PMI Rational Expressions & Equations Unit
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1 PMI Rational Expressions & Equations Unit Variation Class Work 1. y varies inversely with x. If y = 1 when x =, find y when x = 6.. y varies inversely with x. If y = 8 when x =, find x wheny =.. y varies inversely with x. If y = whenx =, find y whenx =.. Circumference varies directly with area of a circle and inversely with the radius. Find the constant of variation if C = 18 when A = 7 and r =. What is radius when C = and A = 0?. y varies jointly with x and z. If y = when x = and z =, find y when x = 6 and z =. 6. y varies jointly with x and z. If y = when x = and z =, find x when y = 8 and z =. 7. V varies jointly with r and h. If V = π when h = 6 and r =, find r when V = 18π and h =. Variation Homework 8. y varies inversely with x. If y = 9 whenx =, find y whenx = y varies inversely with x. If y = 8 when x =, find x when y =. 10. y varies inversely with x. If y =, when x =, find y when x =. 11. Area of a triangle varies jointly with its height and base. Find the constant of variation if A = 16 when h = and b = 8. What is base when A = 9 and h =? 1. y varies jointly with x and z. If y = when x = 6 and z =, find y when x = 6 and z =. 1. y varies jointly with x and z. If y = 0 when x = and z =, find x wheny = 60 and z =. 1. V varies jointly with r and h. V = 6π when h = and r =, find r whenv = 80π and h =. Alg II - Rationals ~1~ NJCTL.org
2 Reducing Rational Expressions Class Work Simplify. 1. x b 1b 17. 1x x a b 1a b j k 6 m 8 16j k 6 m c n + n 6n + n. h 10 h 8.. v v + v. f + 7f + 1 f f 1. s 0s + s 16 8s 6. d 7d + 6 d 8d + 7. x x 1x x x x 8. x 6x x 7x + 18x 9. m m + m + 0. p + 7p 6 p + p a + a 60 a 10. x + y x y. 1a b 1 abc c. 1x + 7x x x + x. p p p 6. 6x + 1x + 9x 1x + 6x 6x Alg II - Rationals ~~ NJCTL.org
3 Reducing Rational Expressions Homework Simplify. 7. 9y c 16c 9. 1x 8x 0. 8a b 1a b j k 6 m 8 1j k 6 m 10. 1c n n 1n n. 1h 8 9h 6. v v + v 9 6. f + 1f + 7 f + f 7 7. rs 0rs + 1rs 1r rs 8. d + d + 1 d + d + 9. x x x x 0. a b 1ac 1bc 1. m mn + n m + n. 8mp m p 1. 1x 10x + x x + x x. 6k l kl + l k l. x y xy x y x y 6. 1k sk ks 6ks 18k 7. 6p 1p p 1p 8. x + x 7x Alg II - Rationals ~~ NJCTL.org
4 Multiplying & Dividing Rational Expressions Class Work Perform the indicated operation. Write answer in simplified form. 9. 8a 11 1a 60. 1b c 1c 8b d + e 61. f f + (d + e) 6. g 7g + 1 g 9 g + g + g + 6g + 6. g + h h 1 g g 10 h + g h h h 1 6h 6. j 10j k + l k + l k + kl + l k l 67. m + m + m + 6 m + 7m + 10 m 68. n n 69. p + p + p 1 p + 7p + 10 p + p + 1 p 1 p + p q r 16q r 8q6 r q r 1q r 18q r 8 qr q r 71. x 7 x 9 x x + x 7. m 6m + m 1 m + 1 m 9 7. m n mn n m mn + n n 7. a ab + b a + b a b 1a 1b 7. 6m + 9m + m + 7m + 6m + m + 18 m + m x x x x x 1 x + x 77. A rectangular shaped dartboard has dimensions (x +) by (x+1) inches. On the board is a square with sides (x+1) inches. What is the probability that a randomly thrown dart that hits board lands in the square? Alg II - Rationals ~~ NJCTL.org
