Chapter 9 Rational Epressions and Equations Lesson 9- Multipling and Dividing Rational Epressions Pages 76 78. Sample answer: 3. Never; solving the equation using cross products leads to 5 0, which is never true. 5. a b ( ), 6 6( ). To multipl rational numbers or rational epressions, ou multipl the numerators and multipl the denominators. To divide rational numbers or rational epressions, ou multipl b the reciprocal of the divisor. In either case, ou can reduce our answer b dividing the numerator and the denominator of the results b an common factors.. 6. 9m n 3 7. 3c 0b 8. 5 9. 6 5 0. p 5 p. cd. ( ) 3( ) 3. D 5. n 7m. 6. 5c b 3 7. s 3 8. 5 t 9. 0. 3. a a. 3 Glencoe/McGraw-Hill 3 Algebra Chapter 9
3. bc 7a 5. p. f 6. z 8 7. b 3 8. 3 9. 3 30. 3 3. 3. 5( 3) ( ) 33. w 3 w 3. 3(r ) r 3 35. (a 5) (a )(a ) 36. 3n m 37. p 39. 38. m n m n 0.. 3. d, or 3. a b or b. 687 3,9 5. 687 m 3,9 a 6. units 7. ( 5)m 9. A rational epression can be used to epress the fraction of a nut miture that is peanuts. Answers should include the following. The rational epression 8 3 is in simplest form because the numerator and the denominator have no common factors. 8. 50. C a Glencoe/McGraw-Hill Algebra Chapter 9
5. A Sample answer: 8 3 could be used to represent the fraction that is peanuts if pounds of peanuts and pounds of cashews were added to the original miture. 5. (, ), (5, ) 53. ( 7, ) 3 5. ( 3) ; parabola 3 ( 3) 55. ( 7) ( ) 9 hperbola ; 56. even; 8 8 ( 7) 9 ( ) 57. odd; 3 58. even; 0 59., 60. 6, 3 6. 0, 5 63. 65. 9 6..99 0 s or about 8 min 9 s 6. 3, 9 6 66. Glencoe/McGraw-Hill 5 Algebra Chapter 9
67. 5 68. 6 69. 8 70. 6 Lesson 9- Adding and Subtracting Rational Epressions Pages 8 8. Catalina; ou need a common denominator, not a common numerator, to subtract two rational epressions. 3a. Alwas; since a, b, and c are factors of abc, abc is alwas a common denominator of 3b.Sometimes; if a, b, and c have no common factors, then abc is the LCD of 3c. Sometimes; if a and b have no common factors and c is a factor of ab, then ab is the LCD of 3d. Sometimes; if a and c are factors of b, then b is the LCD of 3e. Alwas; since 5. 80ab 3 c 7. a b c. a b c. a b c bc ac ab abc abc abc, bc ac ab alwas. abc 3 a b c. a b c. the sum is. Sample answer: d d, d. 6. ( )( ) 8. a 5b 90ab Glencoe/McGraw-Hill 6 Algebra Chapter 9
9. 37 m 0. 5d 6 (d ). 3a 0 (a 5)(a ). 8 5 3 9 ( )( ) 3. units 5. 80 z 7. 36p 3 q 9. ( )( ). (n )(n 3)(n ). 70s t 6. 0a 3 b 3 c 3 8. (w 3) 0. (t 3)(t )(t ). 6 8b ab 3. 3 v. 5 7r r 5. 5 3 6. 9 3 7. 5b 7a 3 5a b 8. 3 0q 9. 0w 3 90w 30. 3 8 3. a 3 a 3. 5m 3(m )(m ) 33. ( 9) ( 3)( 3) 3. 7 38 ( 7)( ) 35. d 0 (d )(d )(d ) 36. h 5 (h )(h 5) 37. 6 ( ) ( 3) 38. 0 39. ( )( ) 0. b.. s s 3. a 7 a. 3 ( ) 5. ohms 6. h Glencoe/McGraw-Hill 7 Algebra Chapter 9
7. h 8. 8( ) ( ) h 9. 5. Subtraction of rational epressions can be used to determine the distance between the lens and the film if the focal length of the lens and the distance between the lens and the object are known. Answers should include the following. To subtract rational epressions, first find a common denominator. Then, write each fraction as an equivalent fraction with the common denominator. Subtract the numerators and place the difference over the common denominator. If possible, reduce the answer. 53. C md (d L) (d L) or md (d L ) q 0 60 could be used to determine the distance between the lens and the film if the focal length of the lens is 0 cm and the distance between the lens and the object is 60 cm. 50. Sample answer: 5. B 5., 5z Glencoe/McGraw-Hill 8 Algebra Chapter 9
55. a(a ) a 56. 8 6 8 9 8 57. ( 3) 58..5 ft 59. 6 6 6 0 8 60. 0 5 0 5 5 5 0 5 0 5 9 5 6. 6 0 8 6 6 ( ) ( 5) 6 5 Glencoe/McGraw-Hill 9 Algebra Chapter 9
