Mini-Lecture 6.1 Rational Functions and Multiplying and Dividing Rational Expressions

Learning Objectives: Mini-Lecture 6. Rational Functions and Multiplying and Dividing Rational Epressions. Find the domain of a rational function.. Simplify rational epressions.. Multiply rational epressions. 4. Divide rational epressions. 5. Use rational functions in applications. 6. Key vocabulary: undefined, domain, simplify, reciprocal. Eamples:. Find the domain of each rational epression. + 6 + a) f( ) = b) g ( ) = c) 4 5 5 h ( ) = 7 + d) 9 p ( ) = 7 + 5 e) f( ) = +. Simplify each rational epression. a) 6 b) 8 9 + c) 0+ 5 5 d) 8 8 e) y y + 5y + 6 + 0y + f) y 64 4y 6. Multiply and simplify. 8 a) 5 5 b) 4y 49 8 y c) + 7+ 0 + + 8+ 5 7 8 4. Divide and simplify. 4 a) b) 5 40 + 5 6 + 9+ 8 + 7+ c) 4 64 + 8+ 5. Use rational functions in applications. A company s cost per book for printing particular books is given by the rational functions 0.8 + 5000 C( ) =. Find the cost per book for printing a) 00 books b) 000 books Teaching Notes: Many students need a review of simplifying, multiplying and dividing numerical fractions before attempting algebraic ones. Many students have trouble with problems where the factors in the numerator and denominator have opposite signs. Refer to the end-of-section eercises for applied problems. Refer students to the Simplifying/ Multiplying/ Dividing Rational Epressions chart in the tet. Answers: a) { is a real number}, b) { is a real number and 0}, c) { is a real number and 7}, d) { is a real number and 5 }, e) { is a real number and y +, }; a) -, b) -9, c) -5, d) -, e) 7 y + 7, f) y + 4y+ 6 4 ; a) 4 9, b), c) 5 + ( 7) 9 ; 4a) 5, b) + 4, c) ; 5a) $7.47, b) $.47 + M - Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Mini-Lecture 6. Adding and Subtracting Rational Epressions Learning Objectives:. Add or subtract rational epressions with common denominators.. Identify the Least Common Denominator (LCD) of two or more rational epressions.. Add or subtract rational epressions with unlike denominators. 4. Key vocabulary: least common denominator. Eamples:. Add or subtract as indicated. a) 8 + b) 9 + + 8 5 7 7 + + 6+ 8 + 6+ 8 c). Find the LCD of the rational epressions in each list. a) c) ' 7 4 9, + 6 b), 7y y 6 5 d), ' y + y+ y 8. Add or subtract as indicated. If possible, simplify your answer. a) 5 9 7 b) 6y 5y + c) 6 + r r 8 d) 4 4 + + + + 4 + 4+ 48 e) 7 8 4 f) + +. Teaching Notes: 0 6 g) + 5 + + 5 Most students need a review of adding, subtracting, and finding LCDs of numerical fractions before attempting algebraic ones. Many students find this section difficult. Some students need to see more eamples for objective. Etra time spent here is well worth it and pays off with greater success in objective. Refer students to the Adding or Subtracting Rational Epressions with Common/Different Denominators and Finding the Least Common Denominator charts in the tet. Answers: a), b) -, c) ; a) 77, b) 7y, c) (-)(+), d) 8(+y) (-y); a) + 4 d) 4, e) 4 + 7+ 0 ( )( + 6)( + 8), f) 7 6, g) 4 + 40 ( + 5) 9 0y 7+, b), c) 4r, rr ( ) M - Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Mini-Lecture 6. Simplifying Comple Fractions Learning Objectives:. Simplify comple fractions by simplifying the numerator and denominator and then dividing.. Simplify comple fractions by multiplying by a common denominator.. Simplify epressions with negative eponents. 4. Key vocabulary: comple rational epression. Eamples:. Simplify each comple fraction by simplifying the numerator and denominator and then dividing. a) + 8 5 4 8 b) + 4 4 + 4 c) 9 + 9 a 9 9 a d) 6 5y y 4 5 y e) 9 + 4 8 f) 4 5 + 5 5 + 5 g) 4 + 5 5 5 h) 9 + + 5 + 7 + + + 5. Simplify selected problems from a) through h) by multiplying the least common denominator.. Simplify.. a) + y b) + 5y 7 y c) 6 + (6 y) Teaching Notes: Stronger students tend to prefer using the multiply by LCD method. Many students need to be reminded of how to deal with negative eponents before attempting objective. Refer students to the Simplifying a Comple Fraction: Method / Method charts in the tet. Answers: a) 5 7, b) + a + 9, c), d) 4+5y, e) 4 a 4 8 b) y + 5 y 6y+, c) 7y 6y, f) 5 0, g) 4 0, h) 6; a-h) same as a-h); a) + y+, y M-4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Learning Objectives: Mini-Lecture 6.4 Dividing Polynomials: Long Division and Synthetic Division. Divide a polynomial by a monomial.. Divide by a polynomial.. Use synthetic division to divide a polynomial by a binomial. 