Complex Fractions2

1. Section 6.3
Complex
Rational
Expressions
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2. Objective #1
Simplify complex rational expressions by
multiplying by 1.
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3. Simplifying Complex Fractions
Complex rational expressions, also called complex
fractions, have numerators or denominators containing one
or more fractions.
5
x
x

5
1 1

5
x
Woe is me,
for I am
complex.
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4. Complex Rational Expressions
Simplifying a Complex Rational
Expression by Multiplying by 1 in the
Form LCD
T
LCD
1) Find the LCD of all rational expressions within the
complex rational expression.
2) Multiply both the numerator and the denominator of the
complex rational expression by this LCD.
3) Use the distributive property and multiply each term in the
numerator and denominator by this LCD. Simplify each term.
No fractional expressions should remain within the numerator
and denominator of the main fraction.
4) If possible, factor and simplify.
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5. Simplifying Complex Fractions
x


x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 5
EXAMPLE

Simplify: .
1
1
5
5
5
x
x

SOLUTION
The denominators in the complex rational expression are 5 and
x. The LCD is 5x. Multiply both the numerator and the
denominator of the complex rational expression by 5x.





 

x
5
 


x
x
x
x
x
x
1
1
5
5
5
5
1
1
5
5
5 Multiply the numerator and
denominator by 5x.

6. Simplifying Complex Fractions
 Divide out common factors.
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x
  
1
x x
x
x
x
x
1
5
5
5
5
5
5
5
  

Use the distributive
property.
CONTINUED
x
  
1
x x
x
x
x
x
1
5
5
5
5
5
5
5
  
2

25 
5

x
x
Simplify.
 x  5  x

5

1   
5

x
Factor and simplify.

7. Simplifying Complex Fractions
Simplify.
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 7
CONTINUED
5 
1

x
Simplify. 5   x

8. Simplifying Complex Fractions
1
x x


x x x x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 8
EXAMPLE
Simplify: .
6
1
6


SOLUTION
The denominators in the complex rational expression are x + 6
and x. The LCD is (x + 6)x. Multiply both the numerator and
the denominator of the complex rational expression by (x + 6)x.
Multiply the numerator and
denominator by (x + 6)x.
 
  6
1
6
1
6
6
6
1
6
1









 x x
x x

9. Simplifying Complex Fractions
Use the distributive
property.
CONTINUED
   
  

 6 6
1
6
  
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 9
6
1
6
 
 

x x
x
x x
x
x x
Divide out common factors.
   

 6 6
1
6
6
1
6
 
 

x x
x
x x
x
x x
Simplify.
 6

x x
 
  6 
6

x x
x x
6 36
6
 
x 2 
x Simplify.


10. Simplifying Complex Fractions
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 10
CONTINUED
6

6x 2 
36x Subtract.

Factor and simplify.
 
6  
1
6  
6

x x

1

 6 Simplify.

x x

11. x x x 
y
 1  1  1

  

 1  1  1

xy  y y x 
y y
x y x y x y x y
  
   
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11
1a. Simplify:
2
2
1
1
x
y
x
y


Multiply the numerator and denominator by the LCD of 2. y
2
2
2
2 2 2 2 2
2
2 2 2
y
y y y y
x y x x y
y
y y y
2
2 2
( )
( )( )
Objective #1: Example

12. x x x 
y
 1  1  1

  

 1  1  1

xy  y y x 
y y
x y x y x y x y
  
   
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 12
1a. Simplify:
2
2
1
1
x
y
x
y


Multiply the numerator and denominator by the LCD of 2. y
2
2
2
2 2 2 2 2
2
2 2 2
y
y y y y
x y x x y
y
y y y
2
2 2
( )
( )( )
Objective #1: Example

13. 
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Objective #1: Example
1b. Simplify:
1 1
7
7
x 
x
Multiply the numerator and denominator
by the LCD of x(x  7).
1 1 1 1 ( 7) ( 7)
7 ( 7) 7 7
7 ( 7) 7 7 ( 7)
( 7) 7 7
7 ( 7) 7 ( 7) 7 ( 7)
1 1
( 7) ( 7)
x x x x
x x x x x x x x
x x x x
x x x x
x x x x x x
x x x x
 
  
      
 
    
  
  

  
 

14. 
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Objective #1: Example
1b. Simplify:
1 1
7
7
x 
x
Multiply the numerator and denominator
by the LCD of x(x  7).
1 1 1 1 ( 7) ( 7)
7 ( 7) 7 7
7 ( 7) 7 7 ( 7)
( 7) 7 7
7 ( 7) 7 ( 7) 7 ( 7)
1 1
( 7) ( 7)
x x x x
x x x x x x x x
x x x x
x x x x
x x x x x x
x x x x
 
  
      
 
    
  
  

  
 

15. Objective #2
Simplify complex rational expressions by dividing.
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16. Simplifying Complex Fractions
Simplifying a Complex Rational
Expression by Dividing
1) If necessary, add or subtract to get a single rational
expression in the numerator.
2) If necessary, add or subtract to get a single rational
expression in the denominator.
3) Perform the division indicated by the main fraction
bar: Invert the denominator of the complex rational
expression and multiply.
4) If possible, simplify.
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17. Simplifying Complex Fractions
m
2 2
m m m
3
2
4 4
9
2 2
m
2
m
m
m m m
9 
2
m m m m
   
2 2 3 3
m m
2  3 
3
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 17
EXAMPLE
Simplify: .
5 6  
6

 
 


m m
m m
SOLUTION
1) Subtract to get a single rational expression in the
numerator.
2
2 2 4 4  3  3
  2
2 
 