Chapter 9 Practice Quiz Page 8. t t 3. c 6b 3. 5. (w )(3w ) 7. 3 a a b. 6. 8. 7 6a 0b 5a b 3 9. n 9 (n 6)(n ) 0. Lesson 9-3 Graphing Rational Functions Pages 88 90. Sample answer: f() ( 5)( ) 3. and 0 are asmptotes of the graph. The -intercept is 0.5 and there is no -intercept because 0 is an asmptote. 5. asmptote: 5; hole:. Each of the graphs is a straight line passing through ( 5, 0) and (0, 5). However, the graph of f() has a hole at (, 6), and the graph of g() 5 does not have a hole.. asmptote: 6. f() ( )( 5) f() Glencoe/McGraw-Hill 50 Algebra Chapter 9
7. f() f() 8 6 ( )( 3) 8. 0 6 f() f() 5 5 6 0 9. f() 0. f() f() ( ) 8 f() 5. f(). 00 mg f() 6 3. C., C ; 0; 0 0 6 C 6 8 6 5. 0 and 0 C 7. asmptotes:, 9. asmptotes:, hole: 5 6. asmptotes:, hole: 3 8. asmptotes:, hole: 3 0. hole: Glencoe/McGraw-Hill 5 Algebra Chapter 9
. hole:. f() f() 3. f(). f() f() f() 3 5. f() 6. f() 6 f() 5 8 f() 3 7. 8 f() f() 5 8. f() f() 3 ( ) 8 9. f() 30. f() f() ( 3) f() 6 8 Glencoe/McGraw-Hill 5 Algebra Chapter 9
3. f() 3. f() f() 3 f() 36 6 33. f() 3. f() f() f() 3 ( )( 5) 35. f() 36. f() f() f() ( )( 3) 37. f() 38. f() f() f() 6 ( 6) Glencoe/McGraw-Hill 53 Algebra Chapter 9
39. f() f() 0. ( ) f() f() 6 6. The graph is bell-shaped with a horizontal asmptote at f() 0. 3. 0 V f m 7 V f 5 m 7. Since the graph of would be a reflection of the graph of f() 6 over the -ais.. m 7; 7; 5 6 6 a 6 6 b, f() 6 6 6 6 8 m 5. about 0.83 m/s 7. P() 8 P() 6 0 6. Sample answers: f() f() f() ( )( 3), ( ) ( )( 3), 5( ) ( )( 3) 8. the part in the first quadrant Glencoe/McGraw-Hill 5 Algebra Chapter 9
9. It represents her original freethrow percentage of 60%. 5. A rational function can be used to determine how much each person owes if the cost of the gift is known and the number of people sharing the cost is s. Answers should include the following. 00 s 0 50 c 00 50 50 c 50 s c 0 50 00s 50. ; This represents 00%, which she cannot achieve because she has alread missed free throws. 5. A 53. B 55. nl the portion in the first quadrant is significant in the real world because there cannot be a negative number of people nor a negative amount of mone owed for the gift. 3 6 ( 3)( ) 57. (6, ); 5 00 5. 56. 58. 3m m n 5(w ) (w 3) (, 0); 3 ( 6) ( ) 5 9 Glencoe/McGraw-Hill 55 Algebra Chapter 9
59. $65,89 6., 0 63..5 65. 0 60. i 6. 6.. 66. 7 33 Lesson 9- Direct, Joint, and Inverse Variation Pages 95 98 a. inverse b. direct 3. Sample answers: wages and hours worked, total cost and number of pounds of apples; distances traveled and amount of gas remaining in the tank, distance of an object and the size it appears 5. direct; 0.5 7. 9.. 5.8 psi 3. Depth(ft) Pressure(psi) 0 0 0.3 p 0.86 3.9.7. Both are eamples of direct variation. For 5, increases as increases. For 5, decreases as increases.. inverse; 0 6. joint; 8. 5 0. P 0.3d. about 50 ft. direct;.5 Glencoe/McGraw-Hill 56 Algebra Chapter 9
P P 0.3d d 5. joint; 5 7. direct; 3 9. direct; 7 6. inverse; 8. inverse; 0. joint; 3. inverse;.5. V k p 3. V kt 5. 8.5 km 7. 0 9. 6 3. 33. 9.6. directl; 6. 60 8. 6 30. 5 3..5 3..6 35. 0.83 36. 37. 6 39. 00.8 cm 3. m 0sd 3. 860 lb 5. joint 7. I k d 38. 30 mph 0. See students work.. joint. / 5md 6. See students work. 8. I I 6 d d Glencoe/McGraw-Hill 57 Algebra Chapter 9
9. The sound will be heard as 50. intensel. 5. about 7,57 calls 53. no; d 0 55. A direct variation can be used to determine the total cost when the cost per unit is known. Answers should include the following. Since the total cost T is the cost per unit u times the number of units n or C un, the relationship is a direct variation. In this equation u is the constant of variation. Sample answer: The school store sells pencils for 0 each. John wants to bu 5 pencils. What is the total cost of the pencils? ($.00) 57. C 59. asmptotes:, 3 6. 5. about 60 mi 5. Sample answer: If the average student spends $.50 for lunch in the school cafeteria, write an equation to represent the amount s students will spend for lunch in d das. How much will 30 students spend in a week? a.50sd; $375 56. D 58. asmptote: ; hole 60. hole: 3 6. 0.0; C 0.0P P d t t (t )(t ) 63. m (m ) m 5 65. 0.;. 6. 9.3 0 7 66. 3; 7 67. 3 5 ; 3 68. C 69. A 70. S Glencoe/McGraw-Hill 58 Algebra Chapter 9
