1. complex; complex 2. 18; 12; 30; 9; 4; 5; ; 4. 2x; 3; 5x; Copyright 2013 Pearson Education, Inc.

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1 Chapter 7: Rational Epressions 7.5 Chec Points Add to get a single rational epression in the numerator. 8 4 Subtract to get a single rational epression in the denominator Perform the division indicated b the fraction bar. Invert and multipl Subtract to get a single rational epression in the numerator. Add to get a single rational epression in the denominator. Perform the division indicated b the fraction bar. Invert and multipl. Subtract to get a single rational epression in the numerator. Perform the division indicated b the fraction bar Invert and multipl. 4 4 Multipl the numerator and the denominator b the LCD of Multipl the numerator and the denominator b the LCD of. Multipl the numerator and the denominator b the LCD of. 7.5 Concept and Vocabular Chec. comple; comple. 8; ; 0; 9; 4; 5;. 0; ; 4; 4. ; ; 5; 5 Copright 0 Pearson Education, Inc.

2 Introductor Algebra for College Students E Section Eercise Set Add to get a single rational epression in the numerator Add to get a single rational epression in the denominator. Perform the division indicated b the fraction bar. Invert and multipl LCD = LCD = LCD = LCD = Copright 0 Pearson Education, Inc. 5

3 Chapter 7: Rational Epressions LCD = LCD = LCD = LCD = LCD = 54 Copright 0 Pearson Education, Inc.

4 Introductor Algebra for College Students E Section LCD = LCD = LCD = LCD Copright 0 Pearson Education, Inc. 55

5 Chapter 7: Rational Epressions LCD R R R R R R RR RR RR RR R R Let R 0 and R 0, RR ohms R R a , , , , , , , ,984 5 Copright 0 Pearson Education, Inc.

6 Introductor Algebra for College Students E Section 7.5 b. According to the data in the bar graph, or about 4% was spent on human resources in 005. c ,99 75(5) 585(5) 5, ,984 0(5) 49(5) 5,984 According to the model about % was spent on human resources in 005. This overestimates the actual percent b %. 54. Answers will var. 5. does not mae sense; Eplanations will var. Sample eplanation: If ou multipl the numerator b a particular value, ou must also multipl the denominator b the same value. 58. does not mae sense; Eplanations will var. Sample eplanation: You forgot to also multipl the constants, and, b the LCD. 0. false; Changes to mae the statement true will var. A sample change is: It is not comple because it does not contain an secondar fractions.. false; Changes to mae the statement true will var. A sample change is: The numerator simplifies to epression simplifies to. 4. Simplif the given comple fraction. LCD = Because the rational epression can be simplified to, this is what it does to each number ; it cubes.. The Copright 0 Pearson Education, Inc. 57

7 Chapter 7: Rational Epressions. 5 Simplif the first comple rational epression using the LCD method. 5 5 Simplif the second comple algebraic epression b the LCD method. Now subtract The graphs do not coincide. 70. Change the epression on the right to The solution set is Copright 0 Pearson Education, Inc.

8 Introductor and Intermediate Algebra for College Students 4E Section The solution set is The solution set is 5 50 ()( ) 0 0 or 0 The solution set is,. 7. Chec Points.. 8 There are no restrictions on the variable because the variable does not appear in an denominator. The LCD is The solution set is 4. Chec: The restriction is 0. The LCD is The solution set is Copright 0 Pearson Education, Inc. 59

9 Chapter 7: Rational Epressions The restriction is 0. The LCD is ( )( ) 0 0 or 0 The solution set is, ( 5)( 5) 5 5 The restrictions are 5 and 5. The LCD is ( 5)( 5). 4 ( 5)( 5) ( 5)( 5) ( 5)( 5) ( 5)( 5) 5 5 ( 5)( 5) 4( 5)( 5) ( 5)( 5) ( 5)( 5) 5 5 4( 5) ( 5) The solution set is 4. 0 Copright 0 Pearson Education, Inc.