5 Multiplying & Dividing Rational Expressions Homework Perform the indicated operation. Write answer in simplified form a 11 1a b c 0c 8b d + e 80. f f (d + e) 81. g 8g + 1 g g + 10g + g g 6 8. g + h + h 8 g + 10g + 1 h + 7 g 16 h 7h h 1 8h 8. 6j 1j 9 8. k + l k + l k kl + l k l 86. m + 6 m + m 6 m + 7m +1 m n 1 n n + n 1 n 88. p + p + p + 7p + 1 p p + p + p p + p q9 r 1q r 7 0q7 r 1 16q r 10 8q7 r q r q r q 8 r r 90. 7x y x y 18x + 6xy + y 6x xy + y 91. mn + n m mn n mn n 6m 11mn + n 8ac bc 9. 0ac bc a b a b 9. 6x 17x + 6x + 7x x 9x + x x x x 7x x + x 0 x x 9. 8 m m m m m m + m 10m 96. A rectangular shaped dartboard has dimensions (x+6) by (x+) inches. On the board is a square with sides (x+) inches. What is the probability that a randomly thrown dart that hits board lands in the square? Alg II - Rationals ~~ NJCTL.org
6 Adding and Subtracting Rational Expressions Class Work Perform the indicated operation. Write answer in simplified form y y x x 99. z + z w + 7 w + 6 w + 1 w v v + 8 v + v (u + ) u 1 (u + 1) u x + x 10. x x 9 x x + x + x x x 7x + 1 x 107. x x x x x x x x + y x x + 6 x + x 10 x + x m + m 6 + m 10 6 m 111. x x x x x x x 11. x x + x x x + x + x 6 x + x 11. y 9 y + y Alg II - Rationals ~6~ NJCTL.org
7 Adding and Subtracting Rational Expressions Homework Perform the indicated operation. Write answer in simplified form y 11. y x x 116. z 6 + z w + w + w + w v v + 9 v + 6 v (u + 1) u 1 (u + 8) u t 1 t x x x + 6 x x + 6x + 9 x x + x x 18 x x x + x + x 1 x + x 1. x x 16. x + 6x x x 6x 17. 6p p 1 p p 18. 8b + 6b + b b 19. m 6m + 16m 1 8m 6m y 7 6y + y 1 y y y y 1 Alg II - Rationals ~7~ NJCTL.org
8 Solving Rational Equations Class Work Solve for x. Check for extraneous solutions. 11. = x + x 1. = 6 x 1 x + 1. x 1 + x + = 6x x + = x x 1. + = 1 x + x x = 1 x x 9 x = x + x 1 x + x 18. x = 1 x + x x + x = x x x + x x x + = 6 x = x x = 1 x x 1 x x + 1. x x + = x x + 9 Alg II - Rationals ~8~ NJCTL.org
9 Solving Rational Equations Homework Solve for x. Check for extraneous solutions. 1. = x 1 x + 1. = 6 x + x x x = x x + = x x = 1 x x x = 16 x x x = x + x x + x x = x+ x + 1 x 1 x x = x + x + x + x + 1 x + 6x x + x = 7 x 1. + = x 1 x + 1 x = 8 x 1 x 1 x + x x + x = + x Alg II - Rationals ~9~ NJCTL.org
10 Graphing Rational Expressions Class Work Graph the following functions. Clearly indicate any x-intercepts, y-intercepts, asymptotes and holes (discontinuities) on the graph, noting them in the spaces provided below. 17. f(x) = x g(x) = x+ 19. h(x) = x+1 x f(x) = x 1 (x 1)(x+) 161. g(x) = x +x+6 x +x+ 16. h(x) = x x 6 x x+6 Alg II - Rationals ~10~ NJCTL.org
11 Graphing Rational Expressions Homework Graph the following functions. Clearly indicate any x-intercepts, y-intercepts, asymptotes and holes (discontinuities) on the graph, noting them in the spaces provided below. 16. f(x) = x+ 16. g(x) = x 16. f(x) = x+ (x 1)(x+) 166. h(x) = x x 167. g(x) = x +9x+18 x +7x h(x) = x +x 1 x +6x 7 Alg II - Rationals ~11~ NJCTL.org
