The document discusses addition and subtraction of rational expressions. It states that fractions with the same denominator can be directly added or subtracted, while those with different denominators must first be converted to have a common denominator. The document provides an addition/subtraction rule and examples demonstrating how to perform these operations on rational expressions, including converting fractions to equivalent forms with a specified common denominator.

1. Addition and Subtraction of Rational Expressions

2. Addition and Subtraction of Rational Expressions
Only fractions with the same denominator may be added or
subtracted directly.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
b. 3x
2x – 3
– 6 – x
2x – 3
=

3. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
b. 3x
2x – 3
– 6 – x
2x – 3
=

4. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
b. 3x
2x – 3
– 6 – x
2x – 3
=

5. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
b. 3x
2x – 3
– 6 – x
2x – 3
=

6. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
b. 3x
2x – 3
– 6 – x
2x – 3
=
Write the result in the factored form, cancel the common
factor and give the simplified answer.

7. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
– 6 – x
2x – 3
=
Write the result in the factored form, cancel the common
factor and give the simplified answer.

8. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Write the result in the factored form, cancel the common
factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
– 6 – x
2x – 3
= 3x – (6 – x)
2x – 3

9. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Write the result in the factored form, cancel the common
factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
– 6 – x
2x – 3
= 3x – (6 – x)
2x – 3
= 3x – 6 + x
2x – 3

10. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Write the result in the factored form, cancel the common
factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
– 6 – x
2x – 3
= 3x – (6 – x)
2x – 3
= 3x – 6 + x
2x – 3
=
4x – 6
2x – 3

11. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Write the result in the factored form, cancel the common
factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
– 6 – x
2x – 3
= 3x – (6 – x)
2x – 3
= 3x – 6 + x
2x – 3
=
4x – 6
2x – 3
=
2(2x – 3)
2x – 3

12. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Write the result in the factored form, cancel the common
factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
– 6 – x
2x – 3
= 3x – (6 – x)
2x – 3
= 3x – 6 + x
2x – 3
=
4x – 6
2x – 3
=
2(2x – 3)
2x – 3
= 2

13. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator.
Addition and Subtraction of Rational Expressions

14. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions

15. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier Method
Given the fraction , to convert it into denominator D as ,
the new numerator N =
A
B
A
B * D.
N
D

16. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier Method
Given the fraction , to convert it into denominator D as ,
the new numerator N =
A
B
A
B * D.
In practice, we write that
A
B
=> A
B
* D D.
new numerator N
N
D

17. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Example B.
a. Convert to a fraction with denominator 12.5
4
Multiplier Method
Given the fraction , to convert it into denominator D as ,
the new numerator N =
A
B
A
B * D.
In practice, we write that
A
B
=> A
B
* D D.
new numerator N
N
D

18. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Example B.
a. Convert to a fraction with denominator 12.5
4
5
4
=
Multiplier Method
Given the fraction , to convert it into denominator D as ,
the new numerator N =
A
B
A
B * D.
In practice, we write that
A
B
=> A
B
* D D.
new numerator N
N
D

19. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Example B.
a. Convert to a fraction with denominator 12.5
4
5
4
* 12
5
4
= 12
the new numerator
Multiplier Method
Given the fraction , to convert it into denominator D as ,
the new numerator N =
A
B
A
B * D.
In practice, we write that
A
B
=> A
B
* D D.
new numerator N
N
D

20. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Example B.
a. Convert to a fraction with denominator 12.5
4
5
4
* 12
35
4
= 12
the new numerator
Multiplier Method
Given the fraction , to convert it into denominator D as ,
the new numerator N =
A
B
A
B * D.
In practice, we write that
A
B
=> A
B
* D D.
new numerator N
N
D

21. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier Method
Given the fraction , to convert it into denominator D as ,
the new numerator N =
Example B.
a. Convert to a fraction with denominator 12.
A
B
A
B * D.
5
4
5
4
* 12
3 15
12
In practice, we write that
A
B
=> A
B
* D D.
5
4
= 12 =
new numerator N
the new numerator
with the new denominator 12.
N
D

22. b. Convert into an expression with denominator 12xy2.
Addition and Subtraction of Rational Expressions
3x
4y

23. Addition and Subtraction of Rational Expressions
3x
4y
3x
4y
b. Convert into an expression with denominator 12xy2.

24. Addition and Subtraction of Rational Expressions
3x
4y
*12xy23x
4y
=
3x
4y
12xy2
the new numerator
b. Convert into an expression with denominator 12xy2.