4. Use the remainder theorem to evaluate polynomials. Eamples:. Divide. a) 4 8 4 by 4 b) y + 9 y y y c) 5 4 8 y + y 6 y 4 4 y. Divide. ) ( a) ( 5 5 b) + + + ) ( ) ( ) ( c) ( 4 8 7 d) 5 7+ 4 ) + ) ( 0+ + ) ( 6 ). Use synthetic division to divide. a) 4 45 + 5 b) 9 5 7 c) 6 + 4+ 4 + 4 d) 4 + 6 4. Use the remainder theorem to P(c). a) P() = + 8; b) P() = 5 4 + + ; Teaching Notes: Remind students to check their answers by multiplying. Encourage students to write the intermediate step a + b = a + b when dividing by a monomial. c c c Most students understand dividing by a binomial better if a numerical long division problem is shown in parallel. Most students understand the synthetic division process after a couple of eamples. Most students prefer synthetic division over long division. Answers: a) -, b) +y-y 4y, c) 8+ ; a) +7, b) 5-7, c) - -5+, d) --; a) -9, b) +5, c) - ++6, d) + + 4+ 8+ 4a), b) 08 M-5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Mini-Lecture 6.5 Solving Equations Containing Rational Epressions Learning Objectives:. Solve equations containing rational epressions... Key vocabulary: equation versus epression, etraneous solutions. Eamples:. Solve each equation and check the solution. a) y y= 5 b) 5 y + 6 = + y 5 4 c) 4 = 5 d) 5 = + + 0 e) + = f) 4+ 6 = 5 6 g) 4 = + + + 6 6 6 = + 5+ 4 + + + 5+ 4 h) 5 i) = 5 5. Teaching Notes: j) + = + 5 + + 8+ 5 Remind students to always determine the values not allowed for before solving a rational epression. Many students are confused by the concept of an etraneous solution. Show them a simple eample such as : = = = = 0 = 0,; =0is etraneous. Some students prefer to make equivalent fractions with a common denominator, and then set the numerators equal to each other. Refer students to the To Solve an Equation Containing Rational Epressions chart in the tet. Answers: a) {75}, b) 4 ; c) {}, d) {7}, e) {-5,4}, f) {}, g) 5, h) {}; i), j) M-6 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Mini-Lecture 6.6 Rational Equations and Problem Solving Learning Objectives:. Solve an equation containing rational epressions for a specified variable.. Solve problems by writing equations containing rational epressions.. Key vocabulary: ratio, rate, proportion. Eamples:. Solve each equation for the specific variable. a) PV T pv = for V b) t + = for c c) a b c A P = for r + rt d) GMm A = h ( B+ b ) for B e) F = for M f) r a anr S = r for a. Solve. a) Number Two times the reciprocal of a number equals 4 times the reciprocal of 5. Find the number. b) Proportion The ratio of the weight of an object on Earth to an object on Planet X is 4 to 9. If a person weighs 0 pounds on Earth, find his weight on planet X. Round to the nearest whole number. c) Work One pump can drain a pool in 9 minutes. When a second pump is also used, the pool only takes 6 minutes to drain. How long would it take the second pump to drain the pool if it were the only pump in use? d) Rate Ale can run 5 miles per hour on level ground on a still day. One windy day he runs miles with the wind, and in the same amount of time runs 4 miles against the wind. What is the rate of the wind? Teaching Notes: Many students find this section difficult. Most students need to set up a chart to solve work and rate problems. Refer them to the tetbook eamples for samples. Encourage students to check whether their solutions seem reasonable. Refer to students to the Solving an Equation for a Specified Variable chart in the tet. pvt ab Answers: a) V =, b) c tp = b + a, c) A P r =, d) Pt b) 58 pounds, c) 8 minutes, d) mph A bh B =, e) h Fr M =, f) a = S( r) + anr ; a) Gm 5, M-7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Mini-Lecture 6.7 Variation and Problem Solving Learning Objectives:. Solve problems involving direct variation.. Solve problems involving inverse variation.. Solve problems involving joint variation. 4. Solve problems involving combined variation. 