6. y = 4_. 8. D: x > 0; R: y > 0; 9. x 1 (3)(12) = (9) y 2 36 = 9 y 2. 9 = _ 9 y = y x 1 (12)(60) = x 2.
|
|
- Austen Hunt
- 5 years ago
- Views:
Transcription
1 INVERSE VARIATION, PAGES 676 CHECK IT OUT! PAGES 686 a. No; the product y is not constant. b. Yes; the product y is constant. c. No; the equation cannot be written in the form y k. y 5. D: > ; R: y > ; y y y ()(6) (4) y 4 y 4 4 y 4 y 5. y y (4.)(6) (.) y 58. y 58.. y y The child s weight is 8.65 lb. THINK AND DISCUSS, PAGE 6. Possible answer: The function rule will have the form y k, and y will be the same nonzero constant for each ordered pair.. EXERCISES, PAGES 66 GUIDED PRACTICE, PAGE 6. The graph of an inverse variation consists of disconnected branches.. Yes; the product y is constant.. No; the product y is not constant. 4. No; the equation cannot be written in the form y k. 5. Yes; y equals the constant y 4 8. D: > ; R: y > ; 9. y y ()() (9) y 6 9 y y 9 4 y. y y ()(6) (45) The gear has 6 teeth. 7. y 6. y y ()(4) (6) PRACTICE AND PROBLEM SOLVING, PAGES 66. No; the product y is not constant.. Yes; the product y is constant. 4. Yes; the function can be written in the form y k. 5. No; the function cannot be written in the form y k. 6. y 7. y Copyright by Holt, Rinehart and Winston. 9 Holt Algebra
2 8. D: > ; R: y > 9. y y ()(4) (6) y y (.5)(5) (4.5)y y y 4.5 y June can buy about yd. 6 a. y k Helen puts psi pressure. b. y k The area is 57.5 i n.. y y (7)(9) (6) y 6 6 y y 6.5 y. direct; 8 4. inverse; 4 5. neither 6. direct; 5 7. inverse; 8. neither 9. inverse; 5. direct; 5. d ; inverse. direct n. neither 4. inverse 5. y ; D: natural numbers; R: y > 6. y k 6π An inverse variation function, y k and k. So neither nor y can be. 8. Substitute the known values of and y into y k and solve for k. Use this value of k in the equation y k. 9. Inverse operations, additive inverse, etc.; all of the terms involve moving in opposite directions; inverse variation describes a relationship in which the variables move in opposite directions (one increases while the other decreases). 4a. y k 75 k (75)() k 5 k b. y k 5 4. C; when, y 4 and when, y C; the product of y is not constant. c. 4. D; y k k 7 ()(7) k 4 k CHALLENGE AND EXTEND, PAGE the linear function y 46. ft in. 6 in. y y (6)() (6) y 7 6 y y 6 y The force created at P is lb. 47. y y ( 4 )() ( 6 ) y 6 y 44. y y ()(.5) () Brad can buy tickets. 6 y y The strength of the signal is about watts. SPIRAL STANDARDS REVIEW, PAGE D: {}; R: {4,,, }; no 49. D: {4,,,, 4}; R: {,, 5}; yes ( + 6 ) ± ±9 or 5 Copyright by Holt, Rinehart and Winston. 9 Holt Algebra
3 5. d 6d 7 d 6d 7 d 6d (d ) 6 d ± 6 d ±4 d 7 or 5. y + 6y 5 y + y 5 4 y + y ( y + ) y + ± y + ± y 5 or a 4, b, c 6 Discriminant b 4ac ( ) 4(4)(6) 9 (96) 5 > has solutions. RATIONAL FUNCTIONS, PAGES 6464 CHECK IT OUT! PAGES 6467 a. y b. y 4 The ecluded value is. The ecluded value is ; y c. y a. y The ecluded value is 4. b. y c. y ; y 5 a. b ; y 5 4a. D: > ; R: natural numbers > b. THINK AND DISCUSS, PAGE 68. Yes; the function is undefined at 5.. To find the vertical asymptote, set + 9 equal to and solve for. The vertical asymptote is 9. To find the horizontal asymptote, look at the constant term. The horizontal asymptote is y 5.. EXERCISES, PAGES 6964 GUIDED PRACTICE, PAGE 69. ecluded value. y 4 The ecluded value is. + + No solutions, therefore no ecluded values. y 4. y The ecluded value is. 6. y ; y ; y 8. y 5. y The ecluded value is y ; y 9. y ; y Copyright by Holt, Rinehart and Winston. 94 Holt Algebra
4 .. 4a. D: > ; R: < y < b... PRACTICE AND PROBLEM SOLVING, PAGES y 7 6. y 4 4 The ecluded value is. 4 The ecluded value is y 8. y 5 5 The ecluded value is. 5 The ecluded value is ; y 9. y. y ; y ; y. y ; y 9. y a. D: > ; R: natural numbers > 5 b y 4 9. y The ecluded value is.. y The ecluded value is y The ecluded value is The ecluded value is.. 5. Copyright by Holt, Rinehart and Winston. 95 Holt Algebra
5 7. y ; y 9. y + 5 ; y 5 4. B 4. A 4. C 8. y 5 ; y 5 4. y ; y Student B is incorrect. The student identified the value at which there is a vertical asymptote ( ). 45. D: > 5; R: nonnegative values 46a. b. y 6 c. y It will take 5 hours. 5. y > 5 + > 5 > > 5 D: > I and III; II and IV 56a. y 5 c. 54. y > 7 > > 7 > 7 D: > 7 b. D: natural numbers 57. The graph of y k is the reflection of the graph of y k across the ais. 47. translated 6 units right 48. translated 7 units left 49. translated 4 units up 5. y > > > D: > 5. translated units right and 9 units down 5. y + + > + > > D: > 58. A 59. D 6. f() + Copyright by Holt, Rinehart and Winston. 96 Holt Algebra
6 CHALLENGE AND EXTEND, PAGE 64 6a. Yes; the function is a quotient of polynomials. b. D: all real numbers c. R: < y d. no 6. No; the graph of f() has an ecluded value at. The graph of g() has no ecluded values. 6. y + + SPIRAL STANDARDS REVIEW, PAGE t + 5 < (t + ) 4t + 5 < t + 9 4t < t + 4 t < j + < 4j 9 j + 9 < 4j 9 < j < j j > 68. c 5 > c + 7 c > c > c c < 7. 4 ( + )( ) + or or The zeros are ± ( )( ) or The only zero is. 65. (r + ) r 6 r + r 6 r r 8 r (g + ) g 5 5g + g 5 5g g 5 g 5 g (m ) < (6 m) 6m < m 6m < 4 m 8m < 4 m < ( + )( ) + or The zeros are and. 7. Let s represent side length of square piece, in inches. Then dimensions of new piece are (s ) in. by (s + ) in. A lw A (s )(s + ) 78 s 78 s s s 784 6(49) (7) 8 Dimensions of new piece are 8 6 in. by 8 + in. SIMPLIFYING RATIONAL EXPRESSIONS, PAGES CHECK IT OUT! PAGES a. b. b t + 5 c. t + 5 t 5 There are no ecluded values. k k + 7k + k + 7k + (k + )(k + 4) k + or k + 4 k or k 4 The ecluded values are and 4. a. 5 m 5m 5 m 5 m m m m ; m c. n n n n ; n b. b. b 5 b + b + 5 (b + 5)(b 5) (b + 5 ) b 5 b ( ) ( )( + ) ( ) ( +)( ) + b + 5b b + 5b b(b + 5) b or b + 5 b 5 The ecluded values are and 5. b. 6 p + p p + 6p ( p + ) p + 6p r + a. r + 7r + r + (r + )(r + 5) r ( 4) (4 )(4 + ) ( 4) ( 4)(4 + ) 4a. (4 + ) 4 + c. ( ) ( + )( ) + 5. The barrel cactus with a radius of inches has less of a chance to survive. Its surfaceareatovolume ratio is greater than for a cactus with a radius of 6 inches. Copyright by Holt, Rinehart and Winston. 97 Holt Algebra