25. Addition and Subtraction of Rational Expressions
3x
4y
*12xy23x
4y
=
3x
4y
12xy2
3xy
b. Convert into an expression with denominator 12xy2.

26. Addition and Subtraction of Rational Expressions
3x
4y
*12xy23x
4y
=
3x
4y
12xy2 =
9x2y
12xy2
3xy
b. Convert into an expression with denominator 12xy2.

27. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
x + 1
2x + 3
3x
4y
=
3x
4y
12xy2 =
9x2y
12xy2
3xy
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.

28. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
x + 1
2x + 3
x + 1
2x + 3
3x
4y
=
3x
4y
12xy2 =
9x2y
12xy2
3xy
=
x + 1
2x + 3
* (4x2 – 9) (4x2 – 9)
new numerator
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.

29. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
x + 1
2x + 3
x + 1
2x + 3
3x
4y
=
3x
4y
12xy2 =
9x2y
12xy2
3xy
=
x + 1
2x + 3
* (4x2 – 9) (4x2 – 9)
=
x + 1
2x + 3
* (2x + 3)(2x – 3) (4x2 – 9)
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.

30. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
x + 1
2x + 3
x + 1
2x + 3
3x
4y
=
3x
4y
12xy2 =
9x2y
12xy2
3xy
=
x + 1
2x + 3
* (4x2 – 9) (4x2 – 9)
=
x + 1
2x + 3
* (2x + 3)(2x – 3) (4x2 – 9)
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.

31. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
x + 1
2x + 3
x + 1
2x + 3
3x
4y
=
3x
4y
12xy2 =
9x2y
12xy2
3xy
=
x + 1
2x + 3
* (4x2 – 9) (4x2 – 9)
=
x + 1
2x + 3
* (2x + 3)(2x – 3) (4x2 – 9)
= (x + 1)(2x – 3) (4x2 – 9)
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.

32. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
c. Convert into an expression denominator 4x2 – 9.
x + 1
2x + 3
x + 1
2x + 3
3x
4y
=
3x
4y
12xy2 =
9x2y
12xy2
3xy
=
x + 1
2x + 3
* (4x2 – 9) (4x2 – 9)
=
x + 1
2x + 3
* (2x + 3)(2x – 3) (4x2 – 9)
= (x + 1)(2x – 3) (4x2 – 9)
=
2x2 – x – 3
4x2 – 9
b. Convert into an expression with denominator 12xy2.

33. Addition and Subtraction of Rational Expressions
We give two methods of combining rational expressions below.

34. Addition and Subtraction of Rational Expressions
We give two methods of combining rational expressions below.
The first one is an extension of the above Multiplier Method,
the lengthier traditional method is given later.

35. Addition and Subtraction of Rational Expressions
Example C. Calculate 7
12
+
5
8
–
4
9
The Multiplier Method (Adding/Subtracting Fractions)
We give two methods of combining rational expressions below.
The first one is an extension of the above Multiplier Method,
the lengthier traditional method is given later.

36. Addition and Subtraction of Rational Expressions
Example C. Calculate 7
12
+
5
8
–
4
9
The Multiplier Method (Adding/Subtracting Fractions)
The Multiplier Method finds the answer by converting the
entire problem to a new denominator, the LCD of all the terms.
(i.e. * LCD/LCD to the problem.)
We give two methods of combining rational expressions below.
The first one is an extension of the above Multiplier Method,
the lengthier traditional method is given later.

37. Addition and Subtraction of Rational Expressions
Example C. Calculate 7
12
+
5
8
–
4
9
The LCD is 72.
The Multiplier Method (Adding/Subtracting Fractions)
The Multiplier Method finds the answer by converting the
entire problem to a new denominator, the LCD of all the terms.
(i.e. * LCD/LCD to the problem.)
We give two methods of combining rational expressions below.
The first one is an extension of the above Multiplier Method,
the lengthier traditional method is given later.