10 Introductor and Intermediate Algebra for College Students 4E Section The restriction is. The LCD is ( ) The proposed solution,, is not a solution because of the restriction. The solution set is The restriction is (00 )(750) (00 ) 00 75, , If government funding is increased to $750 million, then 75% of pollutants can be removed. b a b ( ) a ( ) a a b a b a a b a a a b a a 8. z z z z z z z z z z z ( ) z z z z z 7. Concept and Vocabular Chec. LCD ( ) 5.. 5( ) ( ) ; 9. true 0. false Copright 0 Pearson Education, Inc.

11 Chapter 7: Rational Epressions 7. Eercise Set. 5 There are no restrictions on the variable because the variable does not appear in an denominator. The LCD is The solution set is There are no restrictions. The LCD is The solution set is The restriction is 0. The LCD is The solution set is The restriction is 0. The LCD is The solution set is. 7 5 The restriction is 0 The LCD is The solution set is The restriction is 0. The LCD is The solution set is. Copright 0 Pearson Education, Inc.

12 Introductor and Intermediate Algebra for College Students 4E Section Restrictions:, LCD = The solution set is Restriction: 0 LCD = The solution set is Restriction: 0 LCD = or The solution set is 4, 4. 9 Restriction: 0 LCD = or The solution set is 4, Restriction: 0 LCD = or 0 7 The solution set is 7, Restrictions: 0 LCD = Copright 0 Pearson Education, Inc.

13 Chapter 7: Rational Epressions or The solution set is,. Restrictions:, LCD = or 0 The solution set is,. ( )( ) Restrictions:, LCD = ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) The solution set is Restrictions: 4 LCD = The solution set is. Restriction: LCD = The proposed solution,, is not a solution because of the restriction. Notice that maes two of the denominators zero in the original equation. Therefore, the given equation has no solution. The solution set is. 4 Copright 0 Pearson Education, Inc.

14 Introductor and Intermediate Algebra for College Students 4E Section Restriction: 0, LCD = or The solution set is, Restrictions: LCD = The proposed solution,, is not a solution because of the restriction. Therefore, the equation Restriction:, 0 LCD = or 0 The proposed solution,, is not a solution because of the restriction. The solution set is.. 5 To find an restrictions and the LCD, factor the denominators. Restrictions: LCD = 4 4 The solution set is. has no solution. The solution set is. Copright 0 Pearson Education, Inc. 5

15 Chapter 7: Rational Epressions Restrictions: 4, LCD = The solution set is Factor the denominators. 5 0 Restrictions:, LCD = The solution set is Factor the last denominator. 5 Restrictions:, LCD = The solution set is 4. V P V P V P PV PV PV V P PV V P PV p q f pqf pqf p q f qf pf pq pf pq qf pf q p f pf q p f a S r S Sr a Sr a S as S a r or S S Gm m F d Fd Gm m Fd m Gm z s sz s z Copright 0 Pearson Education, Inc.

16 Introductor and Intermediate Algebra for College Students 4E Section E I R r IR Ir E Ir E IR E IR r I ff f f f ff ff f f ff f f ff ff f f f f f ff f. Solve Factor the first denominator Restrictions: 4, LCD = The solution set is. 4. Simplif Factor the first denominator Copright 0 Pearson Education, Inc. 7

17 Chapter 7: Rational Epressions LCD = Solve 4 4 Restrictions: 0 LCD = or 0 The solution set is, Solve 4 5 Restrictions:, LCD = The solution set is Copright 0 Pearson Education, Inc.

18 Introductor and Intermediate Algebra for College Students 4E Section ,000 C ; C , LCD = , , ,000 00,000 At an average cost of $405 per wheelchair, 00,000 wheelchairs can be produced. C ; C LCD = For $8 million, 80% of the contaminants can be removed. DA 74. C ; C 500, D 000 A 000A 500 A LCD = A A A A A 500A A A A The child s age is ears old. 7. 0,000 C ; C 790 0, LCD = 0, , , or For earl inventor costs to be $790, the owner should order either approimatel or 50 cases at a time. These solutions correspond to the points,790 50,790 on the graph. and Answers will var. 84. maes sense 8. does not mae sense; Eplanations will var. Sample eplanation: If all potential solutions mae an of the denominators of the rational equation equal to zero, then the equation will have no solution. 88. true 90. false; Changes to mae the statement true will var. A sample change is: You could begin b multipling both sides b the LCD, Restrictions: 7, 7 0 Copright 0 Pearson Education, Inc. 9

