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### 2 1 addition and subtraction i

1. 1. Addition and Subtraction of Rational Expressions
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 22. b. Convert into an expression with denominator 12xy2. Addition and Subtraction of Rational Expressions 3x 4y
23. 23. Addition and Subtraction of Rational Expressions 3x 4y 3x 4y b. Convert into an expression with denominator 12xy2.
24. 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. 25. Addition and Subtraction of Rational Expressions 3x 4y *12xy23x 4y = 3x 4y 12xy2 3xy b. Convert into an expression with denominator 12xy2.
26. 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. 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. 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. 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. 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. 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. 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. 33. Addition and Subtraction of Rational Expressions We give two methods of combining rational expressions below.
34. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 44. Addition and Subtraction of Rational Expressions Example D. Combine 3 4xy2 – 5x 6y Example E. Combine 5 x– 2 – 3 x + 4
45. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 55. Addition and Subtraction of Rational Expressions Example F. Combine x x2 – 2x – x – 1 x2 – 4
56. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 73. Example H. Combine Addition and Subtraction of Rational Expressions x 4x – 2 – x – 1 2x2 + x – 1
74. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 90. Addition and Subtraction of Rational Expressions x 4x – 2 – x – 1 2x2 + x – 1 = x2 + x LCD – 2x – 2 LCD
91. 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. 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. 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. 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. 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