38. Addition and Subtraction of Rational Expressions
Example C. Calculate 7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD,
then put the result over the new LCD denominator.
(i.e. * LCD/LCD.)
7
12
+
5
8
–
4
9
( )
The Multiplier Method (Adding/Subtracting Fractions)
The Multiplier Method finds the answer by converting the
entire problem to a new denominator, the LCD of all the terms.
(i.e. * LCD/LCD to the problem.)
We give two methods of combining rational expressions below.
The first one is an extension of the above Multiplier Method,
the lengthier traditional method is given later.

39. Addition and Subtraction of Rational Expressions
Example C. Calculate 7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD,
then put the result over the new LCD denominator.
(i.e. * LCD/LCD.)
7
12
+
5
8
–
4
9
( )* 72 72
The Multiplier Method (Adding/Subtracting Fractions)
The Multiplier Method finds the answer by converting the
entire problem to a new denominator, the LCD of all the terms.
(i.e. * LCD/LCD to the problem.)
We give two methods of combining rational expressions below.
The first one is an extension of the above Multiplier Method,
the lengthier traditional method is given later.

40. Addition and Subtraction of Rational Expressions
Example C. Calculate
6
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD,
then put the result over the new LCD denominator.
(i.e. * LCD/LCD.)
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
The Multiplier Method (Adding/Subtracting Fractions)
The Multiplier Method finds the answer by converting the
entire problem to a new denominator, the LCD of all the terms.
(i.e. * LCD/LCD to the problem.)
We give two methods of combining rational expressions below.
The first one is an extension of the above Multiplier Method,
the lengthier traditional method is given later.

41. Addition and Subtraction of Rational Expressions
Example C. Calculate
6 9 8
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD,
then put the result over the new LCD denominator.
(i.e. * LCD/LCD.)
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
The Multiplier Method (Adding/Subtracting Fractions)
The Multiplier Method finds the answer by converting the
entire problem to a new denominator, the LCD of all the terms.
(i.e. * LCD/LCD to the problem.)
We give two methods of combining rational expressions below.
The first one is an extension of the above Multiplier Method,
the lengthier traditional method is given later.

42. Addition and Subtraction of Rational Expressions
Example C. Calculate
6 9 8
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD,
then put the result over the new LCD denominator.
(i.e. * LCD/LCD.)
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= (42 + 45 – 32) 72
The Multiplier Method (Adding/Subtracting Fractions)
The Multiplier Method finds the answer by converting the
entire problem to a new denominator, the LCD of all the terms.
(i.e. * LCD/LCD to the problem.)
We give two methods of combining rational expressions below.
The first one is an extension of the above Multiplier Method,
the lengthier traditional method is given later.

43. Addition and Subtraction of Rational Expressions
Example C. Calculate
6 9 8
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD,
then put the result over the new LCD denominator.
(i.e. * LCD/LCD.)
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= (42 + 45 – 32) 72
55
=
The Multiplier Method (Adding/Subtracting Fractions)
The Multiplier Method finds the answer by converting the
entire problem to a new denominator, the LCD of all the terms.
(i.e. * LCD/LCD to the problem.)
We give two methods of combining rational expressions below.
The first one is an extension of the above Multiplier Method,
the lengthier traditional method is given later.
72

44. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
Example E. Combine 5
x– 2
– 3
x + 4

45. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Example E. Combine 5
x– 2
– 3
x + 4

46. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2)
Example E. Combine 5
x– 2
– 3
x + 4

47. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3
Example E. Combine 5
x– 2
– 3
x + 4

48. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3 2xy
Example E. Combine 5
x– 2
– 3
x + 4

49. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3 2xy
9 – 10x2y
12xy2=
Example E. Combine 5
x– 2
– 3
x + 4

50. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3 2xy
9 – 10x2y
12xy2=
Example E. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4), multiplying the problem by LCD/LCD:

51. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3 2xy
9 – 10x2y
12xy2=
Example E. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4), multiplying the problem by LCD/LCD:
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)

52. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3 2xy
9 – 10x2y
12xy2=
Example E. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4), multiplying the problem by LCD/LCD:
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)

53. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3 2xy
9 – 10x2y
12xy2=
Example E. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4), multiplying the problem by LCD/LCD:
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)

54. Addition and Subtraction of Rational Expressions
Example D. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3 2xy
9 – 10x2y
12xy2=
Example E. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4), multiplying the problem by LCD/LCD:
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26
(x – 2)(x + 4)
= 2(x + 13)
(x – 2)(x + 4)
or

55. Addition and Subtraction of Rational Expressions
Example F. Combine x
x2 – 2x
– x – 1
x2 – 4

56. Addition and Subtraction of Rational Expressions
Example F. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.

57. Addition and Subtraction of Rational Expressions
Example F. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)

58. Addition and Subtraction of Rational Expressions
Example F. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
x2 – 4 = (x – 2)(x + 2)

59. Addition and Subtraction of Rational Expressions
Example F. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
x2 – 4 = (x – 2)(x + 2)
Hence the LCD = x(x – 2)(x + 2).