19 Chapter 7: Rational Epressions Multipl both sides b The proposed solution, 7, is not a solution of the original equation because it is on the list of restrictions. Therefore, the given equation has no solution. The solution set is Factor b grouping The solution set is 8. Chec 8 : true (8 )(8 ) (8 )(8 ) 8 8 5(8 ) 9(8 ) The solution set is 0. In hour ou can complete 5 In hours ou can complete 5 of the job. of the job. In hours ou can complete 5 of the job The solution set is,. Chec : Chec : , true 5 5, true 70 Copright 0 Pearson Education, Inc.

20 Introductor and Intermediate Algebra for College Students 4E Section Chec Points. Let = the rate of the current. Then the canoe s rate with the current. and the canoe s rate against the current. Distance Rate Distance Time Rate With the current 0 0 Against the current 0 Use the cross-products principle to solve this equation The rate of the current is miles per hour.. Let = the number of hours for both people to paint a house together. First person Second person Fractional part Time Fractional part of job completed woring of job completed in hour together in hours Woring together, the two people can complete the whole job, so. 8 4 Multipl both sides b the LCD, It will tae hours (or hours 40 minutes) if the wor together. Copright 0 Pearson Education, Inc. 7

21 Chapter 7: Rational Epressions. Let the propert ta on the $40,000 house. Ta on $50, 000 house Ta on $40, 000 house Assessed value ($50, 000) Assessed value ($40, 000) $500 $ $50,000 $40, ,000 40,000 50,000 (500)(40,000) 50,000,470,000,000 50,000,470,000,000 50,000 50, The propert ta is $ Let the total number of deer in the refuge (0)(50) 5 8, , There are about 70 deer in the refuge The missing length is inches. h 5.5.5h 5.5h h.5 h The height of the tower is ards. 7.7 Concept and Vocabular Chec. distance traveled; rate of travel Copright 0 Pearson Education, Inc.

22 Introductor and Intermediate Algebra for College Students 4E Section ad bc 5. similar 7.7 Eercise Set. The times are equal, so To solve this equation, multipl both sides b the LCD, If, 0. The biccle s rate is mph and the motorccle s rate is mph Note: The equation is a proportion, so it can also be solved b using the cross-products principle. This 0 allows ou to sip the first step of the solution process shown above. 4. Let = the car s rate. Then 0 = the truc s rate. Distance Rate Distance Time Rate Truc Car The times are equal, so. 0 Use the cross-products principle to solve this equation If = 0, 0 = 40. The truc s rate is 40 miles per hour and car s rate is 0 miles per hour. Copright 0 Pearson Education, Inc. 7

23 Chapter 7: Rational Epressions. Let = the rate of the current. Then 8 = the boat s rate with the current and 8 = the boat s rate against the current. Distance Rate Distance Time Rate With the current 8 8 Against the current 8 8 The times are equal so,. 8 8 Use the cross-products principle to solve this equation The rate of the current is 4 mph. 8. Let = running. Then 9 = driving rate. Distance Rate Distance Time Rate Running 5 5 Driving The total time is hours, so or 5 0. To solve this equation, multipl both sides b the LCD, If = 5, 9 = 45. The driving rate is 45 mph and the running rate is 5 mph. 74 Copright 0 Pearson Education, Inc.