5. Key vocabulary: constant of variation or constant of proportionality. Eamples:. Find the constant of variation and the direct variation equation for each situation. Then solve as indicated. a) y = 4 when =. Find y when = 9. b) The amount of gas that a helicopter uses is directly proportional to the number of hours spent flying. The helicopter flies for hours and uses 8 gallons of fuel. Find the number of gallons of fuel that the helicopter uses to fly for 5 hours.. Find the constant of variation and the inverse variation equation for each situation. Then solve as indicated. a) y = 4 when =. Find y when = 6. b) The amount of time it takes a swimmer to swim a race is inversely proportional to the swimmer s speed. A swimmer finishes a race in 50 seconds with a speed of feet per second. Find the speed if it takes 5 seconds to finish the race.. Find the constant of variation and the joint or the combined variation equation for each situation. Then solve as indicated. a) r varies jointly as the square of s and the square of t. r = when s = and t =. b) is directly proportional to y and inversely proportional to the cube of z. = when y = and z =. c) The volume V of a given mass of gas varies directly as the temperature T and inversely as the pressure P. A measuring device is calibrated to give V = 00 in when T = 50 and P= 0 lb/ in. What is the volume on this device when the temperature is 70 and the pressure is 0 lb / in? Teaching Notes: Most students will understand the concepts of direct and inverse variation better if real-life eamples are discussed in a qualitative way for problem. Some students are confused by solving for the constant of variation and then using that constant in the original equation and solving for a different variable. 4 4 Answers: a) k =, y =, y =, b) k=6, g=6h, 0 gallons of fuel; a) k =, y =, y =, b) k=50, t= 50, 6 feet s per second; a) k=, r=s t 8y T, b) k=8, =, c) k=, V =, in z P M-8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. Form I Find the domain of each rational epression.. f Name Date.. f.. f( ) 6 4 4. Simplify each rational epression. 4. 5. 6. 7. y 6y 6 6 5 8ab a b 6ab 69 4. 5. 6. 7. Perform each indicated operation and simplify. 8. 9. y y 6 4 46 4 8. 9. m m4 0. m 4 m 5m 6y y. 7z z 0... 4 5 6 0 0. E-7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. Form II Name Date Find the domain of each rational epression.. f.. f( ) 6 4 4.. f( ) 67. Simplify each rational epression. 4. 5. 6. 7. 6 9 7 7 7ab a b 6ab 44 4. 5. 6. 7. Perform each indicated operation and simplify. 8. 9. 6y y 4y 6 6 5 6 8. 9. m m4 0. 4 m m 5m 0... 8 4 7.. E-8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. Form III Name Date Find the domain of each rational epression.. f. f 4... f( ) 5 5 4 4. Simplify each rational epression. 4. 4 4. 5. 6. 7. 5 5 8ab a b 6ab 5. 6. 7. Perform each indicated operation and simplify. 8. 6 7 7 8. 4 9. 7 9. 0. 6 4 0. m m4. 4 m 5m m.. 4 4 9 8 5 6. E-9 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. Form I Name Date Add or subtract as indicated. Simplify each answer.... z z 6 4 4... Find the LCD of the rational epressions in each list. 4., 4 4 4. 5. 4 8, 7 7 5 7, 8 6. 4 6 5. 6. Add or subtract as indicated. Simplify each answer. 7. 4 5 8. 7 7. 8. 9. 0. 5 5 9. 0.. 4 8 7 7.. 4 4 4. E-0 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. Form II Name Date Add or subtract as indicated. Simplify each answer.. y 7y 4 4. 6 6 6. 6 6... Find the LCD of the rational epressions in each list. 9 4., 4 5 5., 4 6. 5, 6 8 6 4. 5. 6. Add or subtract as indicated. Simplify each answer. 7. 8. 6 7 5 7. 8. 9. 7 9. 0. 4 0.. 8 y. y y y.. E- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. Form III Name Date Add or subtract as indicated. Simplify each answer... y 7 7.. 5 6 6.. Find the LCD of the rational epressions in each list., 8z z 4. 4 5. 4, 4 5 6. 4 5,, 6 4. 5. 6. Add or subtract as indicated. Simplify each answer. 7. 8. 5 5 5 5 6 7. 8. 6 4 4 6 6 9. 0. 8 4 56. 5 6 4 4. 4 9. 0... E- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. Form I Name Date Simplify each comple fraction.... 4. 5. 6. 7. 8. 4 0 6 5 6 4 ab b a b 9 4 9 b a a a b b a a b b... 4. 5. 6. 7. 8. E- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. (cont.) Name a 9. a ab a b 0. 9. 0. E-4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. Form II Name Date Simplify each comple fraction.. 4 45 5.. 9.. 4. 5. 4 ab b a b 4 4 6 b a a a b b. 4. 5. 6. 4 6. 7. 8. 9. a a b b 5 9 7. 8. 9. E-5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. (cont.) Name 0. 5 5 5 0. E-6 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. Form III Name Date Simplify each comple fraction.... 4. 5. 6. 7. 8. 7 9 49 yz 6 z y 5 5 6 y 5 6 y ab b a b 4 4 5 4... 4. 5. 6. 7. 8. E-7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6. (cont.) Name 9. 0. 5 5 5 5y y 9. 0. E-8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.4 Form I Name Date Divide. 60 4 50 0 0... 4 y 7y y 7y 4 4 y 6 y 6y 5 4.. 4. 6 9 5. 4 6. 9 7. 7 8. 5 0 9. 8 4 8 0. 6 8 4 4. 6 8. t t. 7 4 0 0 0 4 4. 5. 6. 7. 8. 9. 0.... E-9 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.4 Form II Name Date Divide. 8 4a b 8a b 6ab.. 6 6 y 4 y y y 4. 5 00 5 5. 7 6 6 6. 4 4 7. 5 6 8. 4 9. 