60. Addition and Subtraction of Rational Expressions
Example F. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
x2 – 4 = (x – 2)(x + 2)
Hence the LCD = x(x – 2)(x + 2).
* x( x – 2)(x + 2)x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4

61. Addition and Subtraction of Rational Expressions
Example F. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
x2 – 4 = (x – 2)(x + 2)
Hence the LCD = x(x – 2)(x + 2).
* x( x – 2)(x + 2)
(x + 2) x
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4

62. Addition and Subtraction of Rational Expressions
Example F. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
x2 – 4 = (x – 2)(x + 2)
Hence the LCD = x(x – 2)(x + 2).
* x( x – 2)(x + 2)
(x + 2) x
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4
= [x(x + 2) – x(x – 1)] LCD

63. Addition and Subtraction of Rational Expressions
Example F. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
x2 – 4 = (x – 2)(x + 2)
Hence the LCD = x(x – 2)(x + 2).
* x( x – 2)(x + 2)
(x + 2) x
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4
= [x(x + 2) – x(x – 1)] LCD
= [x2 + 2x – x2 + x)] LCD
=
3x
x (x – 2)(x + 2)

64. Addition and Subtraction of Rational Expressions
Example F. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
x2 – 4 = (x – 2)(x + 2)
Hence the LCD = x(x – 2)(x + 2).
* x( x – 2)(x + 2)
(x + 2) x
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4
= [x(x + 2) – x(x – 1)] LCD
= [x2 + 2x – x2 + x)] LCD
=
3x
x (x – 2)(x + 2)
=
3
(x – 2)(x + 2)

65. Addition and Subtraction of Rational Expressions
Traditionally, we add/subtract fractions by converting each
fraction separately. (The multiplier–method keeps all the
calculation in one place and shortens the process.)

66. Example G. Combine
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
2
3xy
–
x
2y2
Traditionally, we add/subtract fractions by converting each
fraction separately. (The multiplier–method keeps all the
calculation in one place and shortens the process.)

67. Example G. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
2
3xy
–
x
2y2
Traditionally, we add/subtract fractions by converting each
fraction separately. (The multiplier–method keeps all the
calculation in one place and shortens the process.)

68. Example G. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
2
3xy
–
x
2y2
Traditionally, we add/subtract fractions by converting each
fraction separately. (The multiplier–method keeps all the
calculation in one place and shortens the process.)

69. Example G. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
2
3xy
–
x
2y2
2
3xy = 6xy2
x
2y2 =
3x2
6xy2
4y
Traditionally, we add/subtract fractions by converting each
fraction separately. (The multiplier–method keeps all the
calculation in one place and shortens the process.)

70. Example G. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
2
3xy = 6xy2
x
2y2 =
3x2
6xy2
4y
Traditionally, we add/subtract fractions by converting each
fraction separately. (The multiplier–method keeps all the
calculation in one place and shortens the process.)

71. Example G. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
2
3xy = 6xy2
x
2y2 =
3x2
6xy2
2
3xy
–
x
2y2 =
4y
6xy2 –
3x2
6xy2 =Hence
4y – 3x2
6xy2
4y
Traditionally, we add/subtract fractions by converting each
fraction separately. (The multiplier–method keeps all the
calculation in one place and shortens the process.)

72. Example G. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
2
3xy = 6xy2
x
2y2 =
3x2
6xy2
2
3xy
–
x
2y2 =
4y
6xy2 –
3x2
6xy2 =Hence
4y – 3x2
6xy2
This is simplified because the numerator is not factorable.
4y
Traditionally, we add/subtract fractions by converting each
fraction separately. (The multiplier–method keeps all the
calculation in one place and shortens the process.)