24 Introductor and Intermediate Algebra for College Students 4E Section Let = the canoe s average rate in still water. Then = the canoe s rate with the current and = the canoe s rate against the current. Distance Rate Distance Time Rate Downstream Upstream The boat s average rate in still water is 4 miles per hour.. Let = the time in minutes, for both people to wash the car together. Fractional part Time Fractional part of job completed woring of job completed in minute together in minutes You Your sister 0 0 Woring together, the two people complete the whole job, so Multipl both sides b the LCD, It will tae about 7. minutes, which is not enough time. 4. Let = the time, in hours, for all three crews to dispense food and water woring together. First team Second team Third team Fractional part Time Fractional part of job completed woring of job completed in hour together in hours Copright 0 Pearson Education, Inc. 75

25 Chapter 7: Rational Epressions Woring together, the two teams complete one whole job, so Multipl both sides b the LCD, It will tae about 4. hours for all three crews to dispense food and water woring together. To epress this time in minutes and seconds, divide 0 b on a calculator, subtract 4, multipl the decimal b 0 and round to the nearest whole number, 7. It will tae 4 hours 7 minutes to complete the job.. Let = the time, in hours, for both pipes to fill in the pool. First pipe Second pipe Fractional part Time Fractional part of job completed woring of job completed in hour together in hours Using both pipes, it will tae hours to fill the pool. 8. Let = the maintenance bill on a store that is 4800 square feet. Bill on shopping center Bill on store Size of shopping center Size of store $45, 000 $ 80, 000 ft 4800 ft 45, , The maintenance bill for the store is $00. 7 Copright 0 Pearson Education, Inc.

26 Introductor and Intermediate Algebra for College Students 4E Section Let = the total number of bass in the lae. Original number Number of tagged of tagged bass bass in sample Total number Number of bass of bass in sample There are about 00 bass in the lae.. Let the number of people, in billions, who get drun ever da (in 00) (9)(.9) billion people get drun ever da. 4. Let the length of Sain s moustache. 8 inches inches ears 7 ears Sain s moustache was 8 inches long The length of the side mared is 8 inches. Copright 0 Pearson Education, Inc. 77

27 Chapter 7: Rational Epressions 8. Notice that the length of the base of the larger triangle is 5 inches The length of the side mared is.5 inches The length of the side mared is feet.. Let h = the height of the tree. h 8 5 h 58 h h 7.7 The tree s height is about 7.7 feet Answers will var. 4. maes sense 44. maes sense 4. Let = the length of the trail. Distance Distance Rate Time Rate Faster sier 9 9 Slower sier It taes the slower sier 4 hour longer to complete the trail than the faster sier, so. 9 4 To solve this equation, multipl both sides b the LCD, The length of the trail is 4.5 miles. 78 Copright 0 Pearson Education, Inc.

28 Introductor and Intermediate Algebra for College Students 4E Section Let = number of hours to fill the pool with all three pipes woring. First pipe Second pipe Third pipe Fractional part Time Fractional part of job completed woring of job completed in hour together in hours It will tae hours or approimatel 55 minutes to fill the pool. 50. Let = the time, in hours, for the eperienced carpenter to panel the room. Then = the time, in hours, for the apprentice to panel the room. Fractional part Time Fractional part of job completed woring of job completed in hour together in hours Eperienced Carpenter Apprentice Together, the two carpenters complete one whole job, so or. To solve this equation, multipl both sides b the LCD,. 8 If = 8, = 4. Woring alone, it would tae the eperienced carpenter 8 hours and the apprentice 4 hours to panel the room. 5. Let lower interest rate and higher interest rate. Principal Interest rate Interest earned Investment at higher 00 interest rate 00 Investment at lower 75 interest rate 75 Copright 0 Pearson Education, Inc. 79

29 Chapter 7: Rational Epressions Since the principal is the same at each rate, solve the following equation Therefore, the lower interest rate is 7% and the higher interest rate is 8% The solution set is {}. 4 slope = -intercept = 4 Plot (0,4). From this point, move units down and units to the right to reach the point (,). Draw a line through (0,4) and (,). c a. Substitute to find b. c A S P 0, 000, , 000, , Chec Points 5. a. Substitute to find b.. Since W varies directl with t, we have W t. Use the given values to find. W t The equation becomes W t. Find W when t. W t W W An minute shower will use gallons of water. 80 Copright 0 Pearson Education, Inc.