7 THINK AND DISCUSS, PAGE Possible answer: The epression has the ecluded values and, but cannot be identified as an ecluded value in the simplified form, +.. EXERCISES, PAGES GUIDED PRACTICE, PAGE Both the numerator and denominator are polynomials.. 5 m m The ecluded value is.. p p p 5 p p 5 (p 5)(p + ) p 5 or p + p 5 or p The ecluded values are 5 and a 8a 4 a 4 a 7. a a a ; a y + y + ; y ( 8) or 8 8 The ecluded values are and d + d d + 6 d(d + 6) d + 6 d; d y 5 y ; y 5 9. h h + 4 h (h + ) h h + ; h b + 4. b + 5b + 4 b + 4 (b + 4)(b + ) b c + 5c + 6 (c + )(c 4) (c + )(c + ) (c + )(c 4) c + c 4 j 5 j + j 5 (j + 5)(j 5) (j )(j + 5) j 5 j n 6 64 n (n 8) (8 n)(8 + n) (n 8) (n 8)(8 + n) (8 + n) 8 + n 9. 5r r + 4r 5(r ) (r + 6)(r ) 5 (r + 6). a. 5q 5 ( q ) 5(q ) ( q)( + q) 5(q ) (q )( + q) 5 ( + q) 5 + q b h b + b h ( + 4). 6 ( + 4) + 4 ;. s 4 s + 4s + 4 (s + )(s ) (s + ) s s + ( )( + ) ( )( + ) ( + )( + ) + p ( h ) b ( h ) ( b + b ) p 4p 5 p + (p 5)(p + ) p ( ) ( )( 4) 4( ) ( )( 4) ( 7) (7 )(7 + ) ( 7) ( 7)(7 + ) (7 + ) a a + a 5 ( a) (a + 5)(a ) (a ) (a + 5)(a ) a + 5 b b + b Copyright by Holt, Rinehart and Winston. 98 Holt Algebra
8 b. ( b )h b h ( b ) + ( b ) ( b + b )h b b + b h They will be the same. PRACTICE AND PROBLEM SOLVING, PAGES c 5. c + c c + c c(c + ) The ecluded value is. c or c + c The ecluded values are and ( 5)( + ) 5 or + 5 or The ecluded values are 5 and. n n 7n 4 n 7n 4 (n + )(n 4) n + or n 4 n or n 4 The ecluded values are and d + 4 d d + 4 d (d + ) d + 4 d ; d. y 4 y 5 y 4 y 5 y 4 y. 5 y ; y q 6 q 9q + 8 q 6 (q )(q 6) q 4. t t 5t + 6 t (t )(t ) t 9. m m 4 m m 4 ; m 4. t 6t t 8t t 8t t 8 ; t. z z + z (z ) (z + )(z ) z z + 5. p 6p 7 p 4p 5 (p 7)(p + ) (p 5)(p + ) p 7 p ( + )( ) ( + )( + ) + 5 4(5 ) ( 5)( + 5) 4( 5) 8. 4 ( 5)( + 5) v 6 44 v (v ) ( v)( + v) (v ) (v )( + v) ( + v) + v 4a. S V lw + lh + wh lwh ( ) ( )( 4) 4 b b + 8b ( b) (b + 7)(b ) (b ) (b + 7)(b ) b + 7 (lw + lh + wh) lwh b. S A ( ) 84 V A ; S B ( ) V B 8 6 Bo A; the surface area of bo A is 84 i n and the surface area of bo B is i n. The volumes are the same, so the company should use the bo with less surface area, bo A ; ( + 4) 5 + 5( + 4) n + n + 5n n + 5n 46. n ( n + n + 5) n ( n + 5) n + n + 5 n + 5 j 5 j 5 j 5 (j + 5)(j 5) j p + p + 6 p + 7 (p + 6 ) (p + 6) p + 6 a a + a a a w + w 7 6w (w )(w + 7) (w ) w + 7 Copyright by Holt, Rinehart and Winston. 99 Holt Algebra
9 48. n n 56 n 6n + 64 (n + 7)(n 8) (n 8 ) n + 7 n ( + 5 ) already simplified 5a. Crew Size () Workdays Workers b. y 5 S 5a. V 6 s 6 s s ( + ) + + ( + ) ( + ) 5 (5 )(5 + ) ( 5)( + ) ( 5)(5 + ) ( 5)( + ) Construction Days (y) c. The ecluded value is. b. s, 6 s 6 c. s 6, 6 s Set the denominator equal to and solve for the variable. 55. Possible answer: D; 4 or 4 or + ( 4)( + ) ( 4 ) 4 4 The ecluded value is A; bh bh CHALLENGE AND EXTEND, PAGE Sometimes; the rational epression + has as an ecluded value, but the rational epression has no ecluded values Never; the numerator must be a polynomial. 6. Sometimes; the rational function y + has as an asymptote, but the rational function y + has no asymptotes v 6 v 4 v 4v v( v) (v )(v + ) v(v ) (v )(v + ) v v +.5y..5 y.4 (.5y.) (.5 y.4) 5y 5 y 4 5(5y ) (5y + )(5y ) 5 5y ( + ) ( ) The ecluded value is a 7a + a + 9a 5 (a )(a ) (a )(a + 5) a a ( + ) or + The ecluded values are and. SPIRAL STANDARDS REVIEW, PAGE D: {,,, 9}; R: {, 4, 5, 6} ± 96 ±4 The ecluded values are ± D: {4, 5, 5, 9}; R: {7,, } 7. Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
10 h 5 h 75. b 4 already simplified 74. s t 6 t 6 s 76. v w 4 w 4 v TECHNOLOGY LAB: GRAPH RATIONAL FUNCTIONS, PAGE 649 TRY THIS, PAGE 649. When the denominator of y is set ( )( 4) equal to and solved for, then equals or 4. However, when the denominator of y is set 4 equal to, the ecluded value seems to be only 4.. No; for the value of, y is undefined, while y 4 is defined as. a. asymptote b. hole CONCEPT CONNECTION, PAGE 65. It would take the crew 5 construction days.. y ; the number of people in the crew; y the number of construction days; inverse variation. As the crew size increases, the number of construction days decreases. 4. y 6.5 It would take the crew 6.5 construction days. 5. y There are 6 people in the crew. 6. D: natural numbers; R: y > 7. y ; and y READY TO GO ON? PAGE 65. No; the product y is not constant.. Yes; the product y is constant.. yes; y 4. no; cannot be written in the form y k 5. yes; constant product 6. no; cannot be written in the form y k 7. y 6 9. y y (6)(4) (8) y 44 8 y 8 y She can buy 8 calculators.. ; and y. ; and y 8. y 4. ; and y. ; and y Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
