Page of 6 9.5 What you should learn GOAL Add and subtract rational epressions, as applied in Eample 4. GOAL Simplify comple fractions, as applied in Eample 6. Why you should learn it To solve real-life problems, such as modeling the total number of male college graduates in E. 4. Addition, Subtraction, and Comple Fractions GOAL WORKING WITH RATIONAL EXPRESSIONS As with numerical fractions, the procedure used to add (or subtract) two rational epressions depends upon whether the epressions have like or unlike denominators. To add (or subtract) two rational epressions with like denominators, simply add (or subtract) their numerators and place the result over the common denominator. EXAMPLE Perform the indicated operation. Adding and Subtracting with Like Denominators 4 a. + b. º + + a. + = 4 + 5 9 = = 4 b. º = º 4 + + +.......... Add numerators and simplify epression. Subtract numerators. To add (or subtract) rational epressions with unlike denominators, first find the least common denominator (LCD) of the rational epressions. Then, rewrite each epression as an euivalent rational epression using the LCD and proceed as with rational epressions with like denominators. EXAMPLE Adding with Unlike Denominators 5 Add: 6 + 4 º Skills Review For help with LCDs, see p. 908. 5 First find the least common denominator of 6 and 4. º It helps to factor each denominator: 6 = 6 and 4 º = 4 ( º). The LCD is ( º ). Use this to rewrite each epression. 5 5 + 4 = º 6 + 5[( º)] () = + 6 4( º) 6 [( º)] 4( º )() 0 º 0 = + ( º ) ( º) = + 0 º 0 ( º ) 56 Chapter 9 Rational Euations and Functions
Page of 6 EXAMPLE Subtracting With Unlike Denominators + Subtract: º + 4 +4 º 4 Look Back For help with multiplying polynomials, see p. 8. + + º = + 4 +4 º4 ( + ) º ( º) ( +) ( + )( º) ( + ) = º ( +) ( º ) ( º )( + )( + ) = º º º ( + 4) ( + ) ( º ) = º º 6 ( + ) ( º ) EXAMPLE 4 Adding Rational Models Statistics The distribution of heights for American men and women aged 0 9 can be modeled by 0.4 y = American men s heights + 0.008( º 0) 4 0.4 y = American women s heights + 0.008( º 64) 4 where is the height (in inches) and y is the percent (in decimal form) of adults aged 0 9 whose height is ± 0.5 inches. Source: Statistical Abstract of the United States a. Graph each model. What is the most common height for men aged 0 9? What is the most common height for women aged 0 9? b. Write a model that shows the distribution of the heights of all adults aged 0 9. Graph the model and find the most common height. a. From the graphing calculator screen shown at the top right, you can see that the most common height for men is 0 inches (4.%). The second most common heights are 69 inches and inches (4.% each). For women, the curve has the same shape, but is shifted to the left so that the most common height is 64 inches. The second most common heights are 6 inches and 65 inches. b. To find a model for the distribution of all adults aged 0 9, add the two models and divide by. y = 0.4 0.4 + + 0.008( º 0) 4 + 0.008( º 64) 4 From the graph shown at the bottom right, you can see that the most common height is 6 inches. 0. 0. 0 0. 0. 0 women men 54 64 0 80 men and women 54 6 80 9.5 Addition, Subtraction, and Comple Fractions 56
Page of 6 GOAL SIMPLIFYING COMPLEX FRACTIONS A comple fraction is a fraction that contains a fraction in its numerator or denominator. To simplify a comple fraction, write its numerator and its denominator as single fractions. Then divide by multiplying by the reciprocal of the denominator. EXAMPLE 5 Simplifying a Comple Fraction HOMEWORK HELP Visit our Web site www.mcdougallittell.com for etra eamples. INTERNET + Simplify: + + + + = Add fractions in denominator. + 4 + + ( + ) = ( + ) Multiply by reciprocal. + + 4 ( + ) = Divide out common factor. ( + )( + 4) = Write in simplified form. + 4.......... Another way to simplify a comple fraction is to multiply the numerator and denominator by the least common denominator of every fraction in the numerator and denominator. EXAMPLE 6 Simplifying a Comple Fraction object PHOTOGRAPHY The focal length of a camera lens is the distance between the lens and the point where light rays converge after passing through the lens. FOCUS ON APPLICATIONS p lens f image PHOTOGRAPHY The focal length ƒ of a thin camera lens is given by ƒ = p + where p is the distance between an object being photographed and the lens and is the distance between the lens and the film. Simplify the comple fraction. ƒ = p + Write euation. = p p p + Multiply numerator and denominator by p. p = + p Simplify. 564 Chapter 9 Rational Euations and Functions