5 6 0 5 0. 6t 4 69t 00 6t 5. 4 6. 7t t. 6... 4. 5. 6. 7. 8. 9. 0.... E-40 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.4 Form III Name Date Divide. 60 4 0 0 0... 6 y y y y 6 4 6a b 0a b 4ab.. 4. 5 00 0 5. 6 4 6. 8 7. 6 8 6 4 8. t 5 t 5 9. 5 0. 4 7. 4. 7t 8 t t t t t 4. 6 5 0 4 4. 5. 6. 7. 8. 9. 0.... E-4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.5 Form I Name Date Solve.... 4. 5. 6. 7. 8. 9. 4 5 5 z 7 0z z 0 y 4 y y y 4 7 4 4 50 0 5 5 0 4... 4. 5. 6. 7. 8. 9. 0. 8 4 9. 6. 4 7 0... E-4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.5 Form II Name Date Solve.... 4. 5. 6. 7. 5 5 y 4 y y y 6 4 5 5 4 5 5 5 4... 4. 5. 6. 7. 8. 5 9 9. 5 8. 9. 0. 0. 6. 6 9 8 z. 6z z z 6z 4.. E-4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.5 Form III Name Date Solve.... 4. 5. 6. 7. 7 4 5 5 4 y y y y 5 y y y4 y4 4 5 74 74 4... 4. 5. 6. 7. 6 8. 9 8. 9. 0. 5 5 5 4 4 4 9. 0.. 6 6 9 8... E-44 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.6 Form I Solve each equation for the specified variable.. for y. for a a b c Name Date.. Solve.. Find the number that when subtracted from the numerator 4 and subtracted from the denominator of results in a 9. fraction equivalent to 6. 4. If a certain number is subtracted from the numerator and added to the denominator of 7, the new fraction is 9 equivalent to. Find the number. 5. Find the number that when subtracted from the numerator 4. 5. and added to the denominator of 7 results in a fraction equivalent to 9. 6. In the United States, out of 60 homes is heated by wood. At this rate, how many homes in a community of,000 homes are heated by wood? (Source: 005 American Housing Survey for the United States) 7. In a foreign country, out of 0 homes is heated by wood. At this rate, how many homes in a community of,000 homes are heated by wood? 8. In the United States, out of homes is heated by fuel oil. At this rate, how many homes in a community of,000 homes are heated by fuel oil? (Source: 005 American Housing Survey for the United States) 9. Sidney can mow the lawn in 4 hours using her riding mower. Her son can mow the same lawn in hours using a push mower. How long would it take them to mow the lawn working together? 0. Will Wilson takes 6 hours to shoot a movie scene. It takes Steve Nugent twice as long to do the same the job. How long will it take them to shoot the scene, working together? 6. 7. 8. 9. 0. E-45 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.6 (cont.) Name. Pat can wash the family s cars in hours. Pat s son Tom can do the same job in hours. How long will it take them to wash the family s cars working together?. James Lawson takes times as long to go 48 miles upstream as he does to return. If the boat cruises at 0 mph in still water, what is the speed of the current?... John Williams takes 7 9 times as long to go 70 miles. downstream as he does to return. If the boat cruises at 40 mph in still water, what is the speed of the current? E-46 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.6 Form II Solve each equation for the specified variable.. for y y Name Date.. for p m n p. Solve.. Find the number that when added to the numerator and subtracted from the denominator of results in a fraction equivalent to 9 7. 4. Find the number that when added to the numerator and subtracted from the denominator of results in a fraction equivalent to 4 9.. 4. 5. If a certain number is added to the numerator and subtracted from the denominator of 4, the new fraction is 5. equivalent to. Find the number. 6. In the United States, out of homes is heated by fuel oil. At this rate, how many homes in a community of 4,000 homes are heated by fuel oil? (Source: 005 American Housing Survey for the United States) 7. In a foreign country, out of 0 homes is heated by wood. At this rate, how many homes in a community of 6,000 homes are heated by wood? 8. In the United States, out of 60 homes is heated by wood. At this rate, how many homes in a community of 5,000 homes are heated by wood? (Source: 005 American Housing Survey for the United States) 6. 7. 8. E-47 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.6 (cont.) Name 9. It takes Sam 8 hours to build a section of fence. When David helps him, they can build a section in hours. How long would it take David to build a section working alone? 