11 4. D: > ; R: natural numbers > n n The ecluded value is ( + )( + 4) + or + 4 or 4 The ecluded values are and t t + t t + t t(t + ) t or t + t The ecluded values are and ; s +. s 4s 5 s + (s 5)(s + ) ; s and s 5 s 5 p p 8 p 8 p 8 The ecluded value is 8.. n n n n n(n ) n ; n and n (4 ) ( 4 ) ( 4) ( 4 ) 4 ; 4 S. cone S πrl + π r πr(l + r) cylinder πrh + π r πr(h + r) 4 MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS, PAGES CHECK IT OUT! PAGES a. (c 4) 5 45(c 4) 45 (4c + 6) 5(4c + 6) 45(c 4) 5 (4)(c 4) 9 5(c 4) 4 5(c 4) 9 4. m 5 m + 6 m 4m m 5 m + 6 m 4m m 5 (m + ) (m 6)(m + ) (m 5) m 6 m 5 m 6 a. n 5 n + 8n + 6 n + 4n n n n 5 n(n + 4) (n + 4 ) (n 5)(n + ) n + 4 n(n + ) n + 4 n + n p + 4 b. p p p + p p + 6 p + 4 (p 5)(p + ) p(p + ) p + 6 (p + 4)(p 5) p ( p + 6) p p p + 6p 4a. 5 ( 5) ( 5) b. 5 y 5 z 4 y y z 4y 4 y 7 z y z 5 y 4 6 b. 8v w v 4 6v w 4 8v w w 4 6v v 4 6v w 6 8 v 4 w 6 v Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
12 c. + ( + ) ( ) + ( + )( ) ( + )( + ) (5 ) 5. P(blue, blue) ( 5 + )(5 +). The probability is about.. THINK AND DISCUSS, PAGE 655. Possible answer: Dividing by a polynomial is the same as multiplying by the reciprocal of the polynomial.. EXERCISES, PAGES GUIDED PRACTICE, PAGE h j h k j h k h 4 j k h j k 6h 5j k ( ) 4( + ) 6( + ) ( ) ( + ) ( + ) ( ) c 4 d c 5a c d 5a c 4 d c 4 d a 6. 4y y z y z ab c a c a b c 6. p q 5 p 4 q 5p q 8 p 6 q 4 6pq p 5 q 7. ( 4y + 8 y 4) 4y + 8 y 4 (y + )(y ) 4(y + ) (y ) y (5 + ) ( + )(5 + ) m ( m 7m ) 6m + 8 m 6m + 8 m 7m m (m )(m + ) 6(m + ) m(m ) m m 4p. ( 8p + 6 p 5p 4) 4p 8p + 6 p 5p 4 4p (p 7)(p + ) 8(p + ) p(p 7) p 7p. c ( c c ) 4c + 4 c 4c + 4 c c c (c )(c + ) 4(c + ) c(c ) 4 c + c ( + )( + 4) ( 4)( + ) 4 ( + )( + 4) + j 5. j 5j + 6 j 4j + j 4 j (j )(j ) (j )(j ) (j ). a ( a + a + 5) a a ( a + a + 5) a + a + 5a. a + 6ab b 5 + a a b + 5ab a(a + 6b) 5 + a b ab(a + 5) a + 6b b Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
13 6. p + 4pq p 6 q 8 q p ( p + 4q) p ( p + 4q) ( q 4) ( q 4) q q p q + q 4 4 p 6q q 7. r + 5r + 4 r + 8 r 6 r + (r + )(r + 4) (r + 4) (r + 4)(r 4) r + (r + 4) r 4 r + 8 r 4 9. a 4 b a a c 8 c 4 a 4 b 4 8 c a c a c 4 a 4 b c 4 4 a 4 c 4 b c ( 5) ( )( + 5) ( ) 8. y 8 y y + y 49 (y 8)(y + ) ( y ) ( y 49) y 6y 6 y 4 5 y m + m 4 m + 4 m m m m + m m m m 4 m + 4 m ( m + ) m m(m ) (m ) 4(m ) m m 4 ( + 5)( 5) 4 ( m + ) ( 5) 5 a. Let be the number of black tiles, then there are + white tiles, and + ( + ) + tiles in total. P(black, white) ( + ) ( + )( + 9) 5(5 + ) b. P(black, white) ( 5 + )( 5 + 9). The probability is about.. PRACTICE AND PROBLEM SOLVING, PAGES p 6 q p 7 r r p 8 q 7 r 4 4. r t r s t 6s t 8 r 4 s 6 r 4 s t 48 r 4 s t 8 5. y + 5 y + (y + ) (y + 5) y + y m 8m m + 6m 6 4 m 8m 6. ( m + 7m 8) m + 7m 8 m + 6m 6 4m(m ) (m )(m + 8) (m )(m + 8) 4m(m ) 4 m 4m ( 4 6 ) ( a + 4a + ) a + 6 a + 6 a + 4a + (a + )(a + ) (a + ) (a + ) a ( )( + ) ( ) ( + ) + 6 n + 8n 9. n n + 9n + 8 n + 6 6n(n + ) (n )(n + ) (n + )(n + 8) (n + ) n(n ) n + 8 n n n + 8. a b a + 4b 5 a + a b 6 a b + 6 a b a b (a + b) 5 a (a + b) 6 a b(a + b) 5 a (a + b) 5 a + 5 a b. t 5t t (t + )(t ) 5(t + ) 5 (t ). 6 j k 5 4 j k 5j j 6 j k 5 j 5j 4 j k 8 j k 5 j 4 k 9 k j Copyright by Holt, Rinehart and Winston. 44 Holt Algebra
14 . a 4 ( 8a a ) a a 4 8a a a a 4 a 8a a a 4 a a(4 a) a ( + )( ) 6 ( + ) 4( ) 4 + 5a. P(red, blue) ( + 4) ( + ) ( + 4) ( + ) 4( + ) 4( + ) 4(4 + 8 ) b. P(red, blue) 9 4(4 () + 8() ) 6 The probability is Student A is incorrect because individual terms (rather than factors) were divided out y 4 ( ) ( ) 8 y y 6 y y 6 y y 4 y y 4 4 y 4 y y 5 y 4 y y 5 y 4 y 5 y y 4 y y y 4 4 y + y ( + )( ) y ( + ) y 4( ) b; the other epressions are all equivalent to y, but b simplifies to 4. y 8. 5 p q 9. 6 m 8m m 9 p q p m + m m + 4m + p q 6 m 8m + 4m + p 4 q q p m m + m m 9 6m(m ) (m + )(m + ) m (m + ) (m + )(m ) m ( )( ) 4( ) 5 ( 4 6 ) ( + )( ) 4 4( + )( ) 6 4. m m 6m w 4 7 w m 4 m 4m w m m m 4 m 4m w 6m 66 w m( m) m(m 4) (m + ) 6(m ) m (m 4) 4(m + ) m 4 m 4m Multiply by 4m. Then divide out the common m factor m to get a final answer of 4. 45a. I O y b. I I 6 4 I (6)(4) I 64 The height of the image is 64 cm. I O y 64 6 y 4 y (4)() y 8 y The distance between is 8 cm. c. M I 64 cm O 6 cm 4 The magnification of the lens is A; t + 4 t (t + 4) (t + 4) 9 (t + 4 ) C; b a 6a b 5 a b 4 b a ab 5b 7 4 Copyright by Holt, Rinehart and Winston. 45 Holt Algebra
15 48. C; ( + 5)( ) ( 8) ( 4)( 6) ( 4) ( 6)( + ) ( + ) + 9 CHALLENGE AND EXTEND, PAGE ( + 4) ( ) + ( ) ( + 4)( ) ( ) ( + )( ) + + ( + ) 5. c + 5 c 4 c + 6c + 5 c + c + 5 c + 6c + 5 c 4 c + c + 5 c + c 4 c + 6c + 5 c + 5 (c + )(c ) c + (c + )(c + 5) (c )(c + ) c c 5. y z y z y y z z y z z y 4 yz y z z a + a + 6a a + a + 5 a + a + a + 6a + 5 a + 5 a + a + 5 a + 6a + 5 a + a + (a + )(a + 5) a + 5 (a + ) a + SPIRAL STANDARDS REVIEW, PAGE Let represent number of etra laps (a + ) m + m m 9 Since money cannot be negative, m y 7 and y (both have slope ) 59. y 5 7 and y (both have slope 5); y + 5 and y 7 (both have slope ) 6. y 7 and y (both have slope 7) ( + )( ) ( ) Ecluded values: 4 4 ± ( 5) ( 4) ( + )( ) Ecluded values: ± ( ) 6. ( + 4) already simplified Ecluded values: Copyright by Holt, Rinehart and Winston. 46 Holt Algebra