30 Introductor and Intermediate Algebra for College Students 4E Section 7.8. Beginning with, we will use s for the stopping distance and v for the speed of the car. Use the given values to find. s v The equation becomes Find s when v 00. s v s v 8 v s 8 v s 8 00 s 8 00 s 8 s 55 About 55 feet are required to stop a car traveling 00 miles per hour.. Beginning with, we will use l for the length of the string and f for the frequenc. Use the given values to find. f l The equation becomes f l 50 f l Find f when l f l 50 f 0 f 5 A string length of 0 inches will vibrate at 5 ccles per second. 4. Let m the number of minutes needed to solve an eercise set. Let p the number of people woring on the problems. Let the number of problems in the eercise set. Use m to find. p m p Thus, m 8. p Find m when p 8 and m 8 m 4 It will tae 4 minutes for 8 people to solve 4 problems. 5. Find : V hr hr Thus, V hr. hr V V 9 The volume of a cone having a radius of feet and a height of feet is 9 cubic feet. Copright 0 Pearson Education, Inc. 8

31 Chapter 7: Rational Epressions 7.8 Concept and Vocabular Chec. ; constant of variation.. 4. n z 5. z. directl; inversel 7. jointl; inversel 7.8 Eercise Set b a c a C AT C mn p ()() 5 5 5() () mn 45()(4) 0 p 0 0 z ; Solving for : z z z z z z z z z z 8 Copright 0 Pearson Education, Inc.

32 Introductor and Intermediate Algebra for College Students 4E Section 7.8. z w w w z z z w w z 8. z( w) zzw zw z zw z z z zw z 0.. z w z w w w w z z w z w z w M E M 8 A person who weighs 8 pounds on Earth will weigh pounds on the moon d f 9 () 9 () 0.75 d 0.75 f f f f A force of 0 pounds is needed. d s 7.5 (45) 7.5 (05) 7.5 (05) d s (0) (00) The stopping distance is 0 feet. t d (000)4.4 (000) t 0.88 d 5000 The water temperature is 0.88 Celsius at a depth of 5000 meters. Copright 0 Pearson Education, Inc. 8

33 Chapter 7: Rational Epressions 0. a. Use L to find. R L R Thus, L. R b. This is an approimate model. c. 890 L R 890 L 4 The average life span of a mouse is about ears.. a. A mammal with a life span of 50 ears will have a heart rate of about 40 beats per minute. 890 b. L R R 50R R 50 R 8 The model estimates a mammal with a life span of 50 ears will have a heart rate of about 8 beats per minute. c. The data for lions is represented on the graph b the point (7, 5). 4.. i d (00).75 (00) i.4 d The illumination is.4 footcandles at a distance of 50 feet. m i c (5) 5 0 (5) 0(5) (0) m i c 00(40) 80 c c c c c 80c c c 50 The chronological age is Copright 0 Pearson Education, Inc.

34 Introductor and Intermediate Algebra for College Students 4E Section e mv (8)() (8)(9) e 0.5mv 0.5(4)() 0.5(4)() 7 A mass of 4 grams and velocit of centimeters per second has a inetic energ of 7 ergs. t a t a a A ear will seem to be f as 50 (45)(0) 50 (0)(900) 50 8, , 000 8, ( 4)(0) f as 0 0 ()(00) 0 0 of a ear. Yes, the wind will eert a force of 0 pounds on the window Answers will var. 50. z varies jointl as the square of and the square root of. 5. Answers will var. 54. does not mae sense; Eplanations will var. Sample eplanation: With a positive constant of variation, both variables will increase. 58. Since heat varies directl as the square of the voltage and inversel as the resistance, we have v H. To triple the amount of heat generated r while voltage remains constant, we must change the resistance. v H r v H r v H r v H r v H r For the heat to be tripled, the resistance must be multiplied b (or divided b ). The solution set is maes sense Copright 0 Pearson Education, Inc. 85

35 Chapter 7: Rational Epressions Set has each -coordinate paired with onl one - coordinate. 4. r r 5 ( 5) ( 5) ( ah) 7 5a5h7 Chapter 7 Review Eercises Set the denominator equal to 0 and solve for The rational epression is undefined for = 4. 5 Set the denominator equal to 0 and solve for or The rational epression is undefined for and The smallest possible value of is 0, so 8 8 for all real numbers. This means that there is no real number for for which 8 0. Thus, the rational epression is defined for all real numbers or 0 The rational epression is undefined for and... 4 The numerator and denominator have no common factor, so this rational epression cannot be simplified. 8 9 or 8 Copright 0 Pearson Education, Inc.