0. Grant Wilson takes 6 hours to shoot a movie scene. Steven Moss can do the same job in half the time. How long will it take them to shoot the scene, working together?. Juan Ramirez can clean the animal cages at the zoo where he works in 4 hours. His supervisor can do the same job in hours. How long will it take them to clean the animal cages at the zoo if they work together?. Ale Bater takes times as long to go 75 miles upstream as he does to return. If the boat cruises at 0 mph in still water, what is the speed of the current? 9. 0.... Lance Greyson takes 5 as long to go 90 miles. downstream as he does to return. If the boat cruises at 4 mph in still water, what is the speed of the current? E-48 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.6 Form III Name Date Solve each equation for the specified variable.. for m m n p.. Solve. for b a b c.. If a certain number is subtracted from the numerator and. subtracted from the denominator of 4, the new fraction is 9 equivalent to. Find the number. 6 4. If a certain number is added to the numerator and subtracted from the denominator of, the new fraction is equivalent to 9. Find the number. 7 5. Find the number that when added to the numerator and added to the denominator of results in a fraction equivalent to 9 4. 7 6. In the United States, out of homes is heated by fuel oil. At this rate, how many homes in a community of 48,000 homes are heated by fuel oil? (Source: 005 American Housing Survey for the United States) 7. In a foreign country, out of 0 homes is heated by wood. At this rate, how many homes in a community of 0,000 homes are heated by wood? 8. In the United States, out of 60 homes is heated by wood. At this rate, how many homes in a community of 850 homes are heated by wood? (Source: 005 American Housing Survey for the United States) 4. 5. 6. 7. 8. E-49 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.6 (cont.) Name 9. John Williams takes 7 times as long to go 70 miles upstream as he does to return. If the boat cruises at 40 mph in still water, what is the speed of the current? 0. One model airplane travels 0 kilometers per hour faster than another. The faster plane can travel 400 km in the same time the slower plane travels 50 km. Find the speed of both planes.. Lance Jones takes times as long to go 90 miles upstream as he does to return. If the boat cruises at 4 mph in still water, what is the speed of the current?. Ale Wilson takes 6 hours to shoot a movie scene. Pat Morrison can do the same job in one third the time. How long will it take them to shoot the scene, working together?. Jack can mow the lawn in hours using a push mower. Fran can mow the same lawn in 4 hours using her riding mower. How long would it take them to mow the lawn working together? 9. 0.... E-50 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.7 Form I If y varies directly as, find the constant of variation and the direct variation equation for each situation.. y = 6 when = Name Date.. y = when = 6. Solve.. The distance a car travels at a fied rate varies directly as the time. If a car travels 0 miles in hours, how long would it take to travel 600 miles? 4. The volume of a gas at a constant temperature varies inversely as the pressure. Find the volume of a gas under a pressure of 0 pounds if the gas occupies 5 cubic centimeters under a pressure of 0 pounds.. 4. If y varies inversely as, find the constant of variation and the direct variation equation for each situation. 5. y = 0 when = 7 6. y = 0.5 when = 5. 6. Find the constant of variation and the variation equation for each situation. 7. y varies jointly as and the square root of z; y = 0 when = 0, z = 49 8. a varies directly as the square of b and inversely as the cube of c; a = when b = 4, c = 7. 8. Solve. 9. Hooke s law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 0-pound weight attached to the spring stretches the spring 5 inches, find the distance that a 45-pound weight attached to the spring stretches the spring. 0. The period (the time required to complete one swing) of a simple pendulum varies directly as the square root of its length. If a pendulum 6 feet long has a period of seconds, find the period of a pendulum 5 feet long. 9. 0. E-5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.7 Form II Name Date If y varies directly as, find the constant of variation and the direct variation equation for each situation.. y = 9 when =. y = when = 6.. Solve.. The distance a car travels at a fied rate varies directly as the time. If a car travels 5 miles in 5 hours, how long would it take to travel 780 miles? 4. The volume of a gas at a constant temperature varies inversely as the pressure. Find the volume of a gas under a pressure of 5 pounds if the gas occupies 5 cubic centimeters under a pressure of 0 pounds.. 