 

 m m m
 
2

m m
   
  
    
    
   2
2 2
2
3 3 2
2 3 3
3 3 2
  

  

  

m m m
m m m
m m m

18. Simplifying Complex Fractions
3 2
CONTINUED
m m m m
   
4 4 2 18
2
3 2 2
m m m
  
6 4 18

  3   3  
2
   3   3  
2
2
2) Add to get a single rational expression in the denominator.
3
m
m
2 2  
3
m m m
  

m m
9
3 9 3 2 2

m   m 
m
m
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18

m m m
m m m
 3 2  2 3
3
5 6 6

 

 

  m m m m
m m
m m
 m

3 
3
   
 
   
3  3   3

 3 2 3
2 3 3
3 2 3
  

  

  

m m m
m m m
m m m
 3  2  3
   3  2 
3

  

m m m
m m m

19. Simplifying Complex Fractions
3) & 4) Perform the division indicated by the main fraction
bar: Invert and multiply. If possible, simplify.
m m m
  
6 4 18
 m  3  m  3  m

2

9

m
 3 2 3
2
m
2 2
m m m
m
5 6 6
m m m
  
3 2 3
m m m
  
6 4 18
m m m
  
3 2 3
m m m
  
6 4 18
m m m
  
6 4 18
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 19
3
4 4
9
2
2
3 2
2 2
  

 

 
 


m m m
m m
m m
   
   
9
3 3 2
2 2
3 2


  

m
m m m
   
   
9
3 3 2
2 2
3 2


  

m
m m m
 2 2
9
3 2
 

m m
CONTINUED

20. Objective #2: Example
x x
x x
x x
x x
 
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 20
2a. Simplify:
1 1

1 1
1 1
1 1
 
 

 
The LCD of the numerator is (x 1)(x 1).
The LCD of the denominator is (x 1)(x 1).

21. Objective #2: Example
x x
x x
x x
x x
 
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 21
2a. Simplify:
1 1

1 1
1 1
1 1
 
 

 
The LCD of the numerator is (x 1)(x 1).
The LCD of the denominator is (x 1)(x 1).

22. x x x x x x x x x x
x x x x x x x x x x
x x x x x x x x x x
x x x x x x x x x x
 1  1 (  1)(  1) (  1)(  1)  2  1  2  1
  
 1  1 (  1)(  1) (  1)(  1) (  1)(  1) (  1)(  1)
 
 1  1 (  1)(  1) (  1)(  1)    2  1  2  1
 1  1 (  1)(  1) (  1)(  1)  (  1)(  1) (  1)( 
1)
x x x x
    
x x x x
2 1 2 1
x x x
1)( 1) ( 1)( 1)
x x x x x x x x
x x x x
x
x x x x x x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 22
Objective #2: Example
2 2
2 2
2 2
2 1 ( 2 1)
(
x

2 2
2 2 2 2
2 1 2 1 2 1 2 1
( 1)( 1) ( 1)( 1)
4
( 1)( 1) 4 ( 1)( 1) 4
2 x 2 2 ( x 1)( x 1) 2 x 2 2 2( x
2 1)
( x 1)( x
1)
2
x
x
2 1
    
   

         
   
   
   
    
 


CONTINUED

23. x x x x x x x x x x
x x x x x x x x x x
x x x x x x x x x x
x x x x x x x x x x
 1  1 (  1)(  1) (  1)(  1)  2  1  2  1
  
 1  1 (  1)(  1) (  1)(  1) (  1)(  1) (  1)(  1)
 
 1  1 (  1)(  1) (  1)(  1)    2  1  2  1
 1  1 (  1)(  1) (  1)(  1)  (  1)(  1) (  1)( 
1)
x x x x
    
x x x x
2 1 2 1
x x x
1)( 1) ( 1)( 1)
x x x x x x x x
x x x x
x
x x x x x x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 23
Objective #2: Example
2 2
2 2
2 2
2 1 ( 2 1)
(
x

2 2
2 2 2 2
2 1 2 1 2 1 2 1
( 1)( 1) ( 1)( 1)
4
( 1)( 1) 4 ( 1)( 1) 4
2 x 2 2 ( x 1)( x 1) 2 x 2 2 2( x
2 1)
( x 1)( x
1)
2
x
x
2 1
    
   

         
   
   
   
    
 


CONTINUED

24. x

1 
4
 
x x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 24
Objective #2: Example
2b. Simplify:
2
1 2
1 7 10
 
Rewrite the expression without negative exponents.
Then multiply the numerator and denominator
by the LCD of 2. x

25. x

1 
4
 
x x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25
Objective #2: Example
2b. Simplify:
2
1 2
1 7 10
 
Rewrite the expression without negative exponents.
Then multiply the numerator and denominator
by the LCD of 2. x

26. x x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 26
Objective #2: Example
2 2
1 2
2
2
2
2 2 2
2 2 2
2
2 2
2
2
4
1
1 4
1 7 10 7 10 1
4 4
1 1
7 10 7 10 1 1
4 ( 2)( 2) 2
7 10 ( 5)( 2) 5
x x
x x
x
x
x x x
x x x
x x x x x
x x x x
x x x x x

 



   

  
  
      
   
  
    
CONTINUED

27. x x
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 27
Objective #2: Example
2 2
1 2
2
2
2
2 2 2
2 2 2
2
2 2
2
2
4
1
1 4
1 7 10 7 10 1
4 4
1 1
7 10 7 10 1 1
4 ( 2)( 2) 2
7 10 ( 5)( 2) 5
x x
x x
x
x
x x x
x x x
x x x x x
x x x x
x x x x x

 



   

  
  
      
   
  
    
CONTINUED