16 5 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS, PAGES CHECK IT OUT! PAGES a. n n + n n n + n 4n n n a. 5a + a 4 a 4 a 4 5a + (a 4) y b. y + + y y + y + y y(y + ) y + y + y a + 6 a 4 a 4 (a + ) (a )(a + ) a b + 4 b. b + b + b 4 b + b 4 b + 4 (b + ) 4b + b + b 4 b + b 4 a. 5 f h 5 f f h 5f h 5 f h h LCM 5 f f h h 5 f h b. 4 ( 6)( + ) ( 6)( + 5) ( 6)( + 5) LCM ( 6)( + )( + 5) 4a. 4 d d b. a + 4a + 8 d 4 d( d d d ) ( a + a 8 a a(a + 4) d ) (a )(a + 4) + 8 a 8 d 6d a a + 8 a 6 d 6 d a d 6d a 6 d d(4d ) 6 d 4d d 5. Katy s rate against the current is 5, or 4. Katy s rate with the current is 5 +, or 6. total time ( ) + 6( ) + 5 When, 5 5 () 5 4 It will take Katy 5 h or.5 min to kayak the round 4 trip. THINK AND DISCUSS, PAGE 66. Possible answer: Factor the denominators. Then write each factor the greatest number of times it appears in the denominator.. EXERCISES, PAGES GUIDED PRACTICE, PAGE 66 y. + 5y y y y + 5y 6y y y y 4m +. m m + 8m + 5 m + 5 4m + + m + 8m + 5 m + m + 5 m + 5 m + 5 (m + 5)(m + 7) m + 5 m ( + 4)( 4) a 5a 6 a + a + a + a + 7a (5a 6) a + 4 a + a + a + a + (a + ) (a + )(a + ) a ( + ) ( + ) + 7. y y y 6 yz y z LCM y y z 6 y z Copyright by Holt, Rinehart and Winston. 47 Holt Algebra
17 ( + 4)( + 5) ( + 5)( 4) ( + 5)( 4) LCM ( + 5)( + 4)( 4) 9. y 6 (y + 4)(y 4) (y + 9)(y 4) (y + 9)(y 4) LCM (y + 9)(y + 4)(y 4). c 4 c c ( ) 4 c 9 c 4 c 5 c ( ) a. total time r + 4.5r r ( ) r r r ( + ) ( + )( + ) ).5r( 6 r b. total time 6 6 r The total travel time is 6 h. 4 6 PRACTICE AND PROBLEM SOLVING, PAGES y + 4y y y 4y + 4y 8y 8 y y 5. a a + + a a + a + a a + a y a + a + (a + )(a ) a + a ( ) ( )( ) 4 7. m m 6 6m m 6 m 6m m(m 6) m 6 m 6 m c + 8. c c 5 4 c 5 c + (c + 8) c 5 4 c 5 4 c 5 c 5 (c + 5)(c 5) 9. a 9a a c a 4a + a a 9a (5 a 4a + ) a a 5a a (a + )(a ) a a +. 4j k 4 m j k k k k m 5jm 5 5 j m LCM 5 5 j k k k k m j k 4 m. a + 4a 4a(a + ) 7a + 9 9(a + ) LCM 6a(a + ). p p p(p ) pq r p q r r LCM p q r r (p ) pq r (p ). 5 y z 5 y y z y 5 y y y LCM 5 y y y z y z ( ) LCM 5 7 ( ) 5 ( ) 5. y + 7y + (y + )(y + 5) y + 9y + (y + 4)(y + 5) LCM (y + )(y + 4)(y + 5) ( ) (6 + 5) ) ( 5 y y 7. y y 4y + y 9 y(y ) (y )(y ) y y 9 y y y y 9 y y ( ) y (y ) y (y ) y (y ) y (y ) (y ) y + (y ) Copyright by Holt, Rinehart and Winston. 48 Holt Algebra
18 8. t t 4 t t t t 4 t t ( ) t (t + 4) t 4 t 4 t + (t + 4) t 4 t + 4 t ( ) + + ( ) 5 + ( + ) ( ) ( ) 5 ( + ) ( ) 4 ( ). m 4m 8 m m 4m + 4 m 4(m )( m 9. m ) m z + 4 z 7z z + 4 7z z( 7 7z( 7) z + z 9 z ( 4 (m ) m(m ) 4 m 4(m ) 4(m ) m(m ) 4 m 4(m ) m 6m 4 m 4(m ) m 6m 4(m ) a. total time w +.85w w( 7) +.85w( ) 7 7w + 7w 7 7w b. When w, 7 7w 7 7() 7 5 It took him 7 h or 44min to complete his 5 walking. a. total time r r 5 5 5) + 45 r 5 r( r r 7 r b. When r 5, 7 7 r 5 4 The total travel time is 4 h. c. Divide the total distance (5 mi) by the total time. 4) ) y + y y c 49 c 49 c + y (5 + y) 7 c 5 + y 5 + y 7 c 49 c (7 c)(7 + c) 7 + c 6. 6a a b a + 6a a + 4 a( b ) b + b 6a a + b 4 a + ( 6a 4 b b ) a 8. r + r r + r + 9 r + r + r (r + 9) r r 9 r + b + b b (r + )(r ) r + r (8 5) ( 5 ) ( 5) y 4 y y ( y y 4 y y ( y y) y 4 y y y 4 y y y + 4) ( + ( + 4) + ) ( + )( + 4) + 6( + ) (+ )( + 4) ( + 4) + 6( + ) ( + )( + 4) 8 + ( + )( + 4) y 4. y 9 y + y 9 y (y )( y + y + ) y + (y + )(y )( ) y(y + ) (y + )(y ) (y + ) (y + )(y ) y(y + ) (y + ) (y + )(y ) y + y (y + )(y ) 4. Student A is incorrect. The student may have subtracted and then simplified before setting the denominator equal to. 4 is also an ecluded value. Copyright by Holt, Rinehart and Winston. 49 Holt Algebra
19 44. P + π () () + π () ( + π) 9 + π 9.46 The probability of winning a prize is about Possible answer: a. He subtracted from both sides of the equation. b. He subtracted from by finding a common denominator () ; 8 ; In this case, either binomial can be used as the least common denominator. After selecting the binomial for the denominator, change the other binomial to match by multiplying it by. 49. A; 6 p + 6 (p + ) p + 5. A; 4 5. D; ( ) ( + )( ) ( ) ( + )( ) ( + )( ) ( ) ( + )( ) + 4 ( + )( ) ( + ) ( + )( ) 5a. r + 5 ; time to post office: r r ) + 5 r 9 r + 5 r b. r + 5 r r ( 4 r c. When r, 4 ; time to library: 5 r r 4 () 4 9 It took her 4 h or h min to bike. 9 CHALLENGE AND EXTEND, PAGE y + y y + y( y y) + y ( + y)( y) ( y) ( + y)( y) + y ( + y)( y) ( y) ( + y) ( + y)( y) 4y ( + y)( y) ; ±y 54. m m 5m m( 5m 5m) + 4 ( m ) + 5m( m m) 5m m m m m 5m m m 9m + 4 ; m m 55. a y + b z + c yz a y( z + z) z( b y y) + yz( c az yz + by yz + c yz az + by + c ; yz, y and z ) 56. y y y y y ( SPIRAL STANDARDS REVIEW, PAGE 665 y ) a + a a 6 5 a a + 6a ( a 5 a + a ) ( 5) + ( 5) (polynomial does not ( + )( 5) factor further) 59. 5h + 5 h h 6 5 h 5h 6 5( h h ) 5(h 4)(h + ) (since (4)(), 4 + ) 6. s + 8 s 4( s + s ) 4 s (s + ) 6. + m + m 5m ( m 5) + m( m 5) (m + )( m 5) (polynomial does not factor further) t 4 t 4t 4(4 t t ) 4t(4 t t ) 4t(4 t 4t + t ) 4t[4t(t ) + (t )] 4t(4t + )(t ) Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
20 6. d 4d (d 6)(d + ) d 6 or d + d 6 or d 64. g 9g 4 g 9g + 4 (g )(g 4) g or g 4 g or g ( + ) t 8 t 4 ( t 4) t 4 ; t ± n + 5n n + n 6 n(n + 5) ( 4) (n )(n + 5) ( + 4)( 4) n ; n 5 and n n + 4 ; ±4 ALGEBRA LAB: MODEL POLYNOMIAL DIVISION, PAGE 666 TRY THIS, PAGE There is no arrangement of tiles that will form a rectangle with a side length of + because + is not a factor of DIVIDING POLYNOMIALS, PAGES CHECK IT OUT! PAGES a. (8 p 4 p + p) (4 p ) 8 p 4 p + p 4 p 8 p 4 p + p 4 p 4 p 4 p p + p b. (6 + 5) a. + 7k + k k + (k + )(k + 5) k + k + 5 c. s + s + 6 s + 6 (s + 6 ) s + 6 s + 6 b. ( a 8a + ) (a 6) a 6 a 8a + a a 6 a 8a + ( a 6a) a + (a + ) a 4a. ( m + 4m ) (m + ) m + m + 4m m 5 m + m + 4m ( m + 9m) 5m (5m 5) m 5 + m + b. ( y + y + ) (y ) y y + y + y + 6 y y + y + ( y y) 6y + (6y 8) y y b. b 49 b + 7 (b + 7)(b 7) b + 7 b 7 a. ( y 5y ) (y ) y y 5y y + y y 5y ( y 6y) y (y ) y + Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