12 1. Simplify x 10x + 1 a. b. c. d. x 1x x x x (x )(x ) x 6x (x 6)(x + 1) x(x ) Unit Review - Multiple Choice. Simplify x + 1x + 6 a. b. x 9 (x 7)(x ) (x + 7)(x + ) (x + 7)(x ) (x 7)(x + ) x c. x + d. 1 x 10x + 1 x + 11x +. Simplify 6m6 n 9m n m n 9 8m n 7 a. b. c. d. m n m n 11 n m n 11 m. Simplify a. b. c. d. x 6x 1 6x x 6x 1 6x 1 x. Simplify a. b. c. d. + x 16 x + 8x x (x )(x + )(x + ) 6 (x + )(x + ) 6x 8 (x )(x + )(x + ) 6x (x )(x + )(x + ) Alg II - Rationals ~1~ NJCTL.org
13 6. The function h(x) = x x 1 has which of the following discontinuities? x 1 a. Vertical asymptotes at x = ± 1 b. Removable discontinuity at x = ± 1 c. Vertical asymptote at x = 1 ; removable discontinuity at x = 1 d. Vertical asymptote at x = 1 ; removable discontinuity at x = 1 7. h varies inversely with t. If h = 8 when t = 6, find t when h = 16. a. b. 8 c. d The volume of a cone varies jointly to its height and the square of the radius of its base. If V = 18π when h = 6 and r =. What is the radius when V = 1π and h =? a. 1 b. c. 6 d Simplify x+ + x 7 x+ x x a. b. c. d. x x+ x x x 1 x x 8 x x x x 10. Solve: = x x + a. b. 8 c. 1 d. no solution 11. Solve: x x 1 + x 1 = x + 1 a. -1 b. 1 c. 6 d. no solution 1. Solve: + = x 9 x x 6 x + x + 6 a. - b. -8 c. 8 d. no solution Alg II - Rationals ~1~ NJCTL.org
14 Extended Response 1. At math camp the lap pool is a rectangle that is (x 16)ft by (x + )ft, the wading pool is a square with sides (x + )ft. a. How many times larger is the lap pool than the wading pool? b. If the wading pool is (x )ft deep, what is the pool s volume? c. If the lap pool has a depth of (x + )ft, how many times larger is the volume of the lap pool to the wading pool?. Determine each of the following for the graph of the rational function and graph the function. f(x) = x +x 6 x 16x 1 holes: vertical asymptotes: horizontal asymptotes: Alg II - Rationals ~1~ NJCTL.org
15 PMI Rational Expressions, Equations and Functions- SOLUTIONS: Variation: Classwork: 1. k = 8; y = 8. k = ; x = 1. k = 7; y = 8.. k = 0.6. k = ; y = 6 6. k = ; x = 6 7. k = π; r = Variation: Homework: 8. k = 6; y = 6 9. k = 0; x = k = 8; y =. 11. k = 0.; b = 6 1. k = ; y = 6 1. k = ; x = 6 1. k = π; r = Reducing Rational Expressions: Classwork: 1. x x a b 19. 1j m 0. (c 1) 1... v v + f +. f. s s 6. d d x + 7. x (x + 1) x(x + 1) 8. (x 1) 9. (m 1) 0. p p 1. (a + ). x xy + y (x y) (ab + 1). c x 1. x. p + 6. x + (x ) Reducing Rational Expressions: Homework: 7. y x 0. ab 6 1. j m. c. n 1 n 7.. v 1 v + f + 6. f 8 7. s(rs 1)(rs ) s 8. d + 1 d + 9. x + 1 x 0. a + ab + b 6c 1. m + n. m (x 1). x (x + 1) l (k l). k + l x. x y (x ) 6. k s + p 7. (p + ) 8. 1 x + 1 Alg II - Rationals ~1~ NJCTL.org
