3x 2 ,
9x 2 1
ANSWER
9x 2 5. 3 ,
x 2 – x x (
x 2 + –
ANSWER
x 1 + – , x
2x 1 6.
2x 1 + (
x 1
ANSWER

8. Basic Example of Adding Fractions with different denominators.
3 3 ∙
∙ 4 4
LCD: 4•3 = 12

9. 3 2𝑥 + 4 𝑥 2 = 2 𝑥 2 2𝑥 𝑥−2 + 3 𝑥+5 = (𝑥−2)(𝑥+5) Example 3
Add with Unlike Denominators
Perform the indicated operation. a. 2x 3 +
x 2 4 b. x – 2 5
Find LCD:
2𝑥=2∙𝑥
𝑥 2 =𝑥∙𝑥
2∙𝑥∙𝑥=2 𝑥 2
3 2𝑥 𝑥 = 𝑥 2
3𝑥+8
∙ 𝑥 𝑥
∙ 2 2
Find LCD:
(𝑥−2)(𝑥+5)
2𝑥 𝑥− 𝑥 = (𝑥−2)(𝑥+5)
∙ (𝑥+5) (𝑥+5)
∙ (𝑥−2) (𝑥−2)
2𝑥 2 +10𝑥+3𝑥−6
= 2 𝑥 2 +13𝑥−6 (𝑥−2)(𝑥+5) 9

10. Perform the indicated operation.
Checkpoint
Add and Subtract with Unlike Denominators
Perform the indicated operation. 2x 3 7.
x 2 2 +
ANSWER
2x 2 + 3x 4

11. Perform the indicated operation.
Checkpoint
Add and Subtract with Unlike Denominators
Perform the indicated operation.
ANSWER
x 2 + 2x 3 ( ) x – 1 8. 3
x 1 – x +

12. Add: 2𝑥 𝑥 2 −36 + 3 𝑥+6 2𝑥 (𝑥−6)(𝑥+6) + 3 𝑥+6 = (𝑥−6)(𝑥+6) Example 4
Add with Unlike Denominators
Add: 2𝑥 𝑥 2 − 𝑥+6
Find LCD:
𝑥 2 −36= 𝑥+6 𝑥−6
𝑥+6= 1(𝑥+6)
(𝑥+6)(𝑥−6)
2𝑥 (𝑥−6)(𝑥+6) 𝑥 = (𝑥−6)(𝑥+6)
2𝑥+3𝑥−18
∙ (𝑥−6) (𝑥−6)
= 5𝑥−18 (𝑥−6)(𝑥+6) 12

13. Homework:
9.5 p.497 #10, 12, 13, 15, 16-24even, 28, 30, 33, 37, 39

14. The factors of are and and the factors of are and . So, the LCD is .
Example 4
Subtract with Unlike Denominators 2
x 2 – 1 x 4
Subtract
SOLUTION
The factors of are and and the factors of are and So, the LCD is .
x 2 – x 2 ( ) + 1 4 14

15. Example 4 2 x 2 – 1 x 4 = ( ) + ( ) x + 2 2 ( ) x + 1 = – ( ) x – 2 (
Subtract with Unlike Denominators 2
x 2 – 1 x 4 =
Factor denominators. ( ) + ( ) x + 2 2 ( ) x + 1
Rewrite using LCD. = – ( ) x – 2 ( ) x + 1 ( ) x + 2 ( ) x – 2 ( ) x + 2 ( ) x + 1 =
Subtract numerators. x + 2 – ( ) 1 =
Distributive property x + 2 – 2x ( ) 1
Combine
like terms. = x ( ) – 2 + 1 15

16. Perform the indicated operation.
Checkpoint
Add and Subtract with Unlike Denominators
Perform the indicated operation.
ANSWER – x 2 ( ) 1 + 9. 1 –
x 2 x +