EXAMPLE 5Simplify a complex fraction (Method 1)

Let f be the focal length of a thin camera lens, p be the distance between an object being photographed and the lens, and q be the distance between the lens and the film. For the photograph to be in focus, the variables should satisfy the lens equation below. Simplify the complex fraction.

Physics

Lens equation: f 11p

1q+

=

EXAMPLE 5Simplify a complex fraction (Method 1)

SOLUTION

11p

1q+

f = 1qpq

pqp+

= 1q + p

pq

=

q + ppq=

Write denominator as a single fraction.

Divide numerator by denominator.

EXAMPLE 6 Simplify a complex fraction (Method 2)

Simplify:5

x + 41

x + 4 + 2x

SOLUTION

The LCD of all the fractions in the numerator and denominator is x(x + 4).

5x + 4

1x + 4 + 2

x

5x + 4

1x + 4 + 2

x

=x(x+4)x(x+4)

Multiply numerator and denominator by the LCD.

x + 2(x + 4)5x= Simplify.

Simplify.5x

3x + 8=

GUIDED PRACTICE for Examples 5 and 6

x6

x3

–

x5

710

–

– 5x 3 (2x – 7)=

Multiply numberator and denominator by the LCD

Simplify

x6

x3

–

x5

710

–11.

x6

x3

–

x5

710

–

3030=

GUIDED PRACTICE for Examples 5 and 6

2x

–

2x

+

4

3

12.

2 – 4x2 + 3x=

2 (1 – 2x )2 + 3x

= Simplify

Simplify

2x

–

2x

+

4

3=

2x

–

2x

+

4

3

xx Multiply numberator and denominator by the LCD

GUIDED PRACTICE for Examples 5 and 6

3x + 5

2x – 3 + 1

x + 5

13.

3x – 33x + 7=

3(x – 3)3x + 7

=

Multiply numberator and denominator by the LCD

Simplify

Simplify

3x + 5

2x – 3 + 1

x + 5=

3x + 5

2x – 3 + 1

x + 5

(x + 5)(x – 3)

(x + 5)(x – 3)