7. P 73. C 7. A Chapter 9 Practice Quiz Page 98. f () f(). f () f() 6 9 3. 9 5... Lesson 9-5 Classes of Functions Pages 50 50. Sample answer: P. constant ( ), direct variation ( ), identit ( ) d This graph is a rational function. It has an asmptote at. 3. The equation is a greatest integer function. The graph looks like a series of steps. 5. inverse variation or rational. greatest integer 6. constant Glencoe/McGraw-Hill 59 Algebra Chapter 9
7. c 9. identit or direct variation 8. b 0. quadratic. absolute value. A r ; quadratic; the graph is a parabola 3. absolute value 5. rational 7. quadratic 9. b. g 3. constant. square root 6. direct variation 8. constant 0. e. a. direct variation.5.5 Glencoe/McGraw-Hill 60 Algebra Chapter 9
5. square root 6. inverse variation or rational 9 7. rational 8. greatest integer 3[] 9. absolute value 30. quadratic 3. C.5 m 33. a line slanting to the right and passing through the origin 3. direct variation 3. similar to a parabola Glencoe/McGraw-Hill 6 Algebra Chapter 9
35. Cost (cents) 60 0 80 0 0 37a. absolute value 37b. quadratic 37c. greatest integer 37d. square root 6 8 0 unces 36. The graph is similar to the graph of the greatest integer function because both graphs look like a series of steps. In the graph of the postage rates, the solid dots are on the right and the circles are on the left. However, in the greatest integer function, the circles are on the right and the solid dots are on the left. 38. A graph of the function that relates a person s weight on Earth with his or her weight on a different planet can be used to determine a person s weight on the other planet b finding the point on the graph that corresponds with the weight on Earth and determining the value on the other planet s ais. Answers should include the following. The graph comparing weight on Earth and Mars represents a direct variation function because it is a straight line passing through the origin and is neither horizontal nor vertical. The equation V 0.9E compares a person s weight on Earth with his or her weight on Venus. V 80 Venus 60 0 0 0 0 0 60 80 Earth E Glencoe/McGraw-Hill 6 Algebra Chapter 9
39. C. 0. D. f () 3 f () 3. f (). f () f () 5 f () 8 ( )( 3) 5. (8, ); a8, 7 8 b; 8 ; 0 8 6 up; unit 6 0 8; ( ) ( 8) 6. a 3, b ; a, b; ; ; 3 right; units Glencoe/McGraw-Hill 63 Algebra Chapter 9
7. (5, ); a5 ; 3, b; ; 8. right; 3 units c 5 3 5 66 6 57 d 3 8 3 9. impossible 50. (7, 5) 5. a 3, b 5. (, ) 53. 5. 55. 7 6 56. 60a 3 b c 57. 5 3 3 59. 3( )( ) 58. 5(d ) 60. (a 3)(a )(a ) 6. (t 5)(t 6)(t ) Lesson 9-6 Solving Rational Equations and Inequalities Pages 509 5. Sample answer: 5 a 3. Jeff; when Dustin multiplied b 3a, he forgot to multipl the b 3a.. ( );. 3 5., 6 6. 3 7. 6, 8. c Glencoe/McGraw-Hill 6 Algebra Chapter 9
9. v 0 or v 6 0. 9 h.. 3 3. 6, 5. a 0 7. 9. t 0 or t 3. 0 3. 5. 7. 7 9. 3. 3 33. band, 80 members; chorale, 50 members 35. cm 37. 5 ml 39. 6.5 3 3. If something has a general fee and cost per unit, rational equations can be used to determine how man units a person must bu in order for the actual unit price to be a given number. Answers should include the following. 500 5 To solve 6, multipl each side of the equation b to eliminate the rational epression.. 3, 6. m 8. 3 0. 0 b. p 0 or p. 6. 8. 30. 3. or 3..8 cm/g 36. 5 km/h 38. 5 0. 3 7 3. B 5 b bc Glencoe/McGraw-Hill 65 Algebra Chapter 9
3. C Then subtract 5 from each side. Therefore, 500. A person would need to make 500 minutes of long distance calls to make the actual unit price 6. Since the cost is 5 per minute plus $5.00 per month, the actual cost per minute could never be 5 or less.. quadratic 5. square root 6. direct variation 0.8 7. 36 9. 30 5. 37 8. 33.75 50. 5 5. 5 0 or 36 53. 5 0 0 6 5. eb` b f Glencoe/McGraw-Hill 66 Algebra Chapter 9