16 Multiplying & Dividing Rational Expressions Class Work: 9. x 60. bc bc 61. f d+e 6. (g+)(g ) (g+1)(g+) g + 6. h j 66. k 1 k+l m 67. (m+)(m+) (n+) (p+1) 69. (p+)(p+) 70. 9r q x + x m 1 m + 7. n(m + n) 1 7. m 7. (m + ) x 76. (x + )(x 1) 77. (x+1) x+1 Multiplying & Dividing Rational Expressions Homework 78. or c 80. b f+ (d+e) g+ 81. (g+1) 8. (h+)(h+7) (g )(h ) j 8. k+1 k 1 m 86. (m + )(m 1) 87. (n+1)(n+) n p+ 88. (p+) 89. r 10 q 7 x + y 90. m n 91. n(m n) c(a + b) (x )(x ) 9. (x 1)(x + ) 9. 1 x + 9. m(m + ) Adding & Subtracting Rational Expressions Classwork 97. x 98. y 99. z x (x+)(x ) 10. x 6x+ (x )(x+) x+ (x+) (x ) x 17 (x )(x ) x 19 (x )(x )(x+) 108. x +1xy x + x + y 109. x + 7 x(x 1) m m 6 x(x 1) 111. x 11. x x + 1x 9 (x )(x 1)(x + ) 11. 1y + 6y + 9 y + Alg II - Rationals ~16~ NJCTL.org
17 Adding and Subtracting Rational Expressions Homework 11. x 11. y z (v+) 118. v+ 6(u ) 119. u 1 (t+1) 10. (t+1)(t+1) 11. x +x+ (x )(x ) 1. 11x+ (x+) (x ) 1. (x+1) (x+)(x 6) 1. 6x+ (x 1)(x+)(x ) 10x 1. x 1x 16. x(x ) 17. 7p + p (7b ) 18. b m + m + (m 1)(m + 1)(m 1) 10. y + 6y 10y + 7 (y + 1)(y 1)(y 1) Solving Rational Equations Class Work or , ± ± No Solution ± 7 Solving Rational Equations Homework (± 10) No Solution 7±i ± 1. 7 ± Alg II - Rationals ~17~ NJCTL.org
18 Graphing Rational Equations Classwork: 17. f(x) = x g(x) = x+ x-intercepts: None y-intercepts: y = Holes: None Vertical asymptotes: x = 1 Horizontal asymptotes: y = 0 x-intercepts: None y-intercepts: y = 1. Holes: None Vertical asymptotes: x = Horizontal asymptotes: y = h(x) = x+1 x f(x) = x 1 (x 1)(x+) x-intercepts: None y-intercepts: y = 1 Holes: x = 1 Vertical asymptotes: x = 1 Horizontal asymptotes: y = 0 x-intercepts: None y-intercepts: y = 1 Holes: x = 1 Vertical asymptotes: x = Horizontal asymptotes: y = 0 Alg II - Rationals ~18~ NJCTL.org
19 161. g(x) = x +x+6 x +x+ 16. h(x) = x x 6 x x+6 x-intercepts: x = y-intercepts: y = Holes: x = Vertical asymptotes: x = 1 Horizontal asymptotes: y = 1 x-intercepts: x = y-intercepts: y = 1 Holes: x = Vertical asymptotes: x = Horizontal asymptotes: y = 1 Graphing Rational Equations Homework: 16. f(x) = x+ 16. g(x) = x x-intercepts: None y-intercepts: y = Holes: None Vertical asymptotes: x = Horizontal asymptotes: y = 0 x-intercepts: None y-intercepts: y = Holes: None Vertical asymptotes: x = Horizontal asymptotes: y = 0 Alg II - Rationals ~19~ NJCTL.org
20 16. f(x) = x+ (x 1)(x+) 166. h(x) = x x x-intercepts: None y-intercepts: y = 1 Holes: x = Vertical asymptotes: x = 1 Horizontal asymptotes: y = g(x) = x +9x+18 x +7x+6 x-intercepts: None y-intercepts:y = 1 Holes: x = Vertical asymptotes: x = Horizontal asymptotes: y = h(x) = x +x 1 x +6x 7 x-intercepts: x = y-intercepts: y = Holes: x = 6 Vertical asymptotes: x = 1 Horizontal asymptotes: y = 1 x-intercepts: x = y-intercepts: y = Holes: x = 7 Vertical asymptotes: x = 1 Horizontal asymptotes: y = 1 Alg II - Rationals ~0~ NJCTL.org
21 Unit Review - Multiple Choice 1. B.. C.. A.. A.. C. 6. A. 7. B. 8. D. 9. D. 10. C. 11. D. 1. A. Extended Response 1. a. (x - )(x + ) x + b. x + x -16x - 6 c. x +. x-intercepts: (, 0) y-intercepts: (0, 1 ) Hole: x = Vertical Asymptote: x = Horizontal Asymptote: y = 1 Alg II - Rationals ~1~ NJCTL.org
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