36 Introductor and Intermediate Algebra for College Students 4E Chapter 7 Review or or Copright 0 Pearson Education, Inc. 87

37 Chapter 7: Rational Epressions and 9 9 LCD = and LCD = 7 and LCD = 7 5 LCD = LCD = Copright 0 Pearson Education, Inc.

38 Introductor and Intermediate Algebra for College Students 4E Chapter 7 Review LCD = or or 4 4 LCD = or LCD = LCD = Copright 0 Pearson Education, Inc. 89

39 Chapter 7: Rational Epressions 9. 5 LCD = LCD = LCD = LCD = LCD = 90 Copright 0 Pearson Education, Inc.

40 Introductor and Intermediate Algebra for College Students 4E Chapter 7 Review LCD = or LCD = The restriction is 0. The LCD is. 8 The solution set is {}. 4 4 The restriction is 0. The LCD is The solution set is { }. 5 The restriction is 0. The LCD is or 0 The solution set is,. Copright 0 Pearson Education, Inc. 9

41 Chapter 7: Rational Epressions The restriction is 5. The LCD is The onl proposed solution, 5, is not a solution because of the restriction 5. Notice that 5 maes two of the denominators zero in the original equation. The solution set is To find an restrictions and the LCD, all denominators should be written in factored form. 4 8 Restrictions:, LCD = The solution set is {5}. Restriction: 0 LCD = or 0 The solution set is,. 9 Copright 0 Pearson Education, Inc.

42 Introductor and Intermediate Algebra for College Students 4E Chapter 7 Review Restriction: LCD = 5 5 The solution set is 5. Restrictions:, LCD = 4 5 The solution set is t 5 P t 5 50t 5 5 t 5 50t 5 5t5 t5 t 5 5t5 50t5 5t5 750t50 5 5t t t It will tae ears for the population to reach 5 el. Copright 0 Pearson Education, Inc. 9

43 Chapter 7: Rational Epressions C S r r 40 00r r r 00 00r 40 00r 0 0 r 0% 00 0 The marup is 0%. R C P n R C np n n np R C np C R C RnP PV P V T T PV PV TT T T TT PT V P TV PT V PTV PV PV PT V T PV 0. A P T Pr A P Pr T Pr Pr PrT A P PrT P A P rt A P rt A rt rt A P rt 94 Copright 0 Pearson Education, Inc.

44 Introductor and Intermediate Algebra for College Students 4E Chapter 7 Review.. R R R RRR RRR R R R RR RRR RRR R R RR RR RR RR R R R RR R R R R R R R RR R R R ne I R nr ne RnrIRnr R nr IR Inr ne IR ne Inr IR n E Ir IR n E Ir EIr EIr IR n E Ir. Let = the rate of the current. Then 0 the rate of the boat with the current and 0 the rate of the boat against the current. Distance Rate Distance Time Rate Downstream Upstream The times are equal, so The rate of the current is 4 miles per hour Copright 0 Pearson Education, Inc. 95

45 Chapter 7: Rational Epressions 4. Let = the rate of the slower car. Then 0 the rate of the faster car. Distance Rate Distance Time Rate Slow car 0 0 Faster Car If = 0, 0 0. The rate of the slower car is 0 miles per hour and the rate of the faster car is 0 miles per hour. 5. Let = the time, in hours, for both people to paint the fence together. Fractional part Time Fractional part of job completed woring of job completed in hour together in hours Painter Apprentice Woring together, the two people complete one whole job, so. Multipl both sides b the LCD,. 4 It would tae them 4 hours to paint the fence woring together.. Let = number of teachers needed for 5400 students , 00 50, For an enrollment of 5400 students, 4 teachers are needed. 9 Copright 0 Pearson Education, Inc.

