9/14/10 12:39 PM Page 381 Chapter dug84356_ch06a.qxd 6 Rational Expressions Advanced technical developments have made sports equipment faster, lighter, and more responsive to the human body. Behind the more exible skis, lighter bats, and comfortable athletic shoes lies the science of biomechanics, which is the study of human movement and the factors that inuence it. Designing and testing an athletic shoe go hand in hand. While a shoe is being designed, it is tested in a multitude of ways, including long-term wear, rear foot stability, and strength of materials. Testing basketball shoes usually includes an evaluation of the force applied to the ground by the foot during running, jumping, and landing. Many biome chanics laboratories have a 5 6.1 Reducing Rational Expressions 6.2 Multiplication and Division 6.3 Finding the Least Common Denominator 6.4 Addition and Subtraction 6.5 Complex Fractions 6.6 Applications of Ratios and Proportions 6.8 Applications of Rational Expressions player cuts from side to side, as well as the force against the bottom of the shoe. Force exerted in landing from a lay up shot can be as high as 4 3 2 Solving Equations with Rational Expressions 6.7 sure the force exerted when a Force (thousands of pounds) special platform that can mea- 14 times the weight of the body. Side-to-side force is usu ally about 1 to 2 body weights 1 0 50 100 150 200 Weight (pounds) 250 300 in a cutting movement. In Exercises 53 and 54 of Section 6.7 you will see how designers of athletic shoes use proportions to nd the amount of force on the foot and soles of shoes for activities such as running and jumping. dug84356_ch06a.qxd 382 9/14/10 12:39 PM Page 382 6-2 Chapter 6 Rational Expressions 6.1 In This Section U1V Rational Expressions and Functions 2V Reducing to Lowest Terms U U3V Reducing with the Quotient Rule for Exponents U4V Dividing a - b by b - a U5V Factoring Out the Opposite of a Common Factor 6V Writing Rational Expressions U E X A M P L E 1 Reducing Rational Expressions Rational expressions in algebra are similar to the rational numbers in arithmetic. In this section, you will learn the basic ideas of rational expressions. U1V Rational Expressions and Functions A rational number is the ratio of two integers with the denominator not equal to 0. For example, 0 3 -9 -7 , , , and 1 2 4 -6 are rational numbers. Of course, we usually write the last three of these rational num 3 bers in their simpler forms , -7, and 0. A rational expression is the ratio of two poly 2 nomials with the denominator not equal to 0. Because an integer is a monomial, a rational number is also a rational expression. As with rational numbers, if the denom inator is 1, it can be omitted. Some examples of rational expressions are x 2 - 1 3a 2 + 5a - 3 3 , , , and 9x. x+8 a-9 7 A rational expression involving a variable has no value unless we assign a value to the variable. If the value of a rational expression is used to determine the value of a second variable, then we have a rational function. For example, x2 - 1 3a2 + 5a - 3 and w = y= x+8 a-9 are rational functions. We can evaluate a rational expression with or without function notation as we did for polynomials in Chapter 5. Evaluating a rational expression a) Find the value of 4x - 1 for x = -3. x+2 U Calculator Close-Up V To evaluate the rational expression in Example 1(a) with a calculator, rst use Y = to dene the rational expres sion. Be sure to enclose both numera tor and denominator in parentheses. Then nd y1(-3). 3x + 2 b) If R(x) = 2x - 1, nd R(4). Solution a) To nd the value of 4x - 1 for x = -3, replace x by -3 in the rational expression: x+2 4(-3) - 1 -13 = = 13 -3 + 2 -1 So the value of the rational expression is 13. The Calculator Close-Up shows how to evaluate the expression with a graphing calculator using a variable. With a scientic or graphing calculator you could also evaluate the expression by entering (4(-3) - 1)/ (-3 + 2). Be sure to enclose the numerator and denominator in parentheses. b) R(4) is the value of the rational expression when x = 4. To nd R(4), replace x by 4 3x + 2 in R(x) = 2x - 1: 3(4) + 2 R(4) = 2(4) - 1 14 R(4) = = 2 7 So the value of the rational expression is 2 when x = 4, or R(4) = 2 (read \"R of 4 is 2\"). Now do Exercises 1-6 dug84356_ch06a.qxd 9/14/10 12:39 PM Page 383 6-3 6.1 Reducing Rational Expressions 383 5 An expression such as 0 is undened because the denition of rational numbers does not allow zero in the denominator. When a variable occurs in a denominator, any real number can be used for the variable except numbers that make the expression undened. E X A M P L E 2 Ruling out values for x Which numbers cannot be used in place of x in each rational expression? a) x2 - 1 x+8 b) x+2 2x + 1 c) x+5 x2 - 4 Solution a) The denominator is 0 if x + 8 = 0, or x = -8. So -8 cannot be used in place of x. (All real numbers except -8 can be used in place of x.) 1 1 b) The denominator is zero if 2x + 1 = 0, or x = -2. So we cannot use -2 in place 1 of x. (All real numbers except -2 can be used in place of x.) c) The denominator is zero if x 2 - 4 = 0. Solve this equation: x-2=0 x=2 x2 - 4 = 0 (x - 2)(x + 2) = 0 Factor. or x + 2 = 0 Zero factor property or x = -2 So 2 and -2 cannot be used in place of x. (All real numbers except 2 and -2 can be used in place of x.) Now do Exercises 7-14 In Example 2 we determined the real numbers that could not be used in place of the variable in a rational expression. The domain of any algebraic expression in one variable is the set of all real numbers that can be used in place of the variable. For rational expressions, the domain must exclude any real numbers that cause the denom inator to be zero. E X A M P L E 3 Domain Find the domain of each expression. a) x2 - 9 x + 3 b) x x2 - x - 6 c) x-5 4 Solution a) The denominator is 0 if x + 3 = 0, or x = -3. So -3 can't be used for x. The domain is the set of all real numbers except -3, which is written in set notation as {x 1 x e -3}. b) The denominator is 0 if x2 - x - 6 = 0: x2 - x - 6 = 0 (x - 3)(x + 2) = 0 x-3=0 or x + 2 = 0 x=3 or x = -2 So -2 and 3 can't be used in place of x. The domain is the set of all real numbers except -2 and 3, which is written as {x 1 x e -2 and x e 3}. dug84356_ch06a.qxd 384 9/14/10 12:39 PM Page 384 6-4 Chapter 6 Rational Expressions c) Since the denominator is 4, the denominator can't be 0 no matter what number is used for x. The domain is the set of all real numbers, R. Now do Exercises 15-22 Note that if a rational expression is used to dene a function, then the domain of the rational expression is also called the domain of the function. For example, the 2 domain of the function y = x - 9 is the set of all real numbers except -3 or {x I x -3}. x +3 When dealing with rational expressions in this book, we will generally assume that the variables represent numbers for which the denominator is not zero. U2V Reducing to Lowest Terms Rational expressions are a generalization of rational numbers. The operations that we perform on rational numbers can be performed on rational expressions in exactly the same manner. Each rational number can be written in innitely many equivalent forms. For example, 3 6 9 12 15 = = = = = . 5 10 15 20 25 U Helpful Hint V How would you ll in the blank in 3 5 = 10 ? Most students learn to divide 5 into 10 to get 2, and then multiply 3 by 2 to get 6. In algebra, it is better to multiply the numerator and denomi 3 nator of 5 by 2, as shown here. Each equivalent form of 3 is obtained from 3 by multiplying both numerator and 5 5 denominator by the same nonzero number. This is equivalent to multiplying the frac tion by 1, which does not change its value. For example, 3 3 3 2 6 = 1= = 5 5 5 2 10 and 3 3 3 9 = = . 5 5 3 15 If we start with 6 and convert it into 3, we say that we are reducing 6 to lowest terms. 10 5 10 We reduce by dividing the numerator and denominator by the common factor 2: / 2 3 3 6 = = / 10 2 5 5 A rational number is expressed in lowest terms when the numerator and the denomi nator have no common factors other than 1. CAUTION We can reduce fractions only by dividing the numerator and the denom inator by a common factor. Although it is true that 2+4 6 = , 10 2 + 8 we cannot eliminate the 2's, because they are not factors. Removing them from the sums in the numerator and denominator would not result in 3. 