21 5a. ( 4 + ) ( ) ( 4 + ) ( ) ( ) + ( + 4) 4 + (4 + 8) b. (4p + p ) (p + ) ( p + 4p ) (p + ) p + p + p + 4p p p + 6 p + p + p + 4p ( p + p ) p + 4p ( p p) 6p (6p + 6) 7 p p p + THINK AND DISCUSS, PAGE 67. Possible answer: It means the binomial is a factor of the polynomial.. ; the denominator of the remainder cannot equal zero.. EXERCISES, PAGES 6767 GUIDED PRACTICE, PAGE 67. ( 4 ) 4 4. ( 6 a 4 4 a ) 4a 6 a 4 4 a 4a 6 a 4 4a 4 a 4a 4 a a. ( b 4b + 4) b 4. (8 r r + 6) 6r b 4b r r + 6 b b b 4b b + 6r 4 8 r b 7b r r 6r + 6 6r r + b r 5. ( ) (5 m m ) 5 m 5 m m 5 m 5 m m 5 m 5 m 5 m m + m m 8. a a a 4 (a 4)(a + ) a 4 a ( )( + ) y + y y (y )(y + 5) y y + 5. t 6t t (t )(t 4) ( + )( + 5) t t +. p p. ( c + 7c + ) (c + 4) p + 4 (p + 4)(p 5) c + 4 c + 7c + p + 4 c + p 5 c + 4 c + 7c + ( c + 4c) c + (c + ) c + 4. ( s s 5 ) (s 5) s s s s + s 5 s s ( s 5s ) s 5 ( + 7) 4 (s 5) ( 4) s + Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
22 ( ) 6 (6 ) ( r + r + 5) (r ) r r + r + 5 r + 7 r r + r + 5 ( r 6r ) 7r + 5 (7r 5) 56 r r 9. ( n + 8n + 5) (n + 4) n + 4 n + 8n + 5 n + 4 n + 4 n + 8n + 5 ) ( n + 4n) 4n + 5 (4n + 6) n n + 4. ( t t + 4) (t ) t t t + 4 t + t t t + 4) ( t t ) t + 4 (t ) 5 t t. ( 8 n 6n 7 ) (n + ) n + 8 n 6n 7 4n 5 n + 8 n 6n 7 (8 n + 4n) n 7 (n 5) 4n 5 + n + 7. ( a + 4a + ) (a + ) a + a + 4a + a + a + a + 4a + ( a + a) a + (a + 4) a + + a +. ( b b + ) (b + ) b + b b + b b + b b + ( b + b) b + (b 6) 7 b + 7 b +. ( ) ( + ) ( + ) ( + ) ( 6 ) 6 + (6 + 8) 5 (5 45) ( p p 4) (p ) p p p + p 4 p + 4p + 8 p p p + p 4 ( p 6 p ) 4 p + p (4 p 8p) 8p 4 (8p 6) p + 4p p 5. ( m + ) (m ) m m + m + m + m m + m + ( m + m) m + (m ) m + + m Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
23 6. ( + 4 5) (5 + ) (4 + 5) ( + 5) (4 + ) 7 + ( 7 85 ) 85 5 ( ) ( 4 k k 8 ) (k + ) k + 4 k k 8 4 k 4k + k + 4 k + k k 8 (4 k + 4 k ) 4 k k ( 4 k 4k ) k 8 (k + ) 4 k 4k + + k + 8. ( j + 6j + ) (j + 4) j + 4 j + j + 6j + j 4j + j + 4 j + j + 6j + ( j + 4 j ) 4 j + 6j (4 j 6j) j + (j + 88) 86 j 4j j + 4 PRACTICE AND PROBLEM SOLVING, PAGES (9 t + t 6t) t. (5 n n + 5) 5n 9 t + t 6t 5 n n + 5 t 5n 9 t + t 6t 5 n t t t 5n n 5n + 5 5n t + 4 n + n t. (6 p p + 8) 4 p 6 p p p 6 p p 4 p 4p + + p p 4 p. 8 t + t t (4t + )(t ) t 4t + 5. ( 5 + 6) ( ) ( ) + 6 ( + 6) 6. ( m + 8m + 8) (m + ) m + m + 8m + 8 m + 4 m + m + 8m + 8 ( m + 4m) 4m + 8 (4m + 8) m (6 a + 7a ) (a + ) a + 6 a + 7a a a + 6 a + 7a (6 a + 9a) a (a ) a 8. ( 8 ) ( 4) ( ) 8 ( 8) +. 4 r 9r + r (4r )(r ) r 4r 4. g + 7g 6 g + (g )(g + ) g + g Copyright by Holt, Rinehart and Winston. 44 Holt Algebra
24 9. ( + 6) ( ) ( 6 ) (4 8) ( m + 5m + 8) (m + ) m + m + 5m + 8 m + m + m + 5m + 8 ( m + m) m + 8 (m + ) 5 m m + 4. ( 6 ) ( ) ( 6 ) ( ) ( m 4m ) (m ) m m + m 4m m + 5m + m m + m 4m ( m m ) m 4m ( m 5m ) 46m (46m ) m + 5m + + m 4. (6 t + t + 9) (t + 9) t t + t + t + 9 t 6t + 5 t t + t + t + 9 (6 t + 8 t ) 8 t + t ( 8 t 54t ) 75t + 9 (75t + 5) 6 t 6t t ( p 4 7 p + p + ) (p ) p p 4 + p 7 p + p + p + p + p + 7 p p 4 + p 7 p + p + ( p 4 p ) p 7 p ( p 9 p ) p + p ( p 6p) 7p + (7p ) p + p + p p 45. Apply long division n n ( 4) 5 + n (5 ) n 47. ( ) ( + ) ( + ) 5 (5 5) ( ) ( ) + ( 4 ) ( ) + Copyright by Holt, Rinehart and Winston. 45 Holt Algebra
25 48. ( ) ( + ) + + ( + ) 4 (4 ) 49a. The values of y are negative and decreasing. b. The values of y are postitive and decreasing. c. The function is not defined at ( + 5 ) ( + 5)( 5) ( 5)( 9) ( 9)( + 5) (.88) 9 5. A B m + m + m +.5m + (m + )(m + ) (m + ) m + 5. Student B is incorrect. The second term should be positive. 5a. y y y b. The function is undefined at. 54. The binomial is not a factor of the polynomial ( + ) (4 + 6) Possible answer: + ; 76 ; when, +, so the results are equal ( + ) + 8 ( + 8) Yes; there is no remainder when you divide by +, so + must be a factor of C 58. B; ( ) ( + ) ( + ) ( 4) B; 6. A; ( 5) ( 5) + ( 5) ( + )( 5) CHALLENGE AND EXTEND, PAGE (6 y + 4 y ) ( y) 6 y + 4 y y 6 y y y y + y + 4 y y ( ) ( ) ( ) + ( ) 6 + (6 6) ( ) ( ) ( ) ( + + ) + + Copyright by Holt, Rinehart and Winston. 46 Holt Algebra
26 6. ( + ) ( ) + ( + ) ( + ) ( + ) ( + ) ( ) ( + ) ( + 8) ( + ) ( + ) ( + 4) ( ) ( + 4) So the height is ( + )( + ) ( + ) + base height ( + 4) ( + ) The base is m longer than the height. 66a. (π ( )) (π ( + + )) ( ) ( + + ) ( + + ) ( ) H V B + The height of the cylinder is ( + ). b. B π r π ( + + ) r + + r ( + ) r + The radius of the base is ( + ). SPIRAL STANDARDS REVIEW, PAGE Let w represent number of weeks. 8 + w + 4w 6 + w 4w 6 w w It takes weeks. 68. Let represent price. 5 5 or Maimum price is $55; minimum price is $ ( ) y ( ) 4 ( ) + 4( + )( + ) + 4( + ) 7. k + 4 k k + k ( + k) k + ( + k)(k + ) k + + k + 4k k + 5k + ( + ) 4 + k + k + k (k + )(k + ) k 8y 4 7 y 6 7 y 6 7 SOLVING RATIONAL EQUATIONS, PAGES CHECK IT OUT! PAGES b. 4 h + h 4h (h + ) 4h h + h a. n n + 4 (n + 4) n n + 4 n n c. 7 ( 7) 7 6 a. a + + a + 4 a a(a + ) ( a b. j + 4 j j 6 j + j j j(j + ) ( 6 j + a) a + ) a(a + ) ( 4 a + a 4(a + ) a 4a + 4 a 4 j ) j ) j(j + ) ( 6j (j + ) (j + ) 6j j j + 4 6j 4 j 4 Copyright by Holt, Rinehart and Winston. 47 Holt Algebra
