The document discusses adding and subtracting rational expressions. It provides examples of adding rational expressions with the same denominator by adding the numerators, and with different denominators by finding the least common denominator. It also demonstrates subtracting rational expressions by subtracting the numerators while keeping the same denominator. The document emphasizes using parentheses when subtracting to avoid sign errors.

1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
Rational
Expressions and
Applications
14

2. Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
1. Add rational expressions having the same
denominator.
2. Add rational expressions having different
denominators.
3. Subtract rational expressions.
Objectives
14.4 Adding and Subtracting Rational
Expressions

3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Add Rational Expressions Having the Same Denominator
Adding Rational Expressions (Same Denominator)
The rational expressions and (where Q ≠ 0) are
added as follows.
In words: To add rational expressions with the same
denominator, add the numerators and keep the same
denominator.
P
Q
R
Q
P R P R
Q Q Q

 

4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Add. Write each answer in lowest terms.
7 8
(a)
18 18

Add Rational Expressions Having the Same Denominator
7 8
18


15
18

5
6

2 2
2 2
6 4
(b)
1 1
 

 
x x x x
x x
2 2
2
6 4
1
  


x x x x
x
2
2
5 5
1
x x
x



5
( 1
( 1)
( 1) )
x
x
x
x



5
1
x
x


6
3
3
5



5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Add Rational Expressions Having Different Denominators
Adding Rational Expressions (Different Denominators)
Step 1 Find the least common denominator (LCD).
Step 2 Write each rational expression as an
equivalent rational expression with the LCD as
the denominator.
Step 3 Add the numerators to get the numerator of the
sum. The LCD is the denominator of the sum.
Step 4 Write in lowest terms using the fundamental
property.

6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Add. Write the answer in lowest terms.
5 4
6 15

Add Rational Expressions Having Different Denominators
25 8
30 30
 
33
30

11 1
, or 1
10 10


7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Add. Write the answer in lowest terms.
2
2
3 2
16 4
a a
a a


 
Add Rational Expressions Having Different Denominators
  
2
3 2
4 4 4
a a
a a a

 
  
  
 
  
2
2 43
4 4 4 4
a aa
a a a a
 
 
   
     
2 2
3 2 8
4 4 4 4
a a a
a a a a
 
 
   
  
2
8
4 4
a a
a a


 

8. Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Add. Write the answer in lowest terms.
2 2
4 3
8 15 3 10
z z
z z z z


   
Add Rational Expressions Having Different Denominators
4 3
( 3)( 5) ( 2)( 5)
z z
z z z z

 
   
4 ( 3)
( 3)( 5) ( 2)(
( 2) ( 3)
( 2) 35)( )
z zz z
z zz z z z

 
   
 

2 2
4 8 9
( 3)( 5)( 2)
z z z
z z z
  

  
2
5 8 9
( 3)( 5)( 2)
z z
z z z
 

  

9. Slide - 9Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Add. Write the answer in lowest terms.
Add Rational Expressions Having Different Denominators
2
5 7
1 1
y
y y


 
2
5 7
1 ( 1)1
y
y y


 
 
2
5 7
1 1
y
y y

 

 
2
5 7
1
y
y
 


2
12
1
y
y




10. Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Subtract Rational Expressions
Subtracting Rational Expressions (Same Denominator)
The rational expressions and (where Q ≠ 0) are
subtracted as follows.
In words: To subtract rational expressions with the same
denominator, subtract the numerators and keep the same
denominator.
P
Q
R
Q
.
P R P R
Q Q Q

 

11. Slide - 11Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Subtract. Write the answer in lowest terms.
3 1 2 5
2 2
x x
x x
 

 
Subtract Rational Expressions
3 1 (2 5)
2
x x
x
  


3 1 2 5
2
x x
x
  


4
2
x
x




12. Slide - 12Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Subtract Rational Expressions
CAUTION
Sign errors often occur in problems like the previous
one. The numerator of the fraction being subtracted
must be treated as a single quantity.
Be sure to use parentheses after the subtraction
sign.

13. Slide - 13Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Subtract. Write the answer in lowest terms.
2 3 2
3 5 1x x
x x x



Subtract Rational Expressions
2 2
3 5 1
( 1)
x x
x x x

 

2 2
3 (5 1)
( 1
( 1
)
)
( 1)
x
x
x x
x x x

 




2
2
3 3 5 1
( 1)
x x x
x x
  


2
2
3 8 1
( 1)
x x
x x
 



14. Slide - 14Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Subtract. Write the answer in lowest terms.
2 2
3 9
2 7 4 7 12x x x x

   
Subtract Rational Expressions
3 9
(2 1)( 4) ( 3)( 4)x x x x
 
   

3(x  3)
(2x 1)(x  4)(x  3)

9(2x 1)
(x  3)(x  4)(2x 1)

3x  9
(2x 1)(x  4)(x  3)

18x  9
(x  3)(x  4)(2x 1)

3x  9 18x  9
(2x 1)(x  4)(x  3)

15x 18
(2x 1)(x  4)(x  3)