5 Reducing Fractions If a e 0 and c e 0, then ab b = . ac c dug84356_ch06a.qxd 9/14/10 12:39 PM Page 385 6-5 6.1 Reducing Rational Expressions 385 To reduce rational expressions to lowest terms, we use exactly the same procedure as with fractions: Reducing Rational Expressions 1. Factor the numerator and denominator completely. 2. Divide the numerator and denominator by the greatest common factor. Dividing the numerator and denominator by the GCF is often referred to as dividing out or canceling the GCF. E X A M P L E 4 Reducing Reduce to lowest terms. a) 30 42 b) x2 - 9 6x + 18 c) 3x2 + 9x + 6 2x2 - 8 Solution a) 30 2 3 5 / / = Factor. 42 2 3 7 / / = 5 7 Divide out the GCF: 2 3 or 6. b) Since 9 = 9 1 = 1 , it is tempting to apply that fact here. However, 9 is not a common 18 9 2 2 2 factor of the numerator and denominator of x - 9 , as it is in 9 . You must factor 6x + 18 18 the numerator and denominator completely before reducing. (x - 3)(x + 3) x2 - 9 = 6(x + 3) 6x + 18 Factor. x-3 Divide out the GCF: x + 3. 6 This reduction is valid for all real numbers except -3, because that is the domain of the original expression. If x = -3, then x + 3 = 0 and we would be dividing out 0 from the numerator and denominator, which is prohibited in the rule for reducing fractions. = c) 3x2 + 9x + 6 3(x + 2)(x + 1) = 2(x + 2)(x - 2) 2x2 - 8 Factor completely. 3x + 3 Divide out the GCF: x + 2. 2(x - 2) This reduction is valid for all real numbers except -2 and 2, because that is the domain of the original expression. = Now do Exercises 23-46 CAUTION In reducing, you can divide out or cancel common factors only. You + cannot cancel x from x + 3 , because it is not a factor of either x + 3 or x 2 3 x + 2. But x is a common factor in 3x , and 3x = 2 . 2x 2x Note that there are four ways to write the answer to Example 3(c) depending on whether the numerator and denominator are factored. Since 3x + 3 3(x + 1) 3(x + 1) 3x + 3 = = = , 2(x - 2) 2(x - 2) 2x - 4 2x - 4 dug84356_ch06a.qxd 386 9/14/10 12:39 PM Page 386 6-6 Chapter 6 Rational Expressions any of these four rational expressions is correct. We usually give such answers with the denominator factored and the numerator not factored. With the denominator factored you can easily spot the values for x that will cause an undened expression. U3V Reducing with the Quotient Rule for Exponents To reduce rational expressions involving exponential expressions, we use the quotient rule for exponents from Chapter 4. We restate it here for reference. Quotient Rule for Exponents If a e 0, and m and n are any integers, then am = am-n. an E X A M P L E 5 Using the quotient rule in reducing Reduce to lowest terms. a) 3a15 6a7 b) 6x4y2 4xy5 Solution a) 3a15 / 3a15 Factor. 7 = 6a / 3 2 a7 a15-7 Quotient rule = for exponents 2 = b) 6x4y2 2 3x 4y2 / = 4xy5 / 2 2xy 5 3x4-1y2-5 = 2 a8 2 = Factor. Quotient rule for exponents 3x3y-3 3x3 = 3 2 2y Now do Exercises 47-58 The essential part of reducing is getting a complete factorization for the numerator and denominator. To get a complete factorization, you must use the techniques for fac toring from Chapter 5. If there are large integers in the numerator and denominator, you can use the technique shown in Section 5.1 to get a prime factorization of each integer. Reducing expressions involving large integers Reduce 420 to lowest terms. 616 Solution Use the method of Section 5.1 to get a prime factorization of 420 and 616: 2 616 2 308 2 154 2 210 2 420 3 105 7 77 11 5 35 7 6 E X A M P L E dug84356_ch06a.qxd 9/14/10 12:39 PM Page 387 6-7 6.1 387 Reducing Rational Expressions The complete factorization for 420 is 22 3 5 7, and the complete factorization for 616 is 23 7 11. To reduce the fraction, we divide out the common factors: 420 22 3 5 7 = 3 2 7 11 616 3 5 = 2 11 15 = 22 Now do Exercises 59-66 U4V Dividing a - b by b - a In Section 4.5 you learned that a - b = -(b - a) = -1(b - a). So if a - b is divided by b - a, the quotient is -1: a - b = -1(b - a) b-a b-a = -1 We will use this fact in Example 7. E X A M P L E 7 Expressions with a - b and b - a Reduce to lowest terms. a) 5x - 5y 4y - 4x b) m2 - n2 n-m Solution a) Factor out 5 from the numerator and 4 from the denominator and use x - y = -1: y-x 5x - 5y 5(x - y) 5 5 = = (-1) = 4y - 4x 4(y - x) 4 4 Another way is to factor out -5 from the numerator and 4 from the denominator and then use y - x =1: y-x 5x - 5y -5(y - x) -5 5 = = (1) = 4(y - x) 4 4y - 4x 4 -1 b) m2 - n2 (m - n)(m + n) = Factor. n-m n-m m-n = -1(m + n) = -m - n n-m = -1 Now do Exercises 67-74 CAUTION We can reduce a - b to -1, but we cannot reduce a - b. There is no factor b-a a+b that is common to the numerator and denominator of a-b a+b or a+b . a-b U5V Factoring Out the Opposite of a Common Factor If we can factor out a common factor, we can also factor out the opposite of that com mon factor. For example, from -3x - 6y we can factor out the common factor 3 or the common factor -3: -3x - 6y = 3(-x - 2y) or -3x - 6y = -3(x + 2y) To reduce an expression, it is sometimes necessary to factor out the opposite of a common factor. dug84356_ch06a.qxd 388 9/14/10 12:39 PM Page 388 6-8 Chapter 6 Rational Expressions E X A M P L E 8 Factoring out the opposite of a common factor Reduce -3w - 3w2 w2 - 1 to lowest terms. Solution We can factor 3w or -3w from the numerator. If we factor out -3w, we get a common factor in the numerator and denominator: -3w - 3w 2 -3w(1 + w) = Factor. w2 - 1 (w - 1)(w + 1) -3w = Since 1 + w = w + 1, we divide out w + 1. w-1 3w = Multiply numerator and denominator by -1. 1-w The last step is not absolutely necessary, but we usually perform it to express the answer with one less negative sign. Now do Exercises 75-84 The main points to remember for reducing rational expressions are summarized in the following reducing strategy. Strategy for Reducing Rational Expressions 1. Factor the numerator and denominator completely. Factor out a common fac tor with a negative sign if necessary. 2. Divide out all common factors. Use the quotient rule if the common factors are powers. U6V Writing Rational Expressions Rational expressions occur in applications involving rates. For uniform motion, rate is distance divided by time, R = D . For example, if you drive 500 miles in 10 hours, your T 0 rate is 5100 or 50 mph. If you drive 500 miles in x hours, your rate is 500 mph. In work prob x W lems, rate is work divided by time, R = . For example, if you lay 400 tiles in 4 hours, T your rate is 400 or 100 tiles/hour. If you lay 400 tiles in x hours, your rate is 400 tiles/hour. 4 x E X A M P L E 9 Writing rational expressions Answer each question with a rational expression. a) If a trucker drives 500 miles in x + 1 hours, then what is his average speed? b) If a wholesaler buys 100 pounds of shrimp for x dollars, then what is the price per pound? c) If a painter completes an entire house in 2x hours, then at what rate is she painting? Solution 500 a) Because R = D, he is averaging x + 1 mph. T b) At x dollars for 100 pounds, the wholesaler is paying or x dollars/pound. 100 c) By completing 1 house in 2x hours, her rate is 1 2x x 100 dollars per pound house/hour. Now do Exercises 107-112 dug84356_ch06a.qxd 9/17/10 8:06 PM Page 389 6-9 6.1 389 Fill in the blank. 1. A rational number is a ratio of two with the denominator not 0. 2. A rational expression is a ratio of two with the denominator not 0. 3. A rational expression is reduced to lowest terms by the numerator and denominator by the GCF. 4. The rule is used in reducing a ratio of monomials. 5. The expressions a - b and b - a are . 6. If a rational expression is used to determine y from x, then y is a function of x. True or false? 7. A complete factorization of 3003 is 2 3 7 11 13. 