1. 9-5 Adding and Subtracting Rational Expressions
Hubarth
Algebra II

2. Ex. 1 Real-world Connection
An object is 15cm from a camera lens. The object is in focus on the film when the lens is
10cm from the film. Find the focal length of the lens centimeters.
1 𝑓 = 1 𝑑 𝑖 𝑑 𝑜
1 𝑓 = =
= 5 30 = 1 6
Since 1 𝑓 = 1 6 , the focal length of the lens is 6cm

3. Ex. 2 Finding Least Common Multiples
Find the least common multiple of 4 𝑥 2 −36 and 6 𝑥 2 +36𝑥+54.
4 𝑥 2 −36=4 𝑥 2 −9
=(2)(2)(𝑥+3)(𝑥−3)
6 𝑥 2 +36𝑥+54=6 𝑥 2 +6𝑥+9
=(3)(2)(x+3)(x+3)
(2)(2)(3)(𝑥−3)(𝑥+3 ) 2 =12(x−3)(x+3 ) 2
12(𝑥−3)(𝑥+3 ) 2 is the least common multiple

4. Ex. 3 Adding Rational Expression
Simplify 1 𝑥 2 +5𝑥+4 + 5𝑥 3𝑥+3
1 𝑥 2 +5𝑥+4 + 5𝑥 3𝑥+3
= 1 (𝑥+4)(𝑥+1) + 5𝑥 3(𝑥+1)
= 1 (𝑥+4)(𝑥+1) ∙ 𝑥 3(𝑥+1) ∙ 𝑥+4 𝑥+4
= 3 3(𝑥+4)(𝑥+1) + 5𝑥(𝑥+4) 3(𝑥+1)(𝑥+4)
= 3+5𝑥(𝑥+4) 3(𝑥+4)(𝑥+1)
= 5 𝑥 2 +20𝑥+3 3(𝑥+4)(𝑥+1)

5. Ex. 4 Subtracting Rational Expressions
Simplify 7𝑦 5 𝑦 2 −125 − 4 3𝑦+15
7𝑦 5 𝑦 2 −125 − 4 3𝑦+15
= 7𝑦 5(𝑦+5)(𝑦−5) − 4 3(𝑦+5)
= 7𝑦 5(𝑦+5)(𝑦−5) ∙ 3 3 − 4 3(𝑦+5) ∙ 5(𝑦−5) 5(𝑦−5)
= 21𝑦 15(𝑦+5)(𝑦−5) − 20(𝑦−5) 15(𝑦+5)(𝑦−5)
= 21𝑦−20𝑦 (𝑦+5)(𝑦−5)
= −𝑦 (𝑦+5)(𝑦−5)

6. A complex fraction is a fraction that has a fraction in its numerator or denominator or in
both its numerator and denominator.
Ex. 5 Simplifying Complex Fractions
Simplify 1 𝑥 𝑦 +4
Method 1
Method 2
1 𝑥 𝑦 +4
= 1 𝑥 𝑦 +4 ∙ 𝑥𝑦 𝑥𝑦
1 𝑥 𝑦 +4
= 1 𝑥 + 3𝑥 𝑥 5 𝑦 + 4𝑦 𝑦
= 1+3𝑥 𝑥 5+4𝑦 𝑦
= 1 𝑥 +3∙𝑥𝑦 5 𝑦 +4∙𝑥𝑦
1+3𝑥 𝑥 ÷ 5+4𝑦 𝑦
= 𝑦+3𝑥𝑦 5𝑥+4𝑥𝑦
1+3𝑥 𝑥 ∙ 𝑦 5+4𝑦
= 𝑦+3𝑥𝑦 5𝑥+4𝑥𝑦

7. Practice
Suppose an object is 20cm from a camera lens. When the object is properly focused, the
lens is 5cm from the film. Find the focal length of the lens.
4cm
2. Find the least common multiple of the expressions.
5x 2 +15x+10 and 2 x 2 −8
10(𝑥+2)(𝑥−2)(𝑥+1)
3. Simplify 1 𝑥 2 −4𝑥−12 + 3𝑥 4𝑥+8
3 𝑥 2 −18𝑥+4 4(𝑥−6)(𝑥+2)
4. Simplify −2 3 𝑥 2 +36𝑥+105 − 3𝑥 6𝑥+30
−3 𝑥 2 −21𝑥−4 6(𝑥+5)(𝑥+7)
5. Simplify the complex fraction: 𝑥−2 𝑥 − 2 𝑥 𝑥−1 − 1 𝑥+1
𝑥 3 −4 𝑥 2 +𝑥+2 2 𝑥 2 +4𝑥