27 a. 7 7 ( 7) ( )( 7) ( 5)( 7) 5 or 7 5 or The only solution is 5, so 7 is an etraneous solution. b. + 4 ( + )( ) 4( ) ( )( 5) or 5 or c The solutions are and 5, and there are no etraneous solutions ( + ) ( 4) or 4 or () (4) The only solution is 4, so is an etraneous solution. THINK AND DISCUSS, PAGE 676. Possible answer: Some solutions may be etraneous and make the epressions undefined.. and. Possible answer: Because the denominators are the same, the epressions would only be equal when 4. Since is an ecluded value, there is no solution. 4. EXERCISES, PAGES GUIDED PRACTICE, PAGE 676. rational equation. etraneous solution s 6 4 s ( + 4) 5s 4(s 6) + 8 5s 4s 4 8 s 4 5. p + p (p) (p + ) 4p p 5p p ( ) 9( 4) ( + ) (. 8 d d(d + ) ( j 8. + ) ( + ) ( d + d j + 4(j + ) j 4j + 8 j j 8 j ( ) a a 9 5 a 9 ( 5) a 5 + ) ( + ) + ( + ) d) d(d + ) ( 8(d + ) d (d + ) 8d + 6 d d 6 d d 5 d) d + Copyright by Holt, Rinehart and Winston. 48 Holt Algebra
28 . s 6 4 s + s s(s 6) ( 4. 7 r + r(r ) ( 7 5. a 4 s + s) 6s 8(s 6) + (s 6) 6s 8s 48 + s 6 s 54 s 8 s 6) s(s 6) ( 4 r r r + 4(r ) + 4r (r ) 4r 4 + 4r r + 9r 5 a a r ) r(r ) ( r 5 9 (a ) a(a 4) a 6 a 4a a 7a + 6 (a )(a 6) a or a 6 a or a 6 r 5 6. r 6 6r ( r r ) 6r ( 5 6) r 5r r 5r (r )(r + 4) r or r + 4 r or r n 7 n 8. n ( 6 n n) ( 7 n ) 6n 7 n n + 6n 7 (n )(n + 7) n or n + 7 n or n ( )( + ) 4 6 ( )( + ) or + or r ) a a + a ( 5 ) a ( 4 a a + ) 5 4a + a a 4a 5 (a 5)(a + ) a 5 or a + a 5 or a. p + p p ( p) ( p p ) + p + p p p (p + )(p ) p + or p p or p. c 4 c c 4 (c 4) (c )(c 4) c c 5c + 4 c 8c + 6 (c 4 ) c 4 c 4 4 c 4 c c c 4 There is no solution, and 4 is etraneous.. w + w w w ( w ) ( w + w w w ) ( w ) () (w + ) w(w + ) w w + w w w w + w 4 (w )(w + 4) w or w + 4 w or w 4 w + w w w w + w w w + ( 4 ) ( 4 ) The only solution is 4, and is etraneous. Copyright by Holt, Rinehart and Winston. 49 Holt Algebra
29 ( 5) ( ) ( 5) 8 ( 5) ( 7) + ( 5) ( 5)( + 6) 5 or or (5) (6) The only solution is 6, and 5 is etraneous. PRACTICE AND PROBLEM SOLVING, PAGES n n 8( + ) ( ) n (n ) n 9n 6 n n ( ) ( + 4) s s 5 4 s 5 4 5s s c c 4 c c(c ) ( 7 c 7(c ) (c ) 4c 7c 7 c + 4c c 5. 9 m m 5 m m ( 9 m m) m ( 5 m ) 8 5 There is no solution; is etraneous ( + 5) ( + 4) 8 ( 8) There is no solution; is etraneous. c ) c(c ) ( 4 c ). ( ) ( ) ; is etraneous.. r 6 r 6 r ( r ) r ( r ) r 9r 8 r 9r + 8 (r )(r 6) r or r 6 r or r ( 6 ) ( + ) + + ( )( + 4) or + 4 or 4 8 ( 8 ) ( ) ( )( 4) or 4 or ( + 4)( ) ( + ) ( 5)( + ) 5 or ; and are etraneous 7. + ( + ) ( + ) + + ; 4 is etraneous ( ) ( 5 ) ) ( ) ( ( ) () There is no solution, and is etraneous. Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
30 9. t t t + 4 t t(t ) (t + 4)(t ) t 9t t + t t t + (t )(t ) t or t t or t t t t + 4 t t t t + 4 t () + 4 () The only solution is, and is etraneous ( ) ( + ) + ( )( + ) or + or () The only solution is, and is etraneous ( 4) There is no solution, and 4 is etraneous ( + )( + ) ( + 7) ( + 5)( ) 5 or ; is etraneous ( ) ( + ) ; is etraneous b. 9 + f % 9% 45 + f 9 + f 45 + f f.9(45 + f) 9 + f f.f.5 f 5 Clancy would have to make 5 free throws. 45. Karla Andrew Books 8 Stacks Books per Stack ( ) + 6()( ) 8() ( 4) 6( 4)( + ) 4 or ( and 6 are etraneous) Since >, 4. Karla has 4 stacks of books. 46a y b y y ( 5) y ( 4 + y ) 8y 5y + y y 4 The image will apear 4 cm aways from the lens. c y 7y ( 8) 7y ( 4 + y ) 4y y + 7 y 7 The distance between will increase to 7 cm. 47. No; you can only use cross multiplication if the equation is in the form of a proportion ; possible answer: first add the fractions on the left to get 4. Then cross multiply and solve to find that 4. This method may be easier than finding the LCM. 44a. 9 % % Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
31 49. B; ( 4) ( 6) ( + 4) ( + 6) ( + 4) 8 or + 4 or (4) (4) The only solution is, and 4 is etraneous. 5. A; + ( ) ( + ) ( ) ( ) ( + ) ( ) ( + ) or () + + () () The only solution is, and is etraneous. 5. D; 5 + ( 5 ) ( + ) ( )( + 5) or + 5 or 5 CHALLENGE AND EXTEND, PAGE Statements Reasons a. ( + 4) 6 Cross Products Property b. + 6 Distributive Property c. Subtraction Property of Equality d. 4 Division Property of Equality a 7 a ( + 4)( a) 7( a) + (4 a) 4a 7 7a ( + a) + a ( )( a) or a or a Since the equation has no solution, a (j)(j + ) j + j j + 4 (j)(j + ) + 4(j)(j + ) j + j + + j 4 j + 4j 4 j 8j j 9j 5 (j + )(j 5) j, 5 6 a + 8 a + 4 6(a)(a + ) + 8(a)(a + ) 4(a)(a + ) a (a + ) 6a a 4 a + 4a 4 a a 6 a 5a 8 (a + )(a 8) a, 8 Since age must be positive, j 5 and a There is a year difference between Jill s and Angela s ages. Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
32 SPIRAL STANDARDS REVIEW, PAGE y + and y are parallel. 56. y and y + 4 are perpendicular. 57. y and y + are parallel; y is perpendicular to both y and y + and y + are perpendicular. 59. y () + y () () + (): y 6 + ( + y ) Substitute in (): (4) y 7 y 7 y (4, 7) 6. y 7 () 4 y () () + (): 9 + 6y 8 + (8 6y 4) Substitute in (): (5) y 7 5 y 7 y y 6 (5, 6) a 7, b 5, c discriminant (5 ) 4(7)() > solutions 6. y 8 () + y 5 () () (): 4y 6 ( + y 5) 7y y Substitute in (): () (, ) 8 APPLYING RATIONAL EQUATIONS, PAGES CHECK IT OUT, PAGES Let m represent number of minutes. Cindy s part + Sara s part whole lawn 5 m + 4 m ( 5 m + 4 m ) () 4m + 5m 9m m 9 min, or about min s 9. Alcohol (ml) Total (ml) Original 5 5 New 5 + a 5 + a 5 + a 5 + a a.8(5 + a) 5 + a 4 +.8a.a 5 a 75 The chemist should add 75 ml of alcohol.. Distance (mi) Time (h) Rate (mi/h) Ryan t + t + Maya t t t + t (t) (t + ) (t)(t + ) (t) (t + ) (t)(t + ) t t + t t t + t (t 5)(t + 6) t 5, 6 Maya makes the trip in 5 h. THINK AND DISCUSS, PAGE 68. Danielle can weed the garden by herself in h, so with Omar s help it makes sense that they finish in less time than h.. EXERCISES, PAGES GUIDED PRACTICE, PAGE 68. Let h represent number of hours. Summer s part + Louise s part whole room h + 5 h 5 ( h + 5 h ) 5() 5h + h 5 8h 5 h h, or about h 5 min 8 Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