8. A complete factorization of 120 is 23 3 5. x+1 9. We can't replace x by -1 or 3 in --. x-3 2x 10. For any real number x, -- = x. 2 a2 + b2 11. Reducing -- to lowest terms yields a + b. a+b Exercises U Study Tips V If you must miss class, let your instructor know. Be sure to get notes from a reliable classmate. Take good notes in class for yourself and your classmates. You never know when a classmate will ask to see your notes. U1V Rational Expressions and Functions Evaluate each rational expression. See Example 1. 3x - 3 x + 5 1. Evaluate -- for x = -2. 3x + 1 2. Evaluate -- for x = 5. 4 - 4 x 2x + 9 3. If R(x) = --, nd R(3). x -20x - 2 4. If R(x) = --, nd R(-1). x - 8 x - 5 5. If R(x) = --, nd R(2), R(-4), R(-3.02), and x+3 R(-2.96). Note how a small difference in x (-3.02 to -2.96) can make a big difference in R(x). x - 2x - 3 2 6. If R(x) = --, nd R(3), R(5), R(2.05), x-2 and R(1.999). Which numbers cannot be used in place of the variable in each rational expression? See Example 2. x 7. -x+1 7a 9. -3a - 5 2x + 3 11. -x2 - 16 p - 1 13. -2 3x 8. -x - 7 84 10. -3 - 2a 2y + 1 12. 2 y -y-6 m + 31 14. -5 Find the domain of each rational expression. See Example 3. 2 x + x 15. -x-2 x+4 16. -x-5 6.1 Warm-Ups Reducing Rational Expressions dug84356_ch06a.qxd 390 17. 9/14/10 12:39 PM Page 390 6-10 Chapter 6 Rational Expressions x x + 5x + 6 U3V Reducing with the Quotient Rule 2 for Exponents x2 + 2 18. 2 x - x - 12 19. 20. Reduce each expression to lowest terms. Assume that all variables represent nonzero real numbers, and use only positive exponents in your answers. See Example 5. x2 - 4 2 47. 21. 22. x2 - 3 x+9 y8 y5 49. z3 z8 w9 w12 51. 4x7 -2x5 52. -6y3 3y9 53. x - 5 x 48. 50. x2 - 3x 9 x10 x7 -12m9n18 8m6n16 54. -9u9v19 6u9v14 55. 6b10c4 -8b10c7 56. 9x20y -6x25y3 57. 30a3bc 18a7b17 58. 15m10n3 24m12np U2V Reducing to Lowest Terms Reduce each rational expression to lowest terms. Assume that the variables represent only numbers for which the denominators are nonzero. See Example 4. 23. 6 27 24. 14 21 25. 42 90 26. 42 54 27. 36a 90 28. 56y 40 78 30w 30. 6x + 2 6 32. 2x + 4y 6y + 3x 34. 35. 3b - 9 6b - 15 36. 3m + 9w 3m - 6w 37. w2 - 49 w+7 38. a2 - b2 a-b 39. a -1 a2 + 2a + 1 40. x -y x2 + 2xy + y2 67. 3a - 2b 2b - 3a 68. 5m - 6n 6n - 5m 41. 2x2 + 4x + 2 4x2 - 4 42. 2x2 + 10x + 12 3x2 - 27 69. h2 - t 2 t-h 70. r 2 - s2 s-r 43. 3x2 + 18x + 27 21x + 63 44. x 3 - 3x 2 - 4x x 2 - 4x 71. 2g - 6h 9h2 - g2 72. 5a - 10b 4b2 - a2 45. 2a3 + 16 4a + 8 46. w3 - 27 w2 - 3w 73. x2 - x - 6 9 - x2 74. 1 - a2 a +a- 2 29. 31. 33. 2 Reduce each expression to lowest terms. Assume that all variables represent nonzero real numbers, and use only positive exponents in your answers. See Example 6. 59. 60. 68 44y 210 264 616 660 61. 62. 2w + 2 2w 231 168 936 624 63. 64. 5x - 10a 10x - 20a 630x5 300x9 96y2 108y5 65. 924a23 448a19 66. 270b75 165b12 2 U4V Dividing a - b by b - a Reduce each expression to lowest terms. See Example 7. 2 2 dug84356_ch06a.qxd 9/14/10 12:39 PM Page 391 6-11 6.1 U5V Factoring Out the Opposite of a Common Factor 103. Reducing Rational Expressions y3 - 2y2 - 4y + 8 y2\t- 4y + 4 104. 391 mx + 3x + my + 3y m2 - 3m - 18 Reduce each expression to lowest terms. See Example 8. 76. 75. -x - 6 x+6 77. -2y - 6y 3 + 9y -5x - 20 3x + 12 78. y - 16 -8 - 2y 2 79. -3x - 6 3x - 6 80. 8 - 4x -8x - 16 81. -12a - 6 2a2 + 7a + 3 82. -2b2 - 6b - 4 b2 - 1 83. a -b 2b2 - 2ab 84. x -1 x - x2 3 3 3 Reduce each expression to lowest terms. See the Strategy for Reducing Rational Expressions box on page 388. 85. 12 2x 8 4x 86. 4x2 2x9 87. 2x + 4 4x 88. 2x + 4x2 4x 89. a-4 4-a 90. 2b - 4 2b + 4 91. 2c - 4 4 - c2 92. -2t - 4 4 - t2 93. x2 + 4x + 4 x2\t- 4 94. 3x - 6 x - 4x + 4 -2x - 4 95. 2 x + 5x + 6 2x + 2w - ax - aw x3 - xw2 106. x2 + ax - 4x - 4a x2 - 16 -2x - 8 96. 2 x + 2x - 8 7 2q8 + q 6 5 2q + q 99. u - 6u - 16 u2 - 16u + 64 100. v + 3v - 18 v2 + 12v + 36 a3\t- 8 2a - 4 102. 4w2 - 12w + 36 2w3 + 54 98. 2 W U6V riting Rational Expressions Answer each question with a rational expression. Be sure to include the units. See Example 9. 107. If Sergio drove 300 miles at x + 10 miles per hour, then how many hours did he drive? 108. If Carrie walked 40 miles in x hours, then how fast did she walk? 109. If x + 4 pounds of peaches cost $4.50, then what is the cost per pound? 110. If nine pounds of pears cost x dollars, then what is the price per pound? 111. If Ayesha can clean the entire swimming pool in x hours, then how much of the pool does she clean per hour? 2 97. 101. 105. 2 8s12 12s - 16s5 112. If Ramon can mow the entire lawn in x 3 hours, then how much of the lawn does he mow per hour? 6 2 Applications Solve each problem. 113. Annual reports. The Crest Meat Company found that the cost per report for printing x annual reports at Peppy Printing is given by the formula 150 + 0.60x , x where C(x) is in dollars. C(x) = dug84356_ch06a.qxd 392 9/14/10 12:39 PM Page 392 6-12 Chapter 6 Rational Expressions given by the formula, C(p) = 500,000 . 100 - p a) Use the accompanying graph to estimate the cost for removing 90% and 95% of the toxic chemicals. b) Use the formula to nd C(99.5) and C(99.9). c) What happens to the cost as the percentage of pollutants removed approaches 100%? 0.80 0.70 0.60 0.50 Annual cost (hundred thousand dollars) Cost per report (dollars) a) Use the accompanying graph to estimate the cost per report for printing 1000 reports. b) Use the formula to nd C(1000), C(5000), and C(10,000). c) What happens to the cost per report as the number of reports gets very large? 0.40 1 2 3 4 5 Number of reports (thousands) Figure for Exercise 113 5 4 3 2 1 0 90 91 92 93 94 95 96 97 98 99 Percentage of chemicals removed 114. Toxic pollutants. The annual cost in dollars for removing p% of the toxic chemicals from a town's water supply is 6.2 In This Section U1V Multiplication of Rational Numbers 2V Multiplication of Rational U Expressions U3V Division of Rational Numbers 4V Division of Rational U Expressions 5V Applications U Figure for Exercise 114 Multiplication and Division In Section 6.1, you learned to reduce rational expressions in the same way that we reduce rational numbers. In this section, we will multiply and divide rational expressions using the same procedures that we use for rational numbers. U1V Multiplication of Rational Numbers Two rational numbers are multiplied by multiplying their numerators and multiplying their denominators. Multiplication of Rational Numbers If b e 0 and d e 0, then a c ac = . b d bd dug84356_ch06a.qxd 9/14/10 12:39 PM Page 393 6-13 E X A M P L E 6.2 1 Multiplication and Division 393 Multiplying rational numbers 6 Find the product 7 14 . 15 Solution U Helpful Hint V The product is found by multiplying the numerators and multiplying the denominators: Did you know that the line separating the numerator and denominator in a fraction is called the vinculum? 6 14 84 = 7 15 105 = 21 4 Factor the numerator and denominator 21 5 4 Divide out the GCF 21. 5 The reducing that we did after multiplying is easier to do before multiplying. First factor all terms, reduce, and then multiply: = 6 14 2 3 2 7 / / = 15 7 / 7 / 5 3 = 4 5 Now do Exercises 1-8 U2V Multiplication of Rational Expressions Rational expressions are multiplied just like rational numbers: factor, reduce, and then multiply. A rational number cannot have zero in its denominator and neither can a rational expression. Since a rational expression can have variables in its denominator, the results obtained in Examples 2 and 3 are valid only for values of the variable(s) that would not cause a denominator to be 0. E X A M P L E 2 Multiplying rational expressions Find the indicated products. a) 9x 10y 5y 3xy b) -8xy4 15z 3z3 2x5y3 Solution a) 9x 10y 3 3/ 2 5/ /x /y = Factor. 5y 3xy /y 5/ /x 3/y = b) 6 y -8xy4 15z -2 2 2xy4 3 5z / / Factor. 