1. Objectives Add and subtract rational expressions.
Simplify complex fractions.

2. Some rational expressions are complex fractions
Some rational expressions are complex fractions. A complex fraction contains one or more fractions in its numerator, its denominator, or both. Examples of complex fractions are shown below.
Recall that the bar in a fraction represents division. Therefore, you can rewrite a complex fraction as a division problem and then simplify. You can also simplify complex fractions by using the LCD of the fractions in the numerator and denominator.

3. Example 5A: Simplifying Complex Fractions
Simplify. Assume that all expressions are defined.
x + 2
x – 1
x – 3
x + 5
Write the complex fraction as division.
x + 2
x – 1
x – 3
x + 5 ÷
Write as division.
x + 2
x – 1
x + 5
x – 3 
Multiply by the reciprocal.
x2 + 7x + 10
x2 – 4x + 3
(x + 2)(x + 5)
(x – 1)(x – 3) or
Multiply.

4. Example 5B: Simplifying Complex Fractions
Simplify. Assume that all expressions are defined.
x – 1 x 2 3 +
Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator.
x – 1 x
(2x) 2 3 +
The LCD is 2x.

5. Example 5B Continued
Simplify. Assume that all expressions are defined.
(3)(2) + (x)(x)
(x – 1)(2)
Divide out common factors.
x2 + 6
2(x – 1)
x2 + 6
2x – 2 or
Simplify.

6. Simplify. Assume that all expressions are defined.
Check It Out! Example 5a
Simplify. Assume that all expressions are defined.
x + 1
x2 – 1 x
x – 1
Write the complex fraction as division.
x + 1
x2 – 1 x
x – 1 ÷
Write as division.
x + 1
x2 – 1
x – 1 x 
Multiply by the reciprocal.

7. Check It Out! Example 5a Continued
Simplify. Assume that all expressions are defined.
x + 1
(x – 1)(x + 1)
x – 1 x 
Factor the denominator. 1 x
Divide out common factors.

8. Simplify. Assume that all expressions are defined.
Check It Out! Example 5b
Simplify. Assume that all expressions are defined. 20
x – 1 6
3x – 3
Write the complex fraction as division. 20
x – 1 6
3x – 3 ÷
Write as division. 20
x – 1
3x – 3 6 
Multiply by the reciprocal.

9. Check It Out! Example 5b Continued
Simplify. Assume that all expressions are defined. 20
x – 1
3(x – 1) 6 
Factor the numerator. 10
Divide out common factors.

10. Simplify. Assume that all expressions are defined.
Check It Out! Example 5c
Simplify. Assume that all expressions are defined.
x + 4
x – 2 1 2x x +
Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator.
x + 4
x – 2
(2x)(x – 2) 1 2x x +
The LCD is (2x)(x – 2).

11. Check It Out! Example 5c Continued
Simplify. Assume that all expressions are defined.
(2)(x – 2) + (x – 2)
(x + 4)(2x)
Divide out common factors.
3x – 6
(x + 4)(2x)
3(x – 2)
2x(x + 4) or
Simplify.