33 . Chicken (qt) Vegetable (qt) Original New + a + a ( + a) () + a 6 a 4 The chef should add 4 qt of chicken stock.. Distance (mi) Time (h) Rate (mi/h) Connor t t Matt t t + t t + t t t ( + t)(t ) (t) t + t t t t t (t 4)(t + ) t 4, It takes Connor 4 h. PRACTICE AND PROBLEM SOLVING, PAGES Let m represent number of minutes. old vacuum s part + new one s part whole room 9 h + 45 m 9 ( 9 m + 45 m ) 9() m + m 9 m 9 m It takes min. 5. Let h represent number of hours. ft pipe + ft pipe whole reservoir h + 6 h ( h + 6 h ) () h + h h h 4 It takes 4 h to fill the reservoir. 7. Distance (mi) Time (h) Rate (mi/h) Local 6 t + 6 t + Epress 6 t 6 t t + t 6(t) + 5(t)(t + ) 6(t + ) 4t + (t)(t + ) 4(t + ) 4t + t + t 4t + 48 t + t 48 (t + 8)(t 6) t 8, 6 The epress takes 6 h, so it travels at 6 6 mi/h 6 8a. Distance (mi) Rate (mi/h) Time (h) Biking r r Running r 4 r 4 b. + r r 4 4 c. (r 4) + (r) 4(r)(r 4) 5(r 4) + (r) (r)(r 4) 5r + r r 4r r r + (r )(r ) r, r is etraneous since r 4 >. Jen s rate while biking is mi/h. 9a. Area ( m ) Length (m) Width (m) Rect. A 96 l 96 l Rect. B 96 l 96 l 48 l b. 96 l 48 l + 4 c l 48 4l l Length l m Width 96 l 96 8 m 6. Orange Juice (c) Total (c) Original 8 New + a 8 + a + a 8 + a.4 + a.4(8 + a) + a. +.4a.6a. a Maria should add c. Copyright by Holt, Rinehart and Winston. 44 Holt Algebra
34 . machine A: min; machine B: 5 min; machine C: 8 min; machine D: min Let represent number of minutes with machines A and D. machine A + machine D whole job + 6 ( + ) 6() Machines A and D take 7.5 min. Let y represent number of minutes with machines B and C. machine A + machine D whole job 5 y + 8 y 9 ( 5 y + 8 y ) 9() 6y + 5y 9 y 9 y 8. Machines B and C take about 8. min. Machines A and D are faster.. Let n represent the number. n + n 5 4 n 4n (n + 4n n) ( 5 4 n ) 4 n n 4 n ± n The number is or.. B is incorrect. The correct rational equation is h 4 + h 6. a. f + with f, y y y + y (y )(y) (y )(y) + (y ) y y y y y 4y y 4y + (y )(y 4) y, 4 Since 4 6, y. b. (y)(y + ) (y) + (y + ) y + y y + y + y 4y (y )(y + 6) y, 6 Image is cm from lens. (y )(y) y 4. It will take half the time it takes them to mow the lawn individually, or 4 min. This makes sense since they split the work evenly. 5. Student responses will vary. 6. A; New solution has (4 + w) ml of water and (8 + w) ml total volume; so 4 + w 8 + w.7 7. B; m 9 ( m + m + m 45 45) 9() m + m 9 5m 9 m 8 CHALLENGE AND EXTEND, PAGE Let h represent number of hours. drain A + drain B + drain C whole tank h + 6 h + 4 h ( h + 6 h + 4 h ) () 4h + h + h It takes 9h h 9 4 h, or h min. 9. Rate ( job Time (h) h ) Luke t t Eddie t t Ryan t + t + Luke s part + Ryan s part + Eddie s part whole job ( t ) 4 + ( t) 4 + ( t + ) 4 4 t + t + 4 (t + ) t(t + ) ( 4 t + t + 4 t(t + )() (t + )) 4(t + ) + (t + ) + 4t t(t + ) 4t t + + 4t t + t t 7t 6 (t + )(t ) t, Luke: t h; Eddie: t () 6 h; Ryan: t + () + 4 h Copyright by Holt, Rinehart and Winston. 45 Holt Algebra
35 . Cranberry (c) Total (c) Original V V New V + 6 V + 6 V + 6 (V + 6) 6 ( V + 6 ) 6 ( (V + 6) ) V + 6 4V + 4 V There were c of punch to start with. SPIRAL STANDARDS REVIEW, PAGE y 4 4() + (5) yes. 4 + y 4 4() + () yes y 4 4(9) + () yes () + ()() + () ( + ) yes () + ()() + () ( ) yes. 4 + y 4 4() + (8) no y 4 4(5) + () no y 4 4(4) + () no (4 ) + (4)() + ( ) ( ) 9 (4 ) 8 no no; 49 (7), and no; 5 (5), and 7 is not a factor of 4 5 is not a factor of no; 6 (6 ), and 6 is not a factor of ( + )( + ) + + ( ) ( 8)( + ) 8 + ( 8) ( + )( + 4) ( ) ( )( 6) 6 ( ) ( 7)( + ) 7 + ( 7) ( + 5)( 8) ( 5) CONCEPT CONNECTION, PAGE y y ( ) y ( + y ) y y + ( )y y. y Undefined The yvalues are positive for >.. 4. M 7.5 cm 5 cm.5 The magnification of the lens is Magnification remains the same. Copyright by Holt, Rinehart and Winston. 46 Holt Algebra
36 READY TO GO ON? PAGE 685. n + n 5 ( n 5n) n + n 5 n 5n n + n(n 5) n 5 n(n + ) n + n. h 6h g h ( h ) g h 4 g 6h 5 g 4g g h g 4g g ( h ). 6 y 6 y y y 6 m 8m + 6 m 4. m + m m m 8 (m + )(m ) (m 4)(m + ) (m 4 ) (m ) m 4 5. n 6 n n 5 5 n n n 6 n n b c b c 8 b c b c 8 b c b c 8 b c b c b + (4 b + 4b) 4 b + 4b 4 b + 4b 4b(b + ) 9. m + 5 m 4 m 5 4 m 5 m + 5 m 4 m 5 8 m 4 m 5 m 4 y ( + )( + ) ( ) ( + ) ( + )( ) p p 5 p p p ( + 4) ( )( + 7) + 7. t + 4 t t t + t t t 5 t ) t ( ( ) 4 4. (6 d + 4d) d 6 d + 4d d 6 d d + 4d d d + 6. ( 7 4 ) ( + ) ( + )( 4) + 4 m + 6m + 5 m + 5 (m )(m + ) (m + 5)(m + ) m + 5 m m + 5 m + 5 m m + 5 m 4 m + 5. m m 5. (5 4 + ) ( ) ( a + a ) (a ) 8. (4 y 9) (y ) a a + a y 4 y + y 9 a + 5 y + a a + a y 4 y + y 9 ( a a) (4 y 6y) 5a 6y 9 (5a ) (6y 9) a + 5 y + 9. ( + 5 8) ( + ) ( ) ( + 4) 8 ( + ) Copyright by Holt, Rinehart and Winston. 47 Holt Algebra
37 . ( ) or. t The only solution is, and is etraneous. t 4 t + t(t + ) ( t + t) 4 t(t + ) 4 ( t + ) 6(t + ) + 4(t + ) t 6t + + 4t + 8 t t t. 4 n 7 n + n ( 4 n ) n ( 7 n + ) 4 7n + n n + 7n 4 (n )(n + 4) n or n d + n or n 4 d d + 8 (d + )(d + 8) 6(d + 8) d + d + 6 6d 48 d + 6d + 64 (d + 8 ) d + 8 d 8 d + d d There is no solution, and 8 is etraneous ( 6)( 4) 4 ( 6 ) ( ) 5 or or 6. h + h 6 ( h + h ) 6() h + h 6 5h 6 h or. 5 It will take them. h to shovel the walk together. 7. current:.6() 8 ml water, ml total new: (8 + w) ml water, ( + w) ml total 8 + w + w w.7( + w) 8 + w +.7w.w w. Student must add ml of water. STUDY GUIDE: REVIEW, PAGES rational epression. rational function. rational equation 4. inverse variation 5. discontinuous function LESSON, PAGE Yes; the product y is constant. 7. No; the product y is not constant. 8. y 4 9. y. y y (5)(6) () y y 5 y 4. y ; ; y 4. y y ()() (5) 66 5 The price will be $,. LESSON, PAGE 687. y. y ; 4; y ; ; y 5. y ; 7 4 ; y 5 Copyright by Holt, Rinehart and Winston. 48 Holt Algebra