3 5 3= 3z 2x y / 3z3 / 2x5y3 = -20xy4z z3x5y3 Reduce. = -20y z2x4 Quotient rule Now do Exercises 9-18 dug84356_ch06a.qxd 394 9/14/10 12:39 PM Page 394 6-14 Chapter 6 Rational Expressions E X A M P L E 3 Multiplying rational expressions Find the indicated products. a) 2x - 2y 2x 4 x2 - y2 b) x 2 + 7x + 12 x 2x + 6 x2 - 16 c) a+b 8a2 2 6a a + 2ab + b2 Solution a) 2x - 2y / 2(x - y) 2x = 4 x2 - y2 / / 2 2 = b) c) / 2 x (x - y)(x + y) x x+y Factor. Reduce. x2 + 7x + 12 x (x + 3) (x + 4) = 2 2(x + 3) 2x + 6 x - 16 x = 2(x - 4) a+b 8a2 a + b 2 4a2 2 2= 6a a + 2ab + b 2 3a (a + b)2 4a = 3(a + b) x (x - 4)(x + 4) Factor. Reduce. Factor. Reduce. Now do Exercises 19-26 U3V Division of Rational Numbers By the denition of division, a quotient is found by multiplying the dividend by the re c d ciprocal of the divisor. If the divisor is a rational number , its reciprocal is simply . d c Division of Rational Numbers If b e 0, c e 0, and d e 0, then a c a d = . b d b c E X A M P L E 4 Dividing rational numbers Find each quotient. a) 5 1 2 b) 6 3 7 14 b) 6 3 6 14 2 3 2 / / 7 = = =4 7 14 7 3 / 7 3 / Solution a) 5 1 = 5 2 = 10 2 Now do Exercises 27-34 U4V Division of Rational Expressions We divide rational expressions in the same way we divide rational numbers: Invert the divisor and multiply. dug84356_ch06a.qxd 9/14/10 12:39 PM Page 395 6-15 E X A M P L E 6.2 5 Multiplication and Division 395 Dividing rational expressions Find each quotient. 5 5 a) 3x 6x b) x7 (2x 2) 2 c) 4 - x2 x-2 x2 + x x 2 - 1 Solution U Helpful Hint V A doctor told a nurse to give a patient half of the usual 500-mg dose of a drug. The nurse stated in court, \"dividing in half means dividing by 1/2, which means multiply by 2.\" The nurse was in court because the patient got 1000 mg instead of 250 mg and died (true story). Dividing a quantity in half and divid ing by one-half are not the same. a) 5 5 5 6x = 3x 6x 3x 5 = Invert the divisor and multiply. / 5 2 3/ /x Factor. /x 3/ / 5 =2 b) x7 x7 1 (2x 2) = 2 2 2x 2 = c) Divide out the common factors. x5 4 Invert and multiply. Quotient rule 4 - x2 x2 - 1 4 - x2 x-2 2 = 2 2 x +x x-2 x +x x -1 Invert and multiply. -1 = (2 - x) (2 + x) (x + 1)(x - 1) Factor. x(x + 1) x-2 = -1(2 + x)(x - 1) x 2-x = -1 x-2 = -1(x2 + x - 2) x Simplify. = -x 2 - x + 2 x Now do Exercises 35-48 We sometimes write division of rational expressions using the fraction bar. For example, we can write a+b 3 a + b 1 as . 1 3 6 6 No matter how division is expressed, we invert the divisor and multiply. E X A M P L E 6 Division expressed with a fraction bar Find each quotient. a+b 3 a) 1 6 x2 - 1 2 b) x-1 3 a2 + 5 3 c) 2 dug84356_ch06a.qxd 396 9/20/10 11:45 AM Page 396 6-16 Chapter 6 Rational Expressions Solution a+b 3 a) 1 6 a+b 3 1 6 Rewrite as division. a+b 6 3 1 Invert and multiply. a+b 2 3 p Factor. p 3 1 (a + b)2 Reduce. 2a + 2b x2 - 1 2 b) x-1 3 x2 - 1 2 x-1 3 x2 - 1 3 2 x-1 Rewrite as division. Invert and multiply. (x - 1)(x + 1) 3 Factor. x-1 2 3x + 3 Reduce. 2 U Helpful Hint V In Section 6.5 you will see another technique for nding the quotients in Example 6. a2 + 5 3 c) 2 a2 + 5 3 2 a2 + 5 1 3 2 Rewrite as division. a2 + 5 6 Now do Exercises 49-56 U5V Applications We saw in Section 6.1 that rational expressions can be used to represent rates. Note that there are several ways to write rates. For example, miles per hour is written mph, mi/hr, or mi. The last way is best when doing operations with rates because it helps us hr reconcile our answers. Notice how hours \"cancels\" when we multiply miles per hour and hours in Example 7, giving an answer in miles, as it should be. E X A M P L E 7 Using rational expressions with uniform motion Shasta drove 200 miles on I-10 in x hours before lunch. a) Write a rational expression for her average speed before lunch. b) She drives for 3 hours after lunch at the same average speed. Write a rational expression for her distance after lunch. Solution a) Because R D , T her rate before lunch is 200 miles x hours or 200 x mph. dug84356_ch06a.qxd 9/14/10 12:39 PM Page 397 6-17 6.2 Multiplication and Division b) Because D = R T, her distance after lunch is the product of 3 hours (her time): D= 200 x 397 mph (her rate) and 200 mi 600 3 hr = mi x hr x Now do Exercises 77-78 The amount of work completed is the product of rate and time, W = R T. So if a machine washes cars at the rate of 12 per hour and it works for 3 hours, the amount of W work completed is 36 cars washed. Note that the rate is given by R = T . E X A M P L E 8 Using rational expressions with work It takes x minutes to ll a bathtub. a) Write a rational expression for the rate at which the tub is lling. b) Write a rational expression for the portion of the tub that is lled in 10 minutes. Solution W a) The work completed in this situation is 1 tub being lled. Because R = T , the rate 1 tub 1 at which the tub is lling is x min or x tub/min. b) Because W = R T, the work completed in 10 minutes or the portion of the tub that 1 is lled in 10 minutes is the product of x tub/min (the rate) and 10 minutes (the time): W= 1 tub 10 10 min = tub x min x Now do Exercises 79-80 Warm-Ups Fill in the blank. 1. 2. expressions are multiplied by multiplying their numerators and multiplying their denominators. can be done before multiplying rational expressions. 3. To rational expressions, invert the divisor and multiply. True or false? 4. One-half of one-fourth is one-sixth. 2 5 10 5. = 3 7 21 6. The product of x-7 6 and is -2. 3 7-x 7. Dividing by 2 is equivalent to multiplying by 8. For any real number a, 9. 2 1 4 = 3 2 3 a a 3= . 3 9 1 . 2 6.2 dug84356_ch06a.qxd 9/17/10 8:07 PM Page 398 Exercises U Study Tips V Personal issues can have a tremendous effect on your progress in any course. If you need help, get it. Most schools have counseling centers that can help you to overcome personal issues that are affecting your studies. 24. 12 4x + 10 8 35 3. 15 24 25. 16a + 8 2a2 + a - 1 4a2 - 1 5a2 + 5 25 56 6. 48 35 26. 6x - 18 4x 2 + 4x + 1 2x - 5x - 3 6x + 3 U1V Multiplication of Rational Numbers Perform the indicated operation. See Example 1. 2 1. 3 5 6 3 2. 4 3 8 4. 4 21 7. 24 2 5 12 51 5. 17 10 7 20 8. 3 35 10 Perform the indicated operation. See Example 4. 27. Perform the indicated operation. See Example 2. 2x 3 5x2 11. 6 5 4x 10. 3 x 2 U3V Division of Rational Numbers U2V Multiplication of Rational Expressions 9. (4x2 + 20x + 25) 3y 7 9x 12. 10 1 4 1 2 30. 32 21 2y 33. 5 2 x 40 3 28. 1 4 1 6 1 2 5 15 7 14 22 34. 9 9 31. 12 5a 3ab 12b 55a 14. 3m 35p 7p 6mp 15. -2 6 21a2 x 7a5 6x 16. 5 5z3w -6y 3 9 -9y 20z 17. 5 15t3y 5 3 2 7 24t w y 20w 18. 22x2y3\tz 6x5 4 33y3z 35. x2 4 37. 5x 2 3 39. 8m3 n4 41. y-6 2 43. x 2 + 4x + 4 8 (x + 2)3 16 x 2 36. 3 2a2 6 2a 10x 21 38. 4u2 3v 14u 15v6 (12mn2) 40. 2p4 3 3q (4pq5) 6-y 6 42. 4-a 5 19. 2x + 2y 7 20. 3 a +a 21. 3a + 3b 10a 15 a2 - b2 44. a2 + 2a + 1 3 a2 - 1 a 22. b3 + b 10 2 5 b +b 45. t 2 + 3t - 10 t 2 - 25 (4t - 8) 46. w2 - 7w + 12 w2 - 4w 2a + 2 6 23. (x2 - 6x + 9) 3 x - 3 3 4 Perform the indicated operation. See Example 5. Perform the indicated operation. See Example 3. 2 32. U4V Division of Rational Expressions 13. 15 6x + 6y 29. 12 (w2 - 9) a2 - 16 3 2 5 15 2 dug84356_ch06a.qxd 9/14/10 12:39 PM Page 399 6-19 6.2 Multiplication and Division 2x - 5 x-1 48. (6y2 - y - 2) 67. 68. rt2 rt2 2 3 2 rt rt 2 3x2 + 16x + 5 x 2 x 9x - 1 70. 2y + 1 3y - 2 2mn4 3m5n7 2 4 6mn2 mn 69. 47. (2x2 - 3x - 5) 4 x 2 + 6x + 5 x x 3x + 3 Perform the indicated operation. See Example 6. x - 2y 5 49. 1 10 3m + 6n 8 50. 3 4 399 71. a2 - 2a + 4 a2 - 4 (a + 2)3 2a + 4 72. w2 - 1 w-1 (w - 1)2 w2 + 2w + 1 73. 2x2 + 19x - 10 4x2 - 1 2 2 x - 100 2x - 19x - 10 74. x3 - 1 9x 2 + 9x + 9 x2 - x x2 + 1 x2 - 4 12 51. x-2 6 6a2 + 6 5 52. 6a + 6 5 x2 + 9 3 53. 5 1 a - 3 54. 4 75. 9 + 6m + m2 9 - 6m + m2 76. 3x + 3w + bx + bw x2 - w2 2 x 2 - y 55. x-y 9 x 2 + 6x + 8 56. x + 2 x + 1 U5V Applications 6 - 2b 9 - b2 Solve each problem. Answers could be rational expressions. Be sure to give your answers with appropriate units. See Examples 7 and 8. 77. Marathon run. Florence ran 26.2 miles in x hours in the Boston Marathon. a) Write a rational expression for her average speed. Miscellaneous Perform the indicated operation. 57. x-1 9 3 1 - x 58. 2x - 2y 1 3 y-x 59. 3a + 3b 1 a 3 60. a-b 2b - 2a b 61. a 1 2 m2 - 9 m + mk + 3m + 3k 2 62. 2 5 2g 3h 1 h 63. 6y (2x) 3 64. 8x (18x) 9 65. a3b4 a5b7 -2ab2 ab 66. -2a2 20a 3a2 15a3 b) She runs at the same average speed for 1 hour in the 2 Cripple Creek Fun Run. Write a rational expression for her distance at Cripple Creek. 