4. If y varies inversely as, find the constant of variation and the direct variation equation for each situation. 5. y = 0 when = 6. y = 0.8 when =.4 5. 6. Find the constant of variation and the variation equation for each situation. 7. y varies jointly as and the square root of z; y = 40 when = 0, z = 49 8. a varies directly as the square of b and inversely as the cube root of c; a = when b = 4, c = 64 7. 8. Solve. 9. The distance that a freely falling body falls varies directly as the square of the time it falls. If a body falls 400 feet in 5 seconds, how far does it fall in seconds? 0. The period (the time required to complete one swing) of a simple pendulum varies directly as the square root of its length. If a pendulum feet long has a period of 4 seconds, find the period of a pendulum feet long. 9. 0. E-5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Additional Eercises 6.7 Form III Name Date If y varies directly as, find the constant of variation and the direct variation equation for each situation.. y = 9 when = 7. y = 0.7 when = 4.. Solve.. The time required for a car to travel a fied distance varies inversely as the rate at which it travels. If it takes 5 hours at 64 mph to travel the distance, how long will it takes at 60 mph? 4. The volume of a gas at a constant temperature varies inversely as the pressure. Find the volume of a gas under a pressure of 4 pounds if the gas occupies cubic centimeters under a pressure of 0 pounds.. 4. If y varies inversely as, find the constant of variation and the direct variation equation for each situation. 5. y = 0 when = 6 5. 6. y = 5 when = 0 6. Find the constant of variation and the variation equation for each situation. 7. y varies jointly as and the square root of z; y = 0 when = 0, z = 6 8. v varies jointly as h and the square of r; v = 07 when h = 8, r = 4 7. 8. Solve. 9. Hooke s law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 0-pound weight attached to the spring stretches the spring inches, find the distance that a 5-pound weight attached to the spring stretches the spring. 0. The period (the time required to complete one swing) of a simple pendulum varies directly as the square root of its length. If a pendulum 0 feet long has a period of seconds, find the period of a pendulum 40 feet long. 9. 0. E-5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Section 6. Rational Functions and Multiplying and Dividing Rational Epressions Objective: Provide practice with multiplying and dividing rational epressions.. Suggested Format: Small Group Time: 0 minutes Find a simplified epression for the area of the figure.. A = lw 9 4. A= bh 9. A = lw + 5+ 6 4 4. Find a simplified epression for the missing side of the rectangle if? + 4 + 5 +. + 7 + 0 A = A- 5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Section 6.7 Variation and Problem Solving Objective: Apply concepts of direct and inverse variation. Suggested Format: Small Group Time: 0 minutes The daily sales, S, of Java Jay s Coffee Shop varies directly as their advertising budget, A, and inversely with the price, P, of a cup of coffee. When the advertising budget is $5 per day the coffee shop sells 80 cups of coffee at $.50 each.. Describe what it means for S to vary directly as A.. Describe what it means for S to vary indirectly as P.. Solve for the constant of variation and use this value to write an equation epressing S in terms of A and P. 4. What are the epected daily sales if the price of a cup of coffee goes up to $.75 per cup but the advertising budget remains at $5 per day? 5. What are the epected daily sales if the advertising budget is increased to $0 per day and the price of coffee remains at $.50 per cup? A- 6 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form A. Find the domain of the rational epression.. f ( ) = + 0 Write each rational epression in lowest terms.. 6 y 6 0 y 4 6.. 5. Perform the indicated operations. Write answers in lowest terms. 4. 5. 6. 7. 8. + 6 + 9 9 + 5 + 4 + 4 6 + 5 5 0 40 + 7 + 0 4 + 4 + 4 6 4 6 40 4 4 60 6 + + 5 + 5 4. 5. 6. 7. 8. 5 6 5 + 6 9. 0. 4 4 + 4 6. 4 + 0 9 + 0 9. 0.. T- 7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form A cont d Simplify each comple fraction... 4 5 7 5 +.. 4. + 5 4. 4 5 Divide. 5. y + y 8 4 6 y 5. 6. 4 + 6 8 + 4 6. 7. Use synthetic division to divide. 7. 4 6 + 5 8 by ( 4) ( ) P = 4 +, use the 8. 8. If ( ) 4 remainder theorem to find P( ). 9. Solve for y. 4y + 6 = 9. T- 8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form A cont d Solve each equation for. 0. 