1. 11.6 Adding and Subtracting Rational Expressions
Add and subtract rational expressions.
To add or subtract rational expressions, you must first rewrite the expression so they have like denominators.

2. Add and subtract with the same denominator
EXAMPLE 1
Add and subtract with the same denominator a. 5 3x 7 + = 12 3x
Add numerators.
3 4
3 x =
Factor and divide out common factor. = 4 x
Simplify. b. 3x
x – 1 – =
x + 5
3x – (x + 5)
x – 1
Subtract numerators.
2x – 5
x – 1 =
Simplify.

3. Find the LCD of rational expressions
EXAMPLE 2
Find the LCD of rational expressions
Find the LCD of the rational expressions.
r + 3
10r2
a , 1 4r
SOLUTION
a. Find the least common multiple (LCM) of 4r and 10r2.
4r = r
The common factors are circled.
10r2 = r r
LCM = 2 r r = 20r2
ANSWER
The LCD of and is 20r2. 1 4r
r + 3
10r2

4. EXAMPLE 2
Find the LCD of rational expressions
Find the LCD of the rational expressions. 5
(x – 3)2
b ,
3x + 4
x2 – x – 6
SOLUTION
b. Find the least common multiple (LCM) of (x – 3)2
and x2 – x – 6.
(x – 3)2 = (x – 3) (x – 3)
x2 – x – 6 = (x – 3) (x + 2)
LCM = (x – 3) (x – 3) (x + 2) = (x – 3)2(x + 2)
ANSWER
The LCD of and is (x – 3)2(x + 2). 5
(x – 3)2
3x + 4
x2 – x – 6

5. EXAMPLE 2
Find the LCD of rational expressions
Find the LCD of the rational expressions.
c , 3
c – 2
c + 8
2c + 7
SOLUTION
c. Find the least common multiple of c – 2 and 2c + 7.
Because c – 2 and 2c + 7 cannot be factored, they don’t have any factors in common. The least common multiple is their product, (c – 2)(2c + 7).
ANSWER
The LCD of and is (c – 2)(2c + 7). 3
c – 2
c + 8
2c + 7

6. Add expressions with different denominators
EXAMPLE 3
Add expressions with different denominators
Find the sum 5
12x3 9
8x2 + 5
12x3 9
8x2 + =
9 3x
8x2 3x
5 2
12x3 2
Rewrite fractions using LCD, 24x3.
27x
24x3 = + 10
Simplify numerators and denominators. =
27x + 10
24x3
Add fractions.

7. Subtract expressions with different denominators
EXAMPLE 4
Subtract expressions with different denominators
Find the difference – 7x
x + 2 10 3x 7x
x + 2 10 3x = –
10(x + 2)
3x(x + 2)
7x (3x)
(x + 2)(3x)
Rewrite fractions using
LCD, 3x(x + 2).
10(x + 2) – 7x(3x)
3x(x + 2) =
Subtract fractions.
– 21x2 + 10x + 20
3x(x + 2) =
Simplify numerator.

8. Subtract expressions with different denominators
EXAMPLE 5
Subtract expressions with different denominators
Find the difference
x + 4
x2 + 3x – 10 –
x – 1
x2 + 2x – 8
x + 4
x2 + 3x – 10 –
x – 1
x2 + 2x – 8 =
x + 4
(x – 2)(x + 5) –
x – 1
(x + 4)(x – 2)
Factor denominators. =
(x + 4)(x + 4)
(x – 2)(x + 5) (x + 4) –
(x – 1)(x + 5)
(x + 4)(x – 2)(x + 5)
Rewrite fractions using
LCD, (x – 2)(x + 5)(x + 4). =
(x + 4)(x + 4) – (x – 1)(x + 5)
(x – 2)(x + 5)(x + 4)
Subtract fractions.

9. Subtract expressions with different denominators
EXAMPLE 5
Subtract expressions with different denominators =
x2 + 8x + 16 – (x2 + 4x – 5)
(x – 2)(x + 5)(x + 4)
Find products in numerator. =
4x + 21
(x – 2)(x + 5)(x + 4)
Simplify.

10. GUIDED PRACTICE
Find the sum or difference. 1. 2 y +
y + 1 =
y + 3 y 2.
4x + 1
2x – 1
2x – 3 –
2x + 4
2x – 1 =
Find the LCD of the rational expressions.
m + 1
7m3 , 1
28m
ANSWER
The LCD of is 28m3.
ANSWER
The LCD is (x + 5)(x – 1)(x + 2). ,
x2 + 2
x2 + 7x + 10 2
x2 + 4x – 5 , 5a
a + 3
a + 6
a – 4
ANSWER
The LCD is (a + 3)(a – 4).

11. GUIDED PRACTICE
Find the sum or difference. 3 2x 7
5x 4 =
15x3 + 14
10x 4 6. + 7. y
y + 1 + 3
y + 2 =
y2 + 5y + 3
( y +1)( y + 2) 8.
2z – 1
z2 + 2z – 8 –
z + 1
z2 – 4 =
z2 – 2z – 6
(z + 4)(z – 2)(z + 2)