78. Driving marathon. Felix drove 800 miles in x hours on Monday. a) Write a rational expression for his average speed. b) On Tuesday he drove for 6 hours at the same average speed. Write a rational expression for his distance on Tuesday. dug84356_ch06a.qxd 400 9/14/10 12:39 PM Page 400 6-20 Chapter 6 Rational Expressions 82. Area of a triangle. If the base of a triangle is 8x + 16 yards and its height is 1 yards, then what is the area of the 79. Filling the tank. Chantal lled her empty gas tank in x minutes. a) Write a rational expression for the rate at which she lled her tank. x+2 triangle? b) Write a rational expression for the portion of the tank that is lled in 2 minutes. 1 x+2 yd 8x + 16 yd 80. Magazine sales. Henry sold 120 magazine subscriptions in x days. a) Write a rational expression for the rate at which he sold the subscriptions. Figure for Exercise 82 Getting More Involved 83. Discussion Evaluate each expression. b) Suppose that he continues to sell at the same rate for 5 more days. Write a rational expression for the number of magazines sold in those 5 days. 1 c) One-half of 81. Area of a rectangle. If the length of a rectangular ag is x meters and its width is 5 meters, then what is the area of x the rectangle? 4x 3 d) One-half of 3x 2 84. Exploration Let R = 5 x b) One-third of 4 a) One-half of 4 6x2 + 23x + 20 2x + 5 . and H = 24x2 + 29x - 4 8x - 1 a) Find R when x = 2 and x = 3. Find H when x = 2 and x = 3. b) How are these values of R and H related and why? m xm Figure for Exercise 81 6.3 In This Section U1V Building Up the Denominator U2V Finding the Least Common Denominator 3V Converting to the LCD U Finding the Least Common Denominator Every rational expression can be written in innitely many equivalent forms. Because we can add or subtract only fractions with identical denominators, we must be able to change the denominator of a fraction. You have already learned how to change the denominator of a fraction by reducing. In this section, you will learn the opposite of reducing, which is called building up the denominator. U1V Building Up the Denominator To convert the fraction 2 3 into an equivalent fraction with a denominator of 21, we factor 21 as 21 = 3 7. Because 2 3 already has a 3 in the denominator, multiply dug84356_ch06a.qxd 9/14/10 12:39 PM Page 401 6-21 6.3 Finding the Least Common Denominator 401 the numerator and denominator of 2 by the missing factor 7 to get a denominator 3 of 21: 2 2 7 14 = = 3 3 7 21 For rational expressions the process is the same. To convert the rational expression 5 x+3 into an equivalent rational expression with a denominator of x2 - x - 12, rst factor x2 - x - 12: x2 - x - 12 = (x + 3)(x - 4) From the factorization we can see that the denominator x + 3 needs only a factor of x - 4 to have the required denominator. So multiply the numerator and denominator by the missing factor x - 4: 5 5(x - 4) 5x - 20 = = x + 3 (x + 3)(x - 4) x2 - x - 12 E X A M P L E 1 Building up the denominator Build each rational expression into an equivalent rational expression with the indicated denominator. a) 3 = ? 12 b) 3 ? = w wx c) 2 ? 3= 3y 12y8 Solution a) Because 3 = 3, we get a denominator of 12 by multiplying the numerator and 1 denominator by 12: 3 3 12 36 3= = = 1 1 12 12 b) Multiply the numerator and denominator by x: 3 3 x 3x = = w w x wx c) Note that 12y8 = 3y3 4y5. So to build 3y3 up to 12y8 multiply by 4y5: 2 2 4y5 8y5 3= 3 5= 3y 3y 4y 12y8 Now do Exercises 1-20 In Example 2 we must factor the original denominator before building up the denominator. E X A M P L E 2 Building up the denominator Build each rational expression into an equivalent rational expression with the indicated denominator. a) 7 ? = 3x - 3y 6y - 6x b) x-2 ? = x + 2 x2 + 8x + 12 402 9/14/10 12:39 PM Page 402 6-22 Chapter 6 Rational Expressions U Helpful Hint V Notice that reducing and building up are exactly the opposite of each other. In reducing you remove a factor that is common to the numera tor and denominator, and in building up you put a common factor into the numerator and denominator. Solution a) Because 3x - 3y = 3(x - y), we factor -6 out of 6y - 6x. This will give a factor of x - y in each denominator: 3x - 3y = 3(x - y) 6y - 6x = -6(x - y) = -2 3(x - y) To get the required denominator, we multiply the numerator and denominator by -2 only: 7 7(-2) = 3x - 3y (3x - 3y)(-2) -14 = 6y - 6x b) Because x2 + 8x + 12 = (x + 2)(x + 6), we multiply the numerator and denominator by x + 6, the missing factor: x - 2 (x - 2)(x + 6) = x + 2 (x + 2)(x + 6) = x2 + 4x - 12 x2 + 8x + 12 Now do Exercises 21-32 CAUTION When building up a denominator, both the numerator and the denomina tor must be multiplied by the appropriate expression. U2V Finding the Least Common Denominator We can use the idea of building up the denominator to convert two fractions with different denominators into fractions with identical denominators. For example, 5 6 1 4 and can both be converted into fractions with a denominator of 12, since 12 = 2 6 and 12 = 3 4: 5 5 2 10 = = 6 6 2 12 1 1 3 3 = = 4 4 3 12 The smallest number that is a multiple of all of the denominators is called the least common denominator (LCD). The LCD for the denominators 6 and 4 is 12. To nd the LCD in a systematic way, we look at a complete factorization of each denominator. Consider the denominators 24 and 30: 24 = 2 2 2 3 = 23 3 30 = 2 3 5 Any multiple of 24 must have three 2's in its factorization, and any multiple of 30 must have one 2 as a factor. So a number with three 2's in its factorization will have enough to be a multiple of both 24 and 30. The LCD must also have one 3 and one 5 in its factorization. We use each factor the maximum number of times it appears in either factorization. So the LCD is 23 3 5: 24 2 3 dug84356_ch06a.qxd 3 5 = 2 2 2 3 5 = 120 30 dug84356_ch06a.qxd 9/14/10 12:39 PM Page 403 6-23 6.3 Finding the Least Common Denominator 403 If we omitted any one of the factors in 2 2 2 3 5, we would not have a multiple of both 24 and 30. That is what makes 120 the least common denominator. To nd the LCD for two polynomials, we use the same strategy. Strategy for Finding the LCD for Polynomials 1. Factor each denominator completely. Use exponent notation for repeated factors. 2. Write the product of all of the different factors that appear in the denominators. 3. On each factor, use the highest power that appears on that factor in any of the denominators. E X A M P L E 3 Finding the LCD If the given expressions were used as denominators of rational expressions, then what would be the LCD for each group of denominators? c) a2 + 5a + 6, a2 + 4a + 4 b) x3yz2, x5y2z, xyz5 a) 20, 50 Solution a) First factor each number completely: 50 = 2 52 20 = 22 5 The highest power of 2 is 2, and the highest power of 5 is 2. So the LCD of 20 and 50 is 22 52, or 100. b) The expressions x 3yz 2, x 5y 2z, and xyz 5 are already factored. For the LCD, use the highest power of each variable. So the LCD is x5y2z 5. c) First factor each polynomial. a2 + 5a + 6 = (a + 2)(a + 3) a2 + 4a + 4 = (a + 2)2 The highest power of (a + 3) is 1, and the highest power of (a + 2) is 2. So the LCD is (a + 3)(a + 2)2. Now do Exercises 33-46 U3V Converting to the LCD When adding or subtracting rational expressions, we must convert the expressions into expressions with identical denominators. To keep the computations as simple as possible, we use the least common denominator. E X A M P L E 4 Converting to the LCD Find the LCD for the rational expressions, and convert each expression into an equivalent rational expression with the LCD as the denominator. a) 4 2 , 9xy 15xz b) 5 1 3 , , 6x2 8x3y 4y2 Solution a) Factor each denominator completely: 9xy = 32xy 15xz = 3 5xz dug84356_ch06a.qxd 404 9/14/10 12:39 PM Page 404 6-24 Chapter 6 Rational Expressions U Helpful Hint V What is the difference between LCD, GCF, CBS, and NBC? The LCD for the denominators 4 and 6 is 12. The least common denominator is greater than or equal to both numbers.The GCF for 4 and 6 is 2. The greatest common fac tor is less than or equal to both num bers. CBS and NBC are TV networks. The LCD is 32 5xyz. Now convert each expression into an expression with this denominator. We must multiply the numerator and denominator of the rst rational expression by 5z and the second by 3y: Same