73. Example H. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1

74. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 =
2x2 + x – 2 =
Example H. Combine

75. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 =
Example H. Combine

76. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Example H. Combine

77. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Example H. Combine

78. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
Example H. Combine

79. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1)
Example H. Combine

80. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
Example H. Combine

81. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
Example H. Combine

82. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) LCD
Example H. Combine

83. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
Example H. Combine

84. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1
=
x – 1
(2x – 1)(x + 1)
Example H. Combine

85. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1
=
x – 1
(2x – 1)(x + 1)
* 2(2x – 1)(x + 1) LCD
Example H. Combine

86. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1
=
x – 1
(2x – 1)(x + 1)
* 2(2x – 1)(x + 1) LCD
Example H. Combine

87. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1
=
x – 1
(2x – 1)(x + 1)
* 2(2x – 1)(x + 1) LCD
= 2(x – 1) LCD
Example H. Combine

88. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1
=
x – 1
(2x – 1)(x + 1)
* 2(2x – 1)(x + 1) LCD
= 2(x – 1) =
2x – 2
LCD LCD
Example H. Combine

89. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2 = x
2(2x – 1) * 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1
=
x – 1
(2x – 1)(x + 1)
* 2(2x – 1)(x + 1) LCD
= 2(x – 1) =
2x – 2
LCD LCD
Hence
x
4x – 2
–
x – 1
2x2 + x – 1
=
x2 + x
LCD
–
2x – 2
LCD
Example H. Combine

90. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
=
x2 + x
LCD –
2x – 2
LCD

91. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
=
x2 + x
LCD –
2x – 2
LCD

92. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
=
x2 + x
LCD –
2x – 2
LCD
=
x2 + x – 2x + 2
LCD

93. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
=
x2 + x
LCD –
2x – 2
LCD
=
x2 + x – 2x + 2
LCD
=
x2 – x + 2
2(2x – 1)(x + 1)
Self–Check:
Do it by the multiplier method to see which way you prefer.
x
2(2x – 1)
–
x – 1
( x + 1)(2x – 1)
[ ]* 2(2x – 1)(x + 1) / LCD

94. Ex. A. Combine and simplify the answers.
Addition and Subtraction of Rational Expressions
x
x – 2
– 2
x – 2
1.
2x
x – 2
+
4
x – 2
2.
3x
x + 3
+ 6
x + 3
3. – 2x
x – 4
+ 8
x – 4
4.
x + 2
2x – 1
–
2x – 1
5.
2x + 5
x – 2
–
4 – 3x
2 – x
6.
x2 – 2
x – 2
– x
x – 27.
9x2
3x – 2 –
4
3x – 28.
Ex. B. Combine and simplify the answers.
3
12
+ 5
6
– 2
3
9. 11
12
+
5
8
– 7
6
10. –5
6
+ 3
8
– 311.
12.
6
5xy2
– x
6y13.
3
4xy2
– 5x
6y
15. 7
12xy
– 5x
8y316.
5
4xy
– 7x
6y214.
3
4xy2
– 5y
12x217.
–5
6 –
7
12+ 2
+ 1 – 7x
9y2
4 – 3x

95. Ex. C. Combine and simplify the answers.
Addition and Subtraction of Rational Expressions
x
2x – 4
– 2
3x – 6
18.
2x
3x + 9
–
4
2x + 6
19.
–3
2x + 1
+ 2x
4x + 2
20. 2x – 3
x – 2
– 3x + 4
5 – 10x
21.
3x + 1
6x – 4
– 2x + 3
2 – 3x22.
–5x + 7
3x – 12+
4x – 3
–2x + 823.
x
x – 2
– 2
x – 3
24. 2x
3x + 1
+ 4
x – 6
25.
–3
2x + 1
+ 2x
3x + 2
26.
2x – 3
x – 2
+
3x + 4
x – 5
27.
3x + 1
+
x + 3
x2 – 428.
x2 – 4x + 4
x – 4
–
x + 5
x2 – x – 2
29.
x2 – 5x + 6
3x + 1
+
2x + 3
9 – x230.
x2 – x – 6
3x – 4
–
2x + 5
x2 + x – 6
31.
x2 + 5x + 6
3x + 4
+
2x – 3
–x2 – 2x + 3
32.
x2 – x
5x – 4
–
3x – 5
1 – x233.
x2 + 2x – 3