1. Adding and Subtracting Rational Expressions:
Multiplication:
Combine right across the multiplication sign.
Division:
Flip the second rational and multiply

2. Adding & Subtracting like Denominators
1. Add or subtract the numerators by combining like terms and put over the denominator
2. Follow the steps for simplifying
Example 1:
𝑥 𝑥 2 −4 − 2 𝑥 2 −4
1 𝑥+2
𝑥−2 𝑥 2 −4
𝑥−2 (𝑥−2)(𝑥+2)

3. Example 2:
9 5−𝑥 + 4 5−𝑥
9+4 5−𝑥
13 5−𝑥

4. Example 3:
𝑥 2 𝑥 − 3𝑥 𝑥 𝑥 3 +27
𝑥 2 −3𝑥+9 𝑥 3 +27
𝑥 2 −3𝑥+9 (𝑥+3)( 𝑥 2 −3𝑥+9)

5. Adding & Subtracting UNLIKE Denominators
Factor the denominators and determine the LCD
Rewrite each fraction over the LCD
Add or subtract the numerators by combining like terms and put over the denominator
Follow the steps for simplifying

6. Example 4:
5𝑥𝑦 − 3𝑥
5𝑥𝑦
∙ 3 3 3𝑥
∙ 5𝑦 5𝑦
15𝑥𝑦 − 15𝑥𝑦

7. Example 5:
64 𝑥 3 − 𝑥
(4𝑥−3)(16 𝑥 2 +12𝑥+9)
∙ 2𝑥 2𝑥 2𝑥
∙ (4𝑥−3)(16 𝑥 2 +12𝑥+9) (4𝑥−3)(16 𝑥 2 +12𝑥+9)
2𝑥(4𝑥−3)(16 𝑥 2 +12𝑥+9) + 2𝑥(4𝑥−3)(16 𝑥 2 +12𝑥+9)

8. Example 6:
15 9𝑥 𝑥
15 9𝑥
∙ 2 2 = 30 18𝑥
5 18𝑥
30 18𝑥 𝑥 = 35 18𝑥

9. 5 6 𝑥 2 + 𝑥 4 𝑥 2 −12𝑥 5 6 𝑥 2 ∙ 2(𝑥−3) 2(𝑥−3) = 10(𝑥−3) 12 𝑥 2 (𝑥−3)
Example 7:
5 6 𝑥 2 + 𝑥 4 𝑥 2 −12𝑥
5 6 𝑥 2
∙ 2(𝑥−3) 2(𝑥−3)
= 10(𝑥−3) 12 𝑥 2 (𝑥−3)
= 3 𝑥 𝑥 2 (𝑥−3)
∙ 3𝑥 3𝑥
𝑥 4𝑥(𝑥−3)

10. 10(𝑥−3) 12 𝑥 2 (𝑥−3) + 3 𝑥 𝑥 2 (𝑥−3)
10 𝑥−3 +3 𝑥 𝑥 2 (𝑥−3)

11. 2(𝑥−5) 2(𝑥+5)(𝑥−5) − 2(𝑥+5) 2(𝑥+5)(𝑥−5) + 4𝑥 2(𝑥+5)(𝑥−5)
Example 3:
1 𝑥+5 − 2 2𝑥−10 + 2𝑥 𝑥 2 −25
1 𝑥+5 − 2 2(𝑥−5) + 2𝑥 (𝑥−5)(𝑥+5)
2(𝑥−5) 2(𝑥+5)(𝑥−5)
− 2(𝑥+5) 2(𝑥+5)(𝑥−5)
+ 2𝑥∙2 2(𝑥+5)(𝑥−5)
2(𝑥−5) 2(𝑥+5)(𝑥−5) − 2(𝑥+5) 2(𝑥+5)(𝑥−5) + 4𝑥 2(𝑥+5)(𝑥−5)
2 𝑥−5 −2 𝑥+5 +4𝑥 2(𝑥+5)(𝑥−5)

12. 2𝑥−10−2𝑥−10+4𝑥 2(𝑥+5)(𝑥−5)
−20+4𝑥 2(𝑥+5)(𝑥−5)
2 (𝑥+5)
4(−5+𝑥) 2(𝑥+5)(𝑥−5)
4(𝑥−5) 2(𝑥+5)(𝑥−5)