denominator 2 2 3y 6y = = 15xz 15xz 3y 45xyz 4 4 5z 20z = = 9xy 9xy 5z 45xyz b) Factor each denominator completely: 6x 2 = 2 3x 2 8x3y = 23x3y 4y2 = 22y 2 The LCD is 23 3 x3y2 or 24x3y2. Now convert each expression into an expression with this denominator: 5 4xy2 20xy2 5 2= 2 2= 24x3y2 6x 6x 4xy 1 1 3y 3y = 3 = 3 8x y 8x y 3y 24x3y2 3 6x3 18x3 3 = 2 = 4y2 4y 6x3 24x3y2 Now do Exercises 47-58 E X A M P L E 5 Converting to the LCD Find the LCD for the rational expressions 5x x2 - 4 and 3 x2 + x - 6 and convert each into an equivalent rational expression with that denominator. Solution First factor the denominators: x2 - 4 = (x - 2)(x + 2) x2 + x - 6 = (x - 2)(x + 3) The LCD is (x - 2)(x + 2)(x + 3). Now we multiply the numerator and denominator of the rst rational expression by (x + 3) and those of the second rational expression by (x + 2). Because each denominator already has one factor of (x - 2), there is no reason to multiply by (x - 2). We multiply each denominator by the factors in the LCD that are missing from that denominator: 5x 5x2 + 15x 5x(x + 3) = = x -4 (x - 2)(x + 2)(x + 3) (x - 2)(x + 2)(x + 3) 2 3 3x + 6 3(x + 2) = = x2 + x - 6 (x - 2)(x + 3)(x + 2) (x - 2)(x + 2)(x + 3) Same denominator Now do Exercises 59-70 dug84356_ch06a.qxd 9/17/10 8:08 PM Page 405 6-25 6.3 405 Fill in the blank. 1. To the denominator of a fraction, we multiply the numerator and denominator by the same nonzero real number. 2. The is the smallest number that is a multiple of all denominators. 3. The LCD is the product of every factor that appears in the factorizations, raised to the power that appears on the factor. 2 2+5 5. -- = -3 3+5 6. The LCD for the denominators 25 3 and 24 32 is 25 32. 1 1 7. The LCD for -- and -- is 60. 10 6 1 1 8. The LCD for -- and -- is x2 - 4. x-2 x+2 1 1 9. The LCD for -- and -- is a2 - 1. a2 - 1 a-1 True or false? 2 25 4. -- = -3 35 Exercises U Study Tips V Try changing subjects or tasks every hour when you study. The brain does not easily assimilate the same material hour after hour. You will learn more from working on a subject one hour per day than seven hours on Saturday. U1V Building Up the Denominator Build each rational expression into an equivalent rational expression with the indicated denominator. See Example 1. 1 ? 1. - = - 3 27 2 ? 2. - = 5 35 3 ? 3. -- = -- 4 16 3 ? 4. -- = -7 28 ? 5. 1 = -- 7 ? 6. 1 = -3x ? 7. 2 = -- 6 ? 8. 5 = -12 5 ? 9. -- = -- x ax x ? 10. -- = -3 3x ? 11. 7 = -2x ? 12. 6 = -4y 5 ? 13. -- = -b 3bt 7 ? 14. -- = -2ay 2ayz -9z ? 15. -- = -2aw 8awz -7yt ? 16. -- = -3x 18xyt ? 2 17. -- = 3 a 15a3 7b ? 18. -- = -12c5 36c8 4 ? 19. -- = -5xy 2 10x 2y 5 ? 5y2 20. -- = -8x3z 24x5z3 6.3 Warm-Ups Finding the Least Common Denominator dug84356_ch06a.qxd 406 9/14/10 12:39 PM Page 406 6-26 Chapter 6 Rational Expressions Build each rational expression into an equivalent rational expression with the indicated denominator. See Example 2. 41. x2 - 16, x 2 + 8x + 16 5 ? 21. = x+3 2x + 6 43. x, x + 2, x - 2 44. y, y - 5, y + 2 45. x 2 - 4x, x 2 - 16, 2x 4 ? 22. = a-5 3a - 15 23. 42. x2 - 9, x 2 + 6x + 9 46. y, y 2 - 3y, 3y 5 ? = 2x + 2 -8x - 8 U3V Converting to the LCD 24. 3 ? = m - n 2n - 2m Find the LCD for the given rational expressions, and convert each rational expression into an equivalent rational expression with the LCD as the denominator. See Example 4. 25. 8a ? = 20b2 - 20b3 5b2 - 5b 47. 1 3 , 6 8 48. 5 3 , 12 20 26. 5x ? = -6x - 9 18x2 + 27x 49. 1 5 , 2x 6x 50. 3 1 , 5x 10x 27. 3 ? = 2 x+2 x - 4 51. 2 1 , 3a 2b 52. y x , 4x 6y 28. a ? = 2 a + 3 a - 9 53. 5 3 , 84a 63b 29. 3x ? = 2 x + 2x + 1 x+1 54. 4b 6 , 75a 105ab -7x ? 30. = 2 2x - 3 4x - 12x + 9 55. ? y-6 = 2 y-4 y + y - 20 56. ? z-6 = 2 z - 2z - 15 z+3 57. x y 1 , , 2 5 3 9y z 12x 6x y 58. 5 1 3b , 3, 6 12a b 14a 2ab3 31. 32. U2V Finding the Least Common Denominator If the given expressions were used as denominators of rational expressions, then what would be the LCD for each group of denominators? See Example 3. See the Strategy for Finding the LCD for Polynomials box on page 403. 33. 12, 16 34. 28, 42 35. 12, 18, 20 36. 24, 40, 48 2 2 37. 6a , 15a 4 3 9 8a b , 5 6a2c 59. 2x 5x , x - 3 x + 2 60. 2a 3a , a - 5 a + 2 3 2 39. 2a b, 3ab , 4a b 40. 4m3nw, 6mn5w8, 9m6nw 3 Find the LCD for the given rational expressions, and convert each rational expression into an equivalent rational expression with the LCD as the denominator. See Example 5. 38. 18x , 20xy 6 1 3 , 5 3x 2 2x dug84356_ch06a.qxd 9/14/10 12:39 PM Page 407 6-27 6.4 61. 2 3 4 , , 2q 2 - 5q - 3 2q 2 + 9q + 4 q 2 + q - 12 70. 407 p -3 2 , , 2p2 + 7p - 15 2p2 - 11p + 12 p2 + p - 20 4 5x , x - y 2y - 2x 63. 69. 4 5 , a-6 6-a 62. Addition and Subtraction x 5x , x 2 - 9 x 2 - 6x + 9 5x -4 , 2 64. 2 x - 1 x - 2x + 1 65. w +2 -2w , w2 - 2w - 15 w2 - 4w - 5 66. z-1 z+1 , z2 + 6z + 8 z2 + 5z + 6 67. Getting More Involved 71. Discussion Why do we learn how to convert two rational expressions into equivalent rational expressions with the same denominator? x 3 -5 , , 6x - 12 x 2 - 4 2x + 4 72. Discussion 68. Which expression is the LCD for 3 2b -5 , , 4b - 9 2b + 3 2b2 - 3b 2 3x - 1 22 3 x2(x + 2) and 2x + 7 ? 2 32 x(x + 2)2 a) 2 3 x(x + 2) 6.4 In This Section U1V Addition and Subtraction of Rational Numbers 2V Addition and Subtraction of U Rational Expressions U3V Applications b) 36x(x + 2) c) 36x2(x + 2)2 d) 23 33x3(x + 2)2 Addition and Subtraction In Section 6.3, you learned how to nd the LCD and build up the denominators of rational expressions. In this section, we will use that knowledge to add and subtract rational expressions with different denominators. U1V Addition and Subtraction of Rational Numbers We can add or subtract rational numbers (or fractions) only with identical denominators according to the following denition. dug84356_ch06a.qxd 408 9/14/10 12:39 PM Page 408 6-28 Chapter 6 Rational Expressions Addition and Subtraction of Rational Numbers If b e 0, then a c a+c a c a-c + = and - = . b b b b b b E X A M P L E 1 Adding or subtracting fractions with the same denominator Perform the indicated operations. Reduce answers to lowest terms. a) 1 7 + 12 12 b) 1 3 4 4 b) 1 3 -2 1 - = =4 4 4 2 Solution a) 1 7 8 / 4 2 2 + = = = 12 12 12 4 3 3 / Now do Exercises 1-8 If the rational numbers have different denominators, we must convert them to equivalent rational numbers that have identical denominators and then add or subtract. Of course, it is most efcient to use the least common denominator (LCD), as in Example 2. E X A M P L E 2 Adding or subtracting fractions with different denominators Find each sum or difference. a) U Helpful Hint V Note how all of the operations with rational expressions are performed according to the rules for fractions. So keep thinking of how you perform operations with fractions, and you will improve your skills with fractions and with rational expressions. 3 7 + 20 12 b) 1 4 6 15 Solution a) Because 20 = 22 5 and 12 = 22 3, the LCD is 22 3 5, or 60. Convert each fraction to an equivalent fraction with a denominator of 60: 3 7 3 3 7 5 + = + 20 12 20 3 12 5 9 35 = + 60 60 44 = 60 4 11 = 4 15 11 = 15 Build up the denominators. Simplify numerators and denominators. Add the fractions. Factor. Reduce. b) Because 6 = 2 3 and 15 = 3 5, the LCD is 2 3 5 or 30: 1 4 1 4 = 6 15 2 3 3 5 = Factor the denominators. 1 5 4 2 Build up the denominators. 2 3 5 3 5 2 dug84356_ch06a.qxd 9/14/10 12:39 PM Page 409 6-29 6.4 Addition and Subtraction = 5 8 30 30 Simplify the numerators and denominators. = -3 30 Subtract. = -1 3 10 3 Factor. 1 10 409 Reduce. =- Now do Exercises 9-18 U2V Addition and Subtraction of Rational Expressions Rational expressions are added or subtracted just like rational numbers. We can add or subtract only when we have identical denominators. All answers should be reduced to lowest terms. Remember to factor rst when reducing, and then divide out any com mon factors. E X A M P L E 3 Rational expressions with the same denominator Perform the indicated operations and reduce answers to lowest terms. a) 2 4 + 3y 3y b) 2x 4 + x+2 x+2 c) 2x + 1 x2 + 2x (x - 1)(x + 3) (x - 1)(x + 3) Solution a) 2 4 6 + = Add the fractions. 3y 3y 3y = b) 2 y Reduce. 