46 Introductor and Intermediate Algebra for College Students 4E Chapter 7 Review 7. Let = number of trout in the lae. Original Number Tagged Deer Number Tagged Deer in Sample Total Number of Deer Total Number Deer in Sample There are 87 trout in the lae The length of the side mared with an is 5 feet. 9. Write a proportion relating the corresponding sides of the large and small triangle. Notice that the length of the base of the larger triangle is 9 ft ft 5 ft The height of the lamppost is.5 feet. 70. Since the profit varies directl as the number of products sold, we have p n. Use the given values to find. p n The equation becomes p 47n When n 05 products, p 47(05) 495. If 05 products are sold, the compan s profit is $495. Copright 0 Pearson Education, Inc. 97

47 Chapter 7: Rational Epressions 7. Since distance varies directl as the square of the time, we have d t. Use the given values to find. d t The equation becomes d t. When t = 0, d A sdiver will fall 00 feet in 0 seconds. 7. Since the pitch of a musical tone varies inversel as its wavelength, we have p. w Use the given values to find. p w The equation becomes p. w 05 When w.4, p The tone s pitch is 440 vibrations per second. 7. Since loudness varies inversel as the square of the distance, we have l d. Use the given values to find. l d The equation becomes l. d When d = 4, l At a distance of 4 feet, the loudness of the stereo is decibels. 74. Since time varies directl as the number of computers and inversel as the number of worers, n we have t. w Use the given values to find. n t w n The equation becomes t. w When n = 40 and w = 5, t. 5 5 It will tae hours for 5 worers to assemble 40 computers. 75. Since the volume varies jointl as height and the area of the base, we have v ha. Use the given values to find The equation becomes v ha. When h = 0 feet and a = 0 square feet, v If the height is 0 feet and the area is 0 square feet, the volume will be 800 cubic feet. 98 Copright 0 Pearson Education, Inc.

48 Introductor and Intermediate Algebra for College Students 4E Chapter 7 Test Chapter 7 Test Set the denominator equal to 0 and solve for or The rational epression is undefined for 9 and Copright 0 Pearson Education, Inc. 99

49 Chapter 7: Rational Epressions LCD = LCD = LCD = LCD = LCD = Copright 0 Pearson Education, Inc.

50 Introductor and Intermediate Algebra for College Students 4E Chapter 7 Test LCD = LCD = 5 Restriction: 0 LCD = The solution set is {} Restriction: 5 LCD = The solution set is { 8}. Copright 0 Pearson Education, Inc. 40

51 Chapter 7: Rational Epressions.. Restrictions:, LCD = or 0 The solution set is 0,. R as a s as asr as a s ar Rs as ar as Rs ar as Rs a Rs Rs a R s Rs Rs Rs Rs Rs a or R s s R. Let = the rate of the current. Then 0 = the rate of the boat with the current and 0 = the rate of the boat against the current. Distance Rate Distance Time Rate Downstream 0 0 Upstream The rate of the current is miles per hour. 4. Let = the time (in minutes) for both pipes to fill the hot tub. 0 0 LCD = It will tae minutes for both pipes to fill the hot tub. 5. Let = number of tule el in the par , There are 000 tule el in the par The length of the side mared with an is. inches. 7. Let C = the current (in amperes). Then R = the resistance (in ohms). Step C R Step To find, substitute 4 for C and 5 for R Step C R 40 Copright 0 Pearson Education, Inc.

52 Introductor and Intermediate Algebra for College Students 4E Cumulative Review Step 4 Substitute 4 for R and solve for C. 0 C When the resistance is 4 ohms, the current is 5.5 amperes. Cumulative Review Eercises (Chapters -7) The solution set is , or 0 The solution set is,. 4 4 Restrictions:, LCD = 4 The solution set is. Copright 0 Pearson Education, Inc. 40

53 Chapter 7: Rational Epressions 5. 9 To solve this sstem b the substitution method, substitute for in the second equation Bac-substitute for into the first equation. The solution set is, To solve this sstem b the addition method, multipl the first equation b 4 and the second equation b. Then add the equations Bac-substitute for in the first equation of the original sstem. 4 The solution set is,. 7. -intercept: -intercept: checpoint: (4,) Draw a line through (,0), (0, ) and (4,). 8. Graph b using its slope, =, and its -intercept,. 9. The graph is a horizontal line with -intercept LCD = Copright 0 Pearson Education, Inc.