4 + = 0... = 5 8 8 + = +.. + 5. = + 7 + 0 + 0 +. 4. The speed of a boat in still water is 0 mph. If 4. the boat travels 40 miles downstream in the same time that it takes to travel 9 miles upstream, find the current of the stream. 5. Suppose that w is directly proportional to y and 5. inversely proportional to v. If w = when v = 0 and y = 0, find w when v = 0 and y = 5. T- 9 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form B Find the domain of each rational function.. + 6. 9 + 5.. Write each rational epression in lowest terms.. 4. y 6 y 48y 6 y 5 4. 4. Perform the indicated operation. Write answers in lowest terms. 5. 4 7 y 4yz 5. 5 6y z 8 y 6. a + b a b a b a + ab 6. 7. 7. + + 5 + 4 8. 9. p + p p p p p + 7 + 9 + + 5 + + 8. 9. 0. + 6 + 0.. 5. T- 0 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form B cont d Divide... y y + y 6y 8 4 + 4 + 4.. 4. Use synthetic division to divide. 4. 4 + ( + + ) by ( ) P = + 7 +, find P(). 5. 5. If ( ) 4 Simplify the comple fractions. 6. 6 9 6. 7. 8. + 5 7. 8. Solve each equation for. 9. 5 + 5 = 9. 0. 8 6 + = 8 0. 0. 0 =. T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form B cont d. = + 0 + 5.. If a number is subtracted from the numerator. of 5 and the same number is added to the denominator of, the result is equivalent to. 5 Find the number. 4. Suppose that w is proportional to the square root 4. of v. If w is 0 when v is 6, find w when v is 9. 5. Suppose that b is directly proportional to c, and 5. inversely proportional to d. If b = 6 when c = and d = 6, find b when c = 8 and d = 8. T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form C. Find the domain of the rational function.. 7 f ( ) = 4 Write each rational epression in lowest terms... 00a bc 4 40a c 40 5.. y 4y + 4. 7 + 4. Perform the indicated operation. Write answers in lowest terms. 5. 6. 7. 8. 5 + 4 5 + 0 + 5 8 + 7 + 64 8 4 6 6 4 + + 8 4 6 6 05 + 5 + 4 + 5. 6. 7. 8. + 4 + 4 4 9. 9. 0. + 0. a + a. +. a a T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form C cont d Simplify each comple fraction... 4 5 8 5 6.. 4. Divide. 4. 5. 0a b + 45a b 0ab 5ab 5. 6. ( 7 5) ( ) + + 6. 7. Use synthetic division to divide. 7. 4 7 + 6 + ( ) ( ) 8. If P( ) = + 5, find ( ) Solve each equation for. P. 8. 9. + = + 0 9. 5 0. 0 = 0. 4. = + 4 +. T- 4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form C cont d. a =. a 4. Three workers can complete a job in hours,. 5 hours, and 6 hours respectively. How long will it the workers to complete the job if they work together? 4. One number is four times another. The sum of 4. their reciprocals is. Find the two numbers. 4 5. Suppose that b is proportional to c, and inversely 5. proportional to the square of d. If b = 45 when c = 40 and d =, find b when c = 0 and d =. T- 5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form D Circle the correct answer.. Find the domain of the rational function f ( ) = { is a real number and,, 4} b. { is a real number and, } { is a real number and, 4, 6} d. { is a real number and, 6} Write each rational epression in lowest terms. 4 5 + 6.. 4 8 y y z 5 z b. yz 4z d. 4yz. ( ) 8 4 6 b. 8 + 4 d. 8 4 + Perform the indicated operations. Write answers in lowest terms. 4. 4 y 8 6 y y 9y y 8 4 8 8 b. 5 8 y 9 5 6 d. 5 6 y 5. + 0 0 + 5 5( 4) b. 5( + 4) 5 ( 4) d. 0 T- 6 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form D cont d 6. 9 0 7 6 + 6 + + + ( ) ( 5)( + 4) b. ( ) ( 5)( + ) ( + 4) ( 5)( 4) d. + 4 7. + + y y y y y 6y ( + ) b. ( y) + + 4 d. ( ) ( ) y 8. ( ) + + 5 + 6 + 6 b. 8 + d. 6 + 9. + + 4 4 + 4 ( 4 )( + 4 ) b. 5 + 4 5 4 4 d. ( + 4 )( 4 ) 0. + 6 + 6 + + b. + + + + 9 d. + + + 7. + + 4 + 8 + ( 4 )( + 6 )( ) 9 ( 4 )( + 6 )( ) b. ( 4 )( + 6 ) 9 d. ( 4 )( + 6 )( ) T- 7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form D cont d y 6y + 8. 4 y 4 + b. y 4 + y y y 6 + d. 4y + 8. ( 6 + 4 + ) ( ) 7 + b. + d. 4 4 + + 5 + + 4. Use synthetic division to divide. + + by +. 4 6 + b. + + + d. + + + + P = + 5 7 + 8. 5. Use the remainder theorem to find P( ) for ( ) 4 46 b. 06 8 d. 8 Simplify each comple fraction. 6. + 4 7 4 b. 4( 6) + 4 7 + 4 4 d. 6 7 T- 8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form D cont d 7. + + + b. + + d. + 8. 4 8 b. d. Solve each equation for. 9. + 4 = 5 b. 6 d. 0. 5 5 + = 0, b. d. 5. = + 9 0 b., 6 d.. + = + 4, b., d. T- 9 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form D cont d. Solve for a + b a = b 6 5 a = b 8 b. a = b b + a = d. 8 b a = 8 4. Suppose that Q is jointly proportional to R and S. If Q = 0 when R = 6 and S = 5, find Q when R = 9 and S =. 