2x 4 2x + 4 + = x+2 x+2 x+2 = Add the fractions. 2(x + 2) Factor the numerator. x+2 =2 Reduce. x 2 + 2x - (2x + 1) x + 2x 2x + 1 = (x - 1)(x + 3) (x - 1)(x + 3) (x - 1)(x + 3) 2 c) Subtract the fractions. = x2 + 2x - 2x - 1 (x - 1)(x + 3) Remove parentheses. = x2 - 1 (x - 1)(x + 3) Combine like terms. = (x - 1)(x + 1) (x - 1)(x + 3) Factor. = x+1 x+3 Reduce. Now do Exercises 19-30 dug84356_ch06a.qxd 410 9/14/10 12:39 PM Page 410 6-30 Chapter 6 Rational Expressions CAUTION When subtracting a numerator containing more than one term, be sure to enclose it in parentheses, as in Example 3(c). Because that numerator is a binomial, the sign of each of its terms must be changed for the subtraction. In Example 4, the rational expressions have different denominators. E X A M P L E 4 Rational expressions with different denominators Perform the indicated operations. a) c) U Helpful Hint V You can remind yourself of the difference between addition and multiplication of fractions with a simple example: If you and your spouse each own 1/7 of Microsoft, then together you own 2/7 of Microsoft. If you own 1/7 of Microsoft, and give 1/7 of your stock to your child, then your child owns 1/49 of Microsoft. 5 2 + 2x 3 a+1 a-2 6 8 b) 4 2 + x 3y xy 3 Solution a) The LCD for 2x and 3 is 6x: 5 2 5 3 2 2x + = + 2x 3 2x 3 3 2x Build up both denominators to 6x. = 15 4x + 6x 6x Simplify numerators and denominators. = 15 + 4x 6x Add the rational expressions. b) The LCD is x 3y 3. 4 2 4 y2 2 x2 + 3= 3 2 + xy3 x 2 3 x y xy xy y Build up both denominators to the LCD. = 4y2 2x2 3 3+ 3 3 xy xy Simplify numerators and denominators. = 4y 2 + 2x 2 x3y3 Add the rational expressions. c) Because 6 = 2 3 and 8 = 23, the LCD is 23 3, or 24: a + 1 a - 2 (a + 1)4 (a - 2)3 = 6 8 6 4 8 3 Build up both denominators to the LCD 24. = 4a + 4 3a - 6 24 24 Simplify numerators and denominators. = 4a + 4 - (3a - 6) 24 Subtract the rational expressions. = 4a + 4 - 3a + 6 24 Remove the parentheses. = a + 10 24 Combine like terms. Now do Exercises 31-46 dug84356_ch06a.qxd 9/14/10 12:39 PM Page 411 6-31 E X A M P L E 6.4 5 Addition and Subtraction 411 Rational expressions with different denominators Perform the indicated operations: a) U Helpful Hint V 1 2 + 2 x - 9 x + 3x b) 2 4 2 5-a a-5 Solution 1 2 1 2 The LCD is x(x - 3)(x + 3). + = + x2 - 9 x2 + 3x (x - 3)(x + 3) x(x + 3) Needs x a) Once the denominators are factored as in Example 5(a), you can simply look at each denominator and ask, \"What factor does the other denomi nator(s) have that is missing from this one?\" Then use the missing factor to build up the denominator. Repeat until all denominators are identical, and you will have the LCD. Needs x - 3 = 1 x 2(x - 3) + (x - 3)(x + 3)x x(x + 3)(x - 3) = x 2x - 6 + x(x - 3)(x + 3) x(x - 3)(x + 3) = 3x - 6 x(x - 3)(x + 3) We usually leave the denominator in factored form. b) Because -1(5 - a) = a - 5, we can get identical denominators by multiplying only the rst expression by -1 in the numerator and denominator: 4 2 4(-1) 2 = 5 - a a - 5 (5 - a)(-1) a - 5 = -4 2 a-5 a-5 = -6 -4 - 2 = -6 a-5 6 =a-5 Now do Exercises 47-64 In Example 6, we combine three rational expressions by addition and subtraction. E X A M P L E 6 Rational expressions with different denominators Perform the indicated operations. x+1 2x + 1 1 + x2 + 2x 6x + 12 6 Solution The LCD for x(x + 2), 6(x + 2), and 6 is 6x(x + 2). x+1 2x + 1 1 x+1 2x + 1 1 + - = + x 2 +\t2x 6x + 12 6 x(x + 2) 6(x + 2) 6 = 6(x + 1) x(2x + 1) 1x(x + 2) + 6x(x + 2) 6x(x + 2) 6x(x + 2) Factor denominators. Build up to the LCD. dug84356_ch06a.qxd 412 9/14/10 12:39 PM Page 412 6-32 Chapter 6 Rational Expressions = 6x + 6 2x2 + x x2 + 2x + 6x(x + 2) 6x(x + 2) 6x(x + 2) Simplify numerators. = 6x + 6 + 2x2 + x - x2 - 2x 6x(x + 2) Combine the numerators. = x2 + 5x + 6 6x(x + 2) = (x + 3)(x + 2) 6x(x + 2) Factor. = x+3 6x Reduce. Combine like terms. Now do Exercises 65-70 U3V Applications We have seen how rational expressions can occur in problems involving rates. In Example 7, we see an applied situation in which we add rational expressions. E X A M P L E 7 Adding work Harry takes twice as long as Lucy to proofread a manuscript. Write a rational expression for the amount of work they do in 3 hours working together on a manuscript. Solution Let x = the number of hours it would take Lucy to complete the manuscript alone and 2x = the number of hours it would take Harry to complete the manuscript alone. Make a table showing rate, time, and work completed: Rate Time Work Lucy 1 msp x hr 3 hr 3 msp x Harry 1 msp 2x hr 3 hr 3 msp 2x Now nd the sum of each person's work. 3 3 2 3 3 + = + x 2x 2 x 2x = 6 3 + 2x 2x 9 2x So in 3 hours working together they will complete = 9 2x of the manuscript. Now do Exercises 81-86 dug84356_ch06a.qxd 9/17/10 8:08 PM Page 413 6-33 6.4 413 Fill in the blank. 1. We can rational expressions only if they have identical denominators. 2. We can any two rational expressions so that their denominators are identical. 3 4 29 5. -- + -- = -5 3 15 4 5 3 6. -- - -- = -5 7 35 5 3 7. -- + -- = 1 20 4 2 3 8. For any nonzero value of x, -- + 1 = --. x x 1 a+1 9. For any nonzero value of a, 1 + -- = --. a a 1 4a - 1 10. For any value of a, a - -- = --. 4 4 True or false? 1 1 2 3. -- + -- = -2 3 5 1 7 1 4. -- - -- = -2 12 12 Exercises U Study Tips V When studying for a midterm or nal, review the material in the order it was originally presented. This strategy will help you to see connections between the ideas. Studying the oldest material rst will give top priority to material that you might have forgotten. U1V Addition and Subtraction of Rational Numbers Perform the indicated operation. Reduce each answer to lowest terms. See Example 1. Perform the indicated operation. Reduce each answer to lowest terms. See Example 2. 4 1 4. -- - -9 9 1 5 5. -- - -6 6 3 7 6. --- - -8 8 7 1 7. --- + -8 8 9 3 8. --- + --20 20 ( ) 7 4 14. -- + -6 15 1 9 15. -- - -8 10 7 1 3. -- - -8 8 5 3 12. -- + -6 10 7 5 13. -- + -16 18 1 3 2. -- + -8 8 1 5 10. -- + -4 6 7 5 11. -- + -10 6 1 1 1. -- + -10 10 1 2 9. -- + -3 9 2 5 16. -- - -15 12 ( ) 3 1 17. --- - --6 8 ( ) 1 1 18. --- - --5 7 6.4 Warm-Ups Addition and Subtraction dug84356_ch06a.qxd 414 9/14/10 12:39 PM Page 414 Chapter 6 Rational Expressions 6-34 U2V Addition and Subtraction of Perform the indicated operation. Reduce each answer to lowest terms. See Examples 5 and 6. Rational Expressions Perform the indicated operation. Reduce each answer to lowest terms. See Example 3. 47. 1 1 + x x+2 19. 1 1 + 2x 2x 20. 1 2 + 3y 3y 48. 1 2 + y y + 1 21. 3 7 + 2w 2w 22. 5x 7x + 3y 3y 49. 2 3 x + 1 x 23. 15 3a + a+5 a+5 24. a + 7 9 - 5a + a-4 a-4 50. 1 2 a-1 a 25. q - 1 3q - 9 - q-4 q - 4 51. 2 1 + a-b a+b 26. 3 - a a - 5 3 3 52. 3 2 + x+1 x-1 4h - 3 h-6 27. h(h + 1) h(h + 1) 53. 3 4 x 2 + x 5x + 5 2t - 9 t - 9 28. t(t - 3) t(t - 3) 54. 3 2 a + 3a 5a + 15 55. 2a a + a2 - 9 a - 3 56. x 3 + x-1 x -1 57. 4 4 + a-b b-a 58. 2 3 + x - 3 3 - x x2 - x - 5 1 - 2x 29. + (x + 1)(x + 2) (x + 1)(x + 2) 2x - 5 x - 2x + 1 + (x - 2)(x + 6) (x - 2)(x + 6) 2 30. Perform the indicated operation. Reduce each answer to lowest terms. See Example 4. 2 2 31. 1 1 + a 2a 32. 1 2 + 3w w 59. 3 2 2a - 2 1 - a 33. x x + 3 2 34. y y + 4 2 60. 5 3 2x - 4 2 - x 35. m +m 5 36. y + 2y 4 61. 1 3 x 2 - 4 x2 - 3x - 10 37. 1 2 + x y 38. 2 3 + a b 62. 2x 3x + 2 x2 - 9 x + 4x + 3 39. 3 1 + 2a 5a 40. 3 5 - 6y 8y 63. 3 4 + 2 x2 + x - 2 x + 2x - 3 41. w-3 w-4 - 9 12 42. y+4 y-2 10 14 64. x+4 x-1 + 2 x - x - 12 x + 5x + 6 43. 2 b -c 4 a 3 44. y + 7b 65. 1 1 1 + + a b c 45. 2 3 + wz2 w2z 46. 66. 1 1 1 + 2+ 3 x x x 1 5 - a5b ab3 2 dug84356_ch06a.qxd 9/14/10 12:39 PM Page 415 6-35 415 6.4 Addition and Subtraction 2 1 1 + x x - 1 x + 2 U3V Applications 68. 1 2 3 + a a + 1 a - 1 69. 5 3 4 - + 2 3a - 9 2a a - 3a 81. Perimeter of a rectangle. Suppose that the length of a rectangle is 3 feet and its width is 5 feet. Find a rational x 2x expression for the perimeter of the rectangle. 70. 5 3 c-4 - 2 6c 4c + 2 2c + c 67. Solve each problem. See Example 7. 82. Perimeter of a triangle. The lengths of the sides of a triangle are 1, 1 , and Match each expression in (a)-(f) with the equivalent expression in (A)-(F). 