54 Introductor and Intermediate Algebra for College Students 4E Mid-Tetboo Chec Point LCD = Let = the amount invested at 5%. Then 4000 = the amount invested at 9% If = 5, then $5 was invested at 5% and $775 at 9%. 0. Let = the length of the shorter piece. Then = the length of the larger piece If 7, then 5. The lengths of the pieces are 7 inches and 5 inches. Mid-Tetboo Chec Point The solution set it {4}.. 5 Multipl both sides b the least common denominator of the fractions, 0: The solution set it {0} The solution set is,7. Copright 0 Pearson Education, Inc. 405

55 Chapter 7: Rational Epressions 4. 5 Multipl the second equation b and add the result to the first equation Bac-substitute 4 for into either of the original equations to find. We choose to use the second equation: The solution set is, Substitute the epression 0 for into the first equation and solve for Bac-substitute for into either of the original equations to find. We choose to use the second equation: The solution set is, Multipl both sides of the equation b the LCD, This proposed solution checs, so the solution set is 8. 5 Multipl both sides of the equation b the LCD, or 0 Both proposed solutions chec, so the solution set is, Copright 0 Pearson Education, Inc.

56 Introductor and Intermediate Algebra for College Students 4E Mid-Tetboo Chec Point The LCD is Multipl the numerator and denominator b the LCD, ( ) Copright 0 Pearson Education, Inc. 407

57 Chapter 7: Rational Epressions.. m The slope is, and the -intercept is. Plot the point 0,. We move up unit and to the right units to the point,0. Find additional points as needed and connect with a line. 5 (, 0) -5 (0,-) m The point-slope equation is m or The slope-intercept equation is Let 0 : 0 The -intercept is, so the line passes through the point 0,. Let 0 : 0 The -intercept is, so the line passes through the point,0. Find additional points to chec and connect them with a straight line. (-,0) 5-5 (0,-) 5 5. The graph is a horizontal line that passes through the point 0, Let = the number The number is Let = the price before the reduction The price of the digital camera before the reduction is $0. 0. Let = the width of the field. = the length of the field The width of the field is 50 ards, and the length of the field is 50 ards.. Let = the amount invested at 7%. = the amount invested at 9%. Principal Rate Interest 7% % , (0,-) 0, Multipl the first equation b 0.07 and add the result to the second equation. 408 Copright 0 Pearson Education, Inc.

58 Introductor and Intermediate Algebra for College Students 4E Mid-Tetboo Chec Point Bac-substitute 7500 for into one of the original equations. We use the first equation , 000, 500 $,500 was invested at 7% and $7500 was invested at 9%.. Let = the number of liters of 40% acid solution. = the number of liters of 70% acid solution. No. of Percent Amount liters of acid of acid 40% Sol 40% % Sol 70% % Mi 50%= Multipl the first equation b 0.4 and add the result to the second equation Bac-substitute 4 for into one of the original equations. We use the first equation. 4 8 The chemist should mi 8 liters of 40% acid solution with 4 liters of 70% acid solution.. Let = the height The triangular sail is feet high. 4. Let = the measure of the nd angle. 0 = the measure of the st angle. 4 0= the measure of the rd angle The st angle measures 5 ; the nd angle measures 5, and the rd angle measures Let = the price of a TV. = the price of a stereo Multipl the first equation b 4, multipl the second equation b, and add the results. 0, Bac-substitute 70 for into one of the original equations. We use the first equation The price of the TV is $50, and the price of the stereo is $70.. Let = the width of the rectangle. = the length of the rectangle of Disregard because the width of a rectangle cannot be negative. So 5 and. The width of the rectangle is 5 meters, and the length is meters. Copright 0 Pearson Education, Inc. 409