38 8.. D: > ; R: y > 9. LESSON, PAGE p r 7 5p r 7 p The ecluded value is. r 7 The ecluded value is 7.. t t t t t t(t ) t or t ( 5)( + ) 5 or + t 5 or The ecluded values are and. The ecluded values are 5 and ( 5)( + 5) 5 or or 5 The ecluded values are 5 and ( 4)( 7) 4 or 7 4 or 7 The ecluded values are 4 and r r 7 r 7 r r r r r r The ecluded value is. 8. k 6 k 9 k k k (k ) k 6 k 9 k k (k ) k or k k or k The ecluded values are and ( )( + 6) + 4 ( )( + 6) or + 6 or 6 The ecluded values are and 6. 9 ( ) ( )( + ) ( ). 6 ( )( + ) + 9 ( )( + ) or + or The ecluded values are and ( + 5) ( )( + 5) ( )( + 5) or + 5 or 5 The ecluded values are and ( + )( + 6) ( 5)( + 6) ( 5)( + 6) 5 or or 6 The ecluded values are 5 and 6. Copyright by Holt, Rinehart and Winston. 49 Holt Algebra
39 . A square A () circle π 4 4 π π LESSON 4, PAGE b 5. b 6 ( b b ) b b 6 b b b (b )(b + ) (b ) b(b + ) b + b 6. 5a b ab a b a b 5 a b 4 a b 5 b b + 8. b + b 4 b + b b 6 b + (b 4)(b + 6) b(b + 6) (b 4)(b + 4) b ( 9) ( + )( ) ( + ) 4( ) n 4 m n mn 6 n 4 m n mn 6 n mn 4 m n 48m n 4 4 m n n m b(b + 4) b + b + 8b ( )( + ) 4 ( + )( ) ( )( + ) 4( + )( ) LESSON 5, PAGE a b 5 a a b a b 5 a b b LCM 5 a a b b a b 4. 6 ( ) 5 5 5( ) LCM 5 ( ) ( ) 4. b b + 8 b b + 8 b 8p 44. p 4p + 8p p 4p n 5 n + 5 n n n 5 (n + 5) n n p 4p h + h h 5 h 5 h h + h h 5 h 5 h ( ) h + h h 5 h h 5 h + h ( h) h 5 h + 5h h 5 LESSON 6, PAGE (5 + 5) n n 5 n 5 (n + )(n 5) n 5 n b b 5 b 7 b b + 4 (5 b) 7 b 5b 7 b 47. 5m + m + m 5m( m m) + m + m 6m + m + m m 6m + m + m 7m + m 49. r + r r ( ) + r r + r + r 4 r 5. 8 ( + )( ) + 5. ( ) ( + ) ( + ) + 6 ( + 6) + Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
40 54. ( + ) ( 5) ( 5) 6 (6 ) ( b 4b + ) (b ) b b + b 4b + b + 6b + 8 b b + b 4b + ( b 6 b ) 6 b 4b ( 6 b b ) 8b + (8b 6) 8 b + 6b b LESSON 7, PAGE r 4r r 4 r, so there are no etraneous solutions b b y y 6( + b) 5b 8 + 6b 5b b 8 8 b b or, so there are no etraneous solutions. 6. ( ) ( ) ( )( + ) or + or 7y 6 y 6 y + 7y y(6y + 7) y or 6y + 7 y 7 6 y, so the only solution is 7, and is 6 an etraneous solution. ; so the only solution is, and is an etraneous solution ( )( + ) or + or, so there are no etraneous solutions ( + 4 ) + 6 6, so there are no etraneous solutions ) ( 6 ( + 4) ( ) 4 + ±, so there are no etraneous solutions. 64. b + 4 b b b 4 b 4 b b b, so there are no etraneous solutions ( 6) 8( 4) ( 4)( + ) 4 or + 4 or ±4, so the only solution is, and 4 is an etraneous solution. 5 + (5 ) ( + ) ( 4)( 5) 4 or 5 4 or 5, so there are no etraneous solutions. Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
41 ( + ) 4( ) ( )( + ) or + or, so there are no etraneous solutions m m 5 7 m 5 9m m 5 + m 5 7 9m + m 5 7 9m + 7(m 5) 9m + 7m 5 m 8 m 9 m 5, so there are no etraneous solutions ( 4)( ) ( 4) ( ) or or ±, so the only solution is, and is an etraneous solution. LESSON 8, PAGE Let h represent number of hours. pipe A + pipe B whole tank h + 8 h 6 ( h + 8 h ) 6() h + h 6 Tank fills in 7 5 5h 6 h h or 7 h min. 7. original:.4(4) 6 ml water, 4 ml total new: (6 + w) ml water, (4 + w) ml total 6 + w 4 + w w.5(4 + w) 6 + w +.5w.5w 4 w 8 Chemist should add 8 ml of water. CHAPTER TEST, PAGE 69. y 8. y + + ; and y 5. y + + ; and y ( + 4)( 4) ( + 4)( ) 4. y y (.6)(5) (.5) y 9.5 y y The club can buy posters. + 5 ; and y 5 4. y + 4 ( )( + 4) or + 4 or 4 The ecluded values are and ( + 5)( ) ( + 5)( 5) 5 5 ( 5)( + 5) 5 or or 5 The ecluded values are ±5. 6. b 4 b b b b b 4 b b b The ecluded value is. 8. b b 5 5 b (b + )(b 5) (b 5) (b + ) b 5 b b 5 The ecluded value is 5. Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
42 . 4 ( ) ( + )( ) 4 +. a b 5 a 5 b 8 a 4 a 4 b 4 a 4 b 4 b. 4 y 4 y y 5 y 4 y 4 5 y y y 6 5 y 6 6 y 5 y ( + )( 4) ( + 4)( ) ( + 4)( 4) ( + )( + ) ( + )( ) ( + )( + ) b 6b b 4 b + b 8b + b 6b 8b + b + b b 4 b(b ) 4(b + ) b (b + ) 6 b ( + 5)( ) ( + )( + ) ( + )( ) ( + 5)( 5) ( + )( + ) ( + )( 5) m m + m m + m 4 m m (b ) 6. b b 5b b + + 5b b + 7 5b 8. 5 (5 ) ) + (. y + 4 y + y y y + 4 y + y y( ) y + 4 y + y y y + 4 y y 4 y ( 4)( + ) ( w + 5w ) (w + 4) w + 4 w + 5w w w + 4 w + 5w ( w + 8w) w (w ) w 6. ( 4 + 9) ( + ) ( + ) (6 ) a. ( ) ( ) ( ) + ( ) ( ) + +. ( 8 t t ) t 8 t t t 8 t t t t 4t 4. k k 5 k + 5 (k 7)(k + 5) k + 5 k 7 Copyright by Holt, Rinehart and Winston. 4 Holt Algebra
43 b The width is 4 cm n ( ) 9( ) or, so there are no etraneous solutions.. n + n 4 n n + 4 (n + 4) n(n ) n + n n n 4n (n 6)(n + ) n 6 or n + n 6 or n n or 4, so there are no etraneous solutions. n 4 ( n 4) (n 4)(n + ) n 8 n n 8 n + n n(n + ) n or n + n n ±, so the only solution is, and is an etraneous solution.. Let h be the number of hours needed. h + h h 6 + h 6 5h 6 h 5 Therefore, they will need 5 h. Copyright by Holt, Rinehart and Winston. 44 Holt Algebra
Solutions Key Rational and Radical Functions
CHAPTER 8 Solutions Key Rational and Radical Functions ARE YOU READY? PAGE 565 1. D. A. B. F 5. C 6. 11 y 5 y 7 11 - y 5-7 7 y - 7 y 8. ( ) - (-) -6 6 10. ( -) ( ) ( - ) 8-1 8 1 z ) 7