Page 4 of 6 GUIDED PRACTICE Vocabulary Check Concept Check Skill Check. Give two eamples of a comple fraction.. How is adding (or subtracting) rational epressions similar to adding (or subtracting) numerical fractions?. Describe two ways to simplify a comple fraction. 4. Why isn t ( + ) the LCD of and? + ( + ) What is the LCD? Perform the indicated operation and simplify. 8 5. + 6. + + 5 + 5 5 Simplify the comple fraction. 6. º º 4 + 5 + + 4 º 5 + 8. 5 5 9. 0. 6 º 8 + 8 + 4. FINANCE For a loan paid back over t years, the monthly payment is given Pi by M = where P is the principal and i is the annual interest rate. º + t i t Pi( + i) Show that this formula is euivalent to M =. ( t + i) º PRACTICE AND APPLICATIONS Etra Practice to help you master skills is on p. 95. HOMEWORK HELP Eample : Es. Eamples, : Es. 8, 6 Eample 4: Es. 4 5 Eample 5: Es. 8 46 Eample 6: Es. 5, 5 OPERATIONS WITH LIKE DENOMINATORS Perform the indicated operation and simplify.. + 4. 6 6 0 º 0 4. º + + 5 5 6 5. + 6. 5 º. + 8 º º º º 5 +8 º 5 FINDING LCDS Find the least common denominator. 4 4 8., 9., 4( + ) 4 º 5 5 + 0.,, 9., + ( º6) 4 º º º8 +., ( º ). º6 º, º º8 +6 +8 LOGICAL REASONING Tell whether the statement is always true, sometimes true, or never true. Eplain your reasoning. 4. The LCD of two rational epressions is the product of the denominators. 5. The LCD of two rational epressions will have a degree greater than or eual to that of the denominator with the higher degree. 9.5 Addition, Subtraction, and Comple Fractions 565
Page 5 of 6 Look Back For help with the negative eponents in Es. 4 and 4, see p.. PHARMACIST In addition to miing and dispensing prescription drugs, pharmacists advise patients and physicians on the use of medications. This includes warning of possible side effects and recommending drug dosages, as discussed in Es. 48 5. CAREER LINK www.mcdougallittell.com INTERNET FOCUS ON CAREERS OPERATIONS WITH UNLIKE DENOMINATORS Perform the indicated operation(s) and simplify. 6 6. +. º º 4 5 8. º + 9. 6 + 4 6( º) 6 + º 9 º 0 5 º 6 0. +. º º º5 º4 + º8 + 4 4 0. º. º 5 + +5 + 8 º 0 + 4. + º + + 5. º 8 º + 8 + 6 º + º 6 4 5 6. + º 4 + º. 0 + + º º 6 SIMPLIFYING COMPLEX FRACTIONS Simplify the comple fraction. º5 0 + º 8. 9. 40. 6 + 4 º + + + º º º 5 º 4 º + º 4 + (4 +) 4. 4. 4. º º + 4 + 4 + +6 + 8 º 44. 45. + º 9 º + 64 46. 5 5 + º º º 6 + º + º 9 º 4. COLLEGE GRADUATES From the 984 85 school year through the 99 94 school year, the number of female college graduates F and the total number of college graduates G in the United States can be modeled by º9,600t + 49,000 560t F = and G = + 98,000 º0.0580t + 0.0048t + where t is the number of school years since the 984 85 school year. Write a model for the number of male college graduates. Source: U.S. Department of Education DRUG ABSORPTION In Eercises 48 5, use the following information. The amount A (in milligrams) of an oral drug, such as aspirin, in a person s bloodstream can be modeled by A = 9t + 0. 0.8t 4 + 0.99t + where t is the time (in hours) after one dose is taken. Source: Drug Disposition in Humans 48. Graph the euation using a graphing calculator. 49. A second dose of the drug is taken hour after the first dose. Write an euation to model the amount of the second dose in the bloodstream. 50. Write and graph a model for the total amount of the drug in the bloodstream after the second dose is taken. 5. About how long after the second dose has been taken is the greatest amount of the drug in the bloodstream? 566 Chapter 9 Rational Euations and Functions
Page 6 of 6 Test Preparation Challenge EXTRA CHALLENGE www.mcdougallittell.com ELECTRONICS In Eercises 5 and 5, use the following information. If three resistors in a parallel circuit have resistances R, R, and R (all in ohms), then the total resistance R t (in ohms) is given by this formula: R t = + + R R R 5. Simplify the comple fraction. 5. You have three resistors in a parallel circuit with resistances 6 ohms, ohms, and 4 ohms. What is the total resistance of the circuit? 54. MULTI-STEP PROBLEM From 988 through 99, the total dollar value V (in millions of dollars) of the United States sound-recording industry can be modeled by V = 5 8+ 4t + 0.05t where t represents the number of years since 988. Source: Recording Industry Association of America a. Calculate the percent change in dollar value from 988 to 989. b. Develop a general formula for the percent change in dollar value from year t to year t +. c. Enter the formula into a graphing calculator or spreadsheet. Observe the changes from year to year for 988 through 99. Describe what you observe from the data. CRITICAL THINKING In Eercises 55 and 56, use the following epressions. +, +, + + + + + + + 4 55. The epressions form a pattern. Continue the pattern two more times. Then simplify all five epressions. 56. The epressions are getting closer and closer to some value. What is it? MIXED REVIEW SOLVING LINEAR EQUATIONS Solve the euation. (Review. for 9.6) 5. º = 5 58. 6 º 0 = º 59. 4 + = º 5 6 60. 8 + 4 = º8 6. º º = 5 6. = º4 +0 6. º5 º 4 = 5 64. + 8 =º 65. =+ 5 6 SOLVING QUADRATIC EQUATIONS Solve the euation. (Review 5., 5. for 9.6) 66. º 5 º 4 = 0 6. 5 º 8 = 4( + ) 68. 6 + º 5 = 0 69. ( º5) = 0. ( +) º=49. ( +6)=º 9.5 Addition, Subtraction, and Comple Fractions 56