71. a) d) A) D) 72. a) d) A) D) 1 +2 y 1 1 + y 2y 3 y y+2 y 1 -x x 1 2 -x x 1 - x3 x2 1-x 2 x 1 2 + y y 2 e) +1 y 3 B) 2y y+2 E) 2y 1 1 b) - 2 x x 1 e) x x 1-x B) x x2 - 1 E) x b) 1 1 + y 2 y f) + 1 2 y+2 C) 2 2y + 1 F) y 1 c) -1 x 1 1 f) 2 x x 1 - x2 C) x x-1 F) x2 x 2x c) 1 x Figure for Exercise 82 83. Traveling time. Janet drove 120 miles at x mph before 6:00 A.M. After 6:00 A.M., she increased her speed by 5 mph and drove 195 additional miles. Use the fact that T = D to complete the following table. R Rate 1 3 73. 2p + 8 2p After x+5 4 12 + 2 w2 + w w - 3w 77. 2 1 - 2 b2 + 4b + 3 b + 5b + 6 78. 3 3 2 - 2 t + 2t 2 t t+2 80. 4 2 2 + + 2 3n n+1 n +n 120 mi mi hr 195 mi Write a rational expression for her total traveling time. Evaluate the expression for x = 60. 9 6 - 2 m -m-2 m - 1 79. Distance 3 3 + 2 a2 + 3a + 2 a + 5a + 6 76. Time mi hr x 75. 2 3x 1 2x Before 3 3 2y 2y + 4 meters. Find a rational expression for the perimeter of the triangle. Perform the indicated operation. Reduce each answer to lowest terms. 74. 2 3x 84. Traveling time. After leaving Moose Jaw, Hanson drove 200 kilometers at x km/hr and then decreased his speed by 20 km/hr and drove 240 additional kilometers. Make a table like the one in Exercise 83. Write a rational expression for his total traveling time. Evaluate the expression for x = 100. 2 85. House painting. Kent can paint a certain house by himself in x days. His helper Keith can paint the same house by himself in x + 3 days. Suppose that they work together on the job for 2 days. To complete the table on the next page, use the fact that the work completed is the product of the dug84356_ch06a.qxd 416 9/14/10 12:39 PM Page 416 6-36 Chapter 6 Rational Expressions Rate Time Kent 1 job x day 2 days Keith Work 1 job 2 days x + 3 day rate and the time. Write a rational expression for the fraction of the house that they complete by working together for 2 days. Evaluate the expression for x = 6. Photo for Exercise 86 Getting More Involved 86. Barn painting. Melanie can paint a certain barn by herself in x days. Her helper Melissa can paint the same barn by herself in 2x days. Write a rational expression for the frac tion of the barn that they complete in one day by working together. Evaluate the expression for x = 5. 87. Writing Write a step-by-step procedure for adding rational expressions. 88. Writing Explain why fractions must have the same denominator to be added. Use real-life examples. Math at Work Gravity on the Moon Hundreds of years before humans even considered traveling beyond the earth, Isaac Newton established the laws of gravity. So when Neil Armstrong made the rst human step onto the moon in 1969, he knew what amount of gravitational force to expect. Let's see how he knew. Newton's equation for the force of gravity between two objects is F = G m1m2, where m1 and m2 are the masses of the objects (in kilograms), d is d2 the distance (in meters) between the centers of the two objects, and G is the gravitational constant 6.67 10-11. To nd the force of gravity for Armstrong on earth, use 5.98 1024 kg for the mass of the earth, 6.378 106 m for the radius of the earth, and 80 kg for Armstrong's mass. We get F = 6.67 10-11 5.98 1024 kg 80 kg 784 Newtons. (6.378 106 m)2 To nd the force of gravity for Armstrong on the moon, use 7.34 1022 kg for the mass of the moon and 1.737 106 m for the radius of the moon. We get F = 6.67 10-11 7.34 1022 kg 80 kg 130 Newtons. (1.737 106 m)2 So the force of gravity for Armstrong on the moon was about one-sixth of the force of gravity for Armstrong on earth. Fortunately, the moon is smaller than the earth. Walking on a planet much larger than the earth would present a real problem in terms of gravitational force. dug84356_ch06a.qxd 9/14/10 12:39 PM Page 417 6-37 6.5 Mid-Chapter Quiz 2a2 - 10a + 12 6 - 3a Perform the indicated operation. 6 21 3xy2 5. 6. 7 10 5z 7. 9. a2 - 9 2a + 4 5a + 10 2a - 6 5 25 9 33 s2 s2 11. + 21 3 8. 10. 8x2z3 8y4 3x - 9 x2 - 6x + 9 8 12 6.5 U1V Complex Fractions U2V Using the LCD to Simplify 5 4 3 + 2 ab ab 15. x 3x + x + 1 x2 + 2x + 1 16. y y y+5 y+2 1 1 1 + + a b c b2 b6 21 3 m2 - 8m + 7 12. (m - 7) 2m In This Section 14. 8x - 2 8 4. w2 - 1 2w + 2 5 5 6 21 17. 2. 417 Chapter 6 13. Reduce to lowest terms. 36 1. 84 3. Sections 6.1 through 6.4 Complex Fractions Miscellaneous. 3x - 6 18. What numbers(s) can't be used in place of x in ? 2x + 1 19. Find the value of 3x - 6 when x = -2. 2x + 1 20. Find R(-1) if R(x) = 6x2 + 3 . 5x - 1 Complex Fractions In this section, we will use the idea of least common denominator to simplify complex fractions. Also we will see how complex fractions can arise in applications. Complex Fractions U3V Applications U1V Complex Fractions A complex fraction is a fraction having rational expressions in the numerator, denominator, or both. Consider the following complex fraction: 1 2 + 2 3 1 5 4 8 Numerator of complex fraction Denominator of complex fraction Since the fraction bar is a grouping symbol, we can compute the value of the numer ator, the value of the denominator, and then divide them, as shown in Example 1. dug84356_ch06b.qxd 418 9/14/10 12:44 PM Page 418 6-38 Chapter 6 Rational Expressions E X A M P L E 1 Simplifying complex fractions Simplify. 1 2 + 2 3 a) 1 5 4 8 2 4 5 b) 1 +3 10 Solution a) Combine the fractions in the numerator: 1 2 1 3 2 2 3 4 + + + 2 3 2 3 3 2 6 6 Combine the fractions in the denominator as follows: 1 5 1 2 5 2 5 4 8 4 2 8 8 8 Now divide the numerator by the denominator: 2 4 5 b) 1 +3 10 7 1 2 + 2 3 6 7 3 6 1 5 8 4 8 20 2 18 18 5 5 5 30 31 1 5 + 10 10 10 3 8 7 6 31 10 18 10 5 31 7 6 3 8 8 3 56 18 28 9 36 31 Now do Exercises 1-12 U2V Using the LCD to Simplify Complex Fractions A complex fraction can be simplied by performing the operations in the numerator and denominator, and then dividing the results, as shown in Example 1. However, there is a better method. All of the fractions in the complex fraction can be eliminated in one step by multiplying by the LCD of all of the single fractions. The strategy for this method is detailed in the following box and illustrated in Example 2. Strategy for Simplifying a Complex Fraction 1. Find the LCD for all the denominators in the complex fraction. 2. Multiply both the numerator and the denominator of the complex fraction by the LCD. Use the distributive property if necessary. 3. Combine like terms if possible. 4. Reduce to lowest terms when possible. E X A M P L E 2 Using the LCD to simplify a complex fraction Use the LCD to simplify 1 2 + 2 3 . 1 5 4 8 dug84356_ch06b.qxd 9/14/10 12:44 PM Page 419 6-39 6.5 Complex Fractions 419 U Calculator Close-Up V Solution You can check Example 2 with a calculator as shown here. The LCD of 2, 3, 4, and 8 is 24. Now multiply the numerator and denominator of the complex fraction by the LCD: 1 2 + 2 3 1 5 4 8 1 2 + 24 2 3 1 4 Multiply the numerator and denominator by the LCD. 5 24 8 1 2 24 + 24 2 3 1 5 24 24 8 4 Distributive property 12 + 16 6 15 Simplify. 28 9 28 9 Now do Exercises 13-20 CAUTION We simplify a complex fraction by multiplying the numerator and denomi nator of the complex fraction by the LCD. Do not multiply the numerator and denominator of each fraction in the complex fraction by the LCD. In Example 3 we simplify a complex fraction involving variables. E X A M P L E 3 A complex fraction with variables Simplify 2 1 x2 1 x . 1 2 U Helpful Hint V Solution When students see addition or subtraction in a complex fraction, they often convert all fractions to the same denominator. This is not wrong, but it is not necessary. Simply multiplying every fraction by the LCD eliminates the denominators of the original fractions. The LCD of the denominators x, x 2, and 2 is 2x 2: 1 2 x 1 1 x2 2 2 1 (2x2) x 1 x2 Multiply the numerator and denominator by 2x2. 1 (2x2) 2 1 2 2x2 2x2 x 1 1 2x2 2x2 x2 2 Distributive property dug84356_ch06b.qxd 420 9/14/10 12:44 PM Page 420 6-40 Chapter 6 Rational Expressions 4x2 2x Simplify. 2 x2 The numerator of this answer can be factored, but the rational expression cannot be reduced. Now do Exercises 21-30 E X A M P L E 4 Simplifying a complex frac