66 b. 99 d. 5. The sum of the reciprocals of two consecutive integers is 5. Find the integers. 56, 4 b., 6, 7 d., 4 T- 0 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form E Circle the correct answer. 4. Find all numbers for which the rational epressions 4 + is undefined. =,,, 6 b. = 0,, 6 = 0,, 4, 6 d. = 4. Solve y + 6 = for y. y y = b. y = 6 y = d. y + 6 y = Write each rational epression in lowest terms.. 4 8 yz y z 4 y z b. 4 z 4y 4y d. z y z 9 5 4 5 4. 4 6 8 b. ( + )( + 4 ) + 8 d. + 8 Perform the indicated operations. Write answers in lowest terms. 5. 4 y 9y 5y 5y y b. y 9 y 5 d. y T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form E cont d 6. + 8 4 y + + 6 + 6 8 + 8 y y y y y y b. d. y y ( ) 7. 4 + 7 6 + 48 54 ( ) + b. ( + ) ( )( ) d. ( + )( ) 8. + 9 9 6 9 b. + d. 9 9 9. + 4 + + + + 8 9 b. + 8 9 6 9 d. + 5 6 9 0. + + 5 0 + + 45 + 4 ( + 5 )( 6 ) b. ( + 5 )( 6 )( ) + d. ( + 5 )( 6 )( ). 4 + 7 + b. + 0 + + + d. 0 + T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form E cont d Simplify each comple fraction.. 4 4 8 6 b. d.. 6 + 9 + 7 9 + ( + )( ) b. ( + )( + 9 ) 6 d. + 9 4. + + 5 + + 5 + b. + 5 + 7 + 5 + d. + 5 7 Divide. 5. y y + y 4y 4 8 4 y b. y + y 8 + 4 y d. y + 6. + 4 + 60 + 5 + 9 + b. + 5 + d. 8 + + 64 + 5 + 5 + T- Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form E cont d 7. Use synthetic division to divide 4 9 + + + by. 8 + 7 + 5 + b. + 8 + 7 + 5 + d. + 8 + 7 4 + + 8 + 7 + 5 0 P = + 9 + + use the remainder theorem to find P( ). 8. If ( ) 4 8 b. 8 4 d. Solve each equation for. 9. 4 + = 6 5 4 b. 0 5 d. 0. + 5 = b. d. 4. 4 = + 9 6 b. 8, d.. 4 0 = + 9 +, 7 b., 7 7, d. 7,. Solve for + = a 6 a = + 4 b. 6 + a = + 6 + a = d. a = 6 T- 4 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form E cont d 4. A plane flies 75 miles with the wind and 95 miles against the wind in the same amount of time. If the speed of the plane in still air is 50 mph, find the speed of the wind. 45 mph b. 65 mph 55 mph d. 60 mph 5. Suppose that B is proportional to C and inversely proportional to D. If B = 5 when C = 60 and D =, find B when C = 40 and D = 4. 5 b. 0 0 d. 5 T- 5 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form F Circle the correct answer. Write each rational epression in lowest terms.. 4 0 6 + 5 + + + b. ( ) + d. ( ) +. + 4 4 y 4y y y y b. 4 7 4 4y d. y Find the domain of the rational function.. f ( ) = + + 8 9 { is a real number and 9,, 0} b. { is a real number and 9, } { is a real number and, 0, 9} d. { is a real number and, 9} 4. Solve for a c a = c + d a d c = b. c = ad c = d d. a a ad c = a + Perform the indicated operations. Write answers in lowest terms. 5. 6 8 7 5 5 ( ) b. 4 5 ( + ) 4 5 + d. 4 5 ( + ) T- 6 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form F cont d 6. 9 + 8 y y y 4 + 4 + ( ) b. d. + 7. + + 0 0 + 9 ( + ) 6 5 b. ( ) 6 4 ( ) ( ) 4 d. 4 6 ( ) 8. 4 + 6 + + 9 ( 4 )( + 6 ) + 5 + b. ( 4 )( + 6 ) + 5 ( 4 )( + 6 ) 5 + d. ( 4 )( + 6 ) 5 + + 4 8 9. + 8 b. ( + ) ( ) 5 8 5 8 d. 8 Divide. 0. a b a b ab 5ab 0 0 5 4a b ab b. 4a b ab ab d. 0a b 0a b. 4 7 + + 5 0 + + 4 + b. 5 + 5 4 d. 5 0 00 + + 0 + 50 + 5 8 + + 5 + 76 + 5 T- 7 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form F cont d. Use synthetic division to divide + 4 7 by +. 4 + + + b. + 5 + + + d. + 5. If ( ) 6 4 + + 46 + + + P = + 8 8 +, use the remainder theorem to find P. 5 b. 5 d. Simplify each comple fraction. 4. + 4 4 4 b. 0 + d. 8 + 6 5. 5 4 + 4 5 85 b. 68 75 68 85 d. 5 75 6. + + b. + d. 7 T- 8 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form F cont d 7. + 4 b. 5 4 + d. 7 8. Add: + + 4 6 + 5 + 4 b. 6 + 5 4 6 5 4 d. 4 6 5 + Solve each equation for. 9. 5 4 = 6, b. 5 6, 5 5, d. 6 6,0, 5 4 0. + = + 5 6 + 6. 6 b. 8 d. 6, 4 + = + + 4 4 7, b. 4,7 7 d. 8, 9. = + 8, 6 b. 6, 9 d. 9 T- 9 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall

Name: Instructor: Date: Section: Chapter 6 Test Form F cont d. One number is 5 times another. The sum of their reciprocals is the fraction 5. Find the two numbers. 5, 6 b. 6, 0, 0 6 d., 4 4. Two trains leave at the same time going in opposite directions. One train travels 5 mph faster than the other. Five hours later, the trains are 675 miles apart. Find the speed of each. 50 mph, 75 mph b. 45 mph, 70 mph 55 mph, 80 mph d. 65 mph, 90 mph 5. Suppose B is proportional to the square of C and inversely proportional to D. If B = 7 when C = 6 and D = 6, find B when C = and D = 6. 8 b. 4 d. 6 T- 40 Copyright 009 Pearson Education, In, publishing as Pearson Prentice Hall