proportions and the multiplier method for solving rational equations

1. Addition and Subtraction of Rational Expressions
Frank Ma © 2011

2. Addition and Subtraction of Rational Expressions
Only fractions with the same denominator may be added or
subtracted directly.

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

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
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.

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
Usually we put 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
+ =

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
Usually we put 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

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
Usually we put 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

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
Usually we put 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

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
Usually we put 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

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
Usually we put 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

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
Usually we put 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

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
Usually we put 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

13. 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
Usually we put 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

14. 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
Usually we put 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

15. 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
Usually we put 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

16. 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
Usually we put 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

17. 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

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

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
Multiplier Method
Given the fraction , to convert it into denominator D,
the new numerator is
A
B
A
B
* 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
Multiplier Method
Given the fraction , to convert it into denominator D,
the new numerator is
Example B.
a. Convert to a fraction with denominator 12.
A
B
A
B
* D.
5
4
In practice, we write this as
A
B
= A
B
* D D
new numerator

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,
the new numerator is
Example B.
a. Convert to a fraction with denominator 12.
A
B
A
B
* D.
5
4
In practice, we write this as
A
B
= A
B
* D D
5
4
=
new numerator

22. 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,
the new numerator is
Example B.
a. Convert to a fraction with denominator 12.
A
B
A
B
* D.
5
4
5
4
* 12
In practice, we write this as
A
B
= A
B
* D D
5
4
= 12
new numerator
new numerator

23. 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,
the new numerator is
Example B.
a. Convert to a fraction with denominator 12.
A
B
A
B
* D.
5
4
5
4
* 12
3
In practice, we write this as
A
B
= A
B
* D D
5
4
= 12
new numerator
new numerator

24. 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,
the new numerator is
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 this as
A
B
= A
B
* D D
5
4
= 12 =
new numerator

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

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

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

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

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

30. 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.

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)
new numerator
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
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.

33. 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.

34. 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.

35. 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.

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

37. Addition and Subtraction of Rational Expressions
We give two methods of combining rational expressions below.
The first one is the traditional method.

38. Addition and Subtraction of Rational Expressions
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

39. Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

40. Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

41. 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.
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

42. 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.
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

43. 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.
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

44. Example C. Combine
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
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

45. Example C. 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.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

46. Example C. 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.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

47. Example C. 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
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

48. Example C. 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
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

49. Example C. 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
6xy2Hence
4y
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

50. Example C. 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
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

51. Example C. 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
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.

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

53. 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 =
2x2 + x – 2 =

54. 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 =

55. 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)

56. 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)

57. 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

58. 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)

59. 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

60. 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

61. 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) LCD

62. 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

63. 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)

64. 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) * 2(2x – 1)(x + 1) LCD

65. 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) * 2(2x – 1)(x + 1) LCD

66. 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) * 2(2x – 1)(x + 1) LCD
= 2(x – 1) LCD

67. 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) * 2(2x – 1)(x + 1) LCD
= 2(x – 1) =
2x – 2
LCD LCD

68. 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) * 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

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

70. 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

71. 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

72. 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)

73. 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)
This is simplified because the numerator is not factorable.

74. 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)
This is simplified because the numerator is not factorable.
Multiplier Method

75. 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)
This is simplified because the numerator is not factorable.
Multiplier Method
The Multiplier Method is the same method used for fractional
numbers.

76. 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)
This is simplified because the numerator is not factorable.
Multiplier Method
The Multiplier Method is the same method used for fractional
numbers. We used the fact that x = (x * LCD) / LCD = x * 1

77. 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)
This is simplified because the numerator is not factorable.
Multiplier Method
The Multiplier Method is the same method used for fractional
numbers. We used the fact that x = (x * LCD) / LCD = x * 1
then compute (x * LCD) using the Distributive Law.

78. 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)
This is simplified because the numerator is not factorable.
Multiplier Method
The Multiplier Method is the same method used for fractional
numbers. We used the fact that x = (x * LCD) / LCD = x * 1
then compute (x * LCD) using the Distributive Law. Following
is an example using this method.

79. Example E. Calculate
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9

80. Example E. Calculate
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72.

81. Example E. Calculate
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.

82. Example E. Calculate
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )

83. Example E. Calculate
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72

84. Example E. Calculate
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication

85. Example E. Calculate
6
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication

86. Example E. Calculate
6 9
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication

87. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication

88. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72

89. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=

90. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y

91. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.

92. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.

93. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2)

94. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. 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

95. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. 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

96. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. 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

97. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. 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=

98. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4).

99. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)

100. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4)

101. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)

102. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [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)

103. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [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

104. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [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)
=
Example H. Combine x
x2 – 2x
– x – 1
x2 – 4
2(x + 13)
(x – 2)(x + 4)
or

105. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [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)
=
Example H. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
2(x + 13)
(x – 2)(x + 4)
or

106. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [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)
=
Example H. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
2(x + 13)
(x – 2)(x + 4)
or

107. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [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)
=
Example H. 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)
2(x + 13)
(x – 2)(x + 4)
or

108. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [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)
=
Example H. 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).
2(x + 13)
(x – 2)(x + 4)
or

109. Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions

110. Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
=
x
x2 – 2x
– x – 1
x2 – 4

111. * x( x – 2)(x + 2)
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4

112. * x( x – 2)(x + 2)
(x + 2)
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4

113. * x( x – 2)(x + 2)
(x + 2) x
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4

114. * x( x – 2)(x + 2)
(x + 2) x
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
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

115. * x( x – 2)(x + 2)
(x + 2) x
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
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

116. * x( x – 2)(x + 2)
(x + 2) x
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
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)

117. * x( x – 2)(x + 2)
(x + 2) x
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
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)

118. 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 – 2
7. 9x2
3x – 2
– 4
3x – 2
8.
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

119. 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 – 3x
22. –5x + 7
3x – 12
+ 4x – 3
–2x + 8
23.
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 – 4
28.
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 – 631.
x2 + 5x + 6
3x + 4
+
2x – 3
–x2 – 2x + 3
32.
x2 – x
5x – 4
–
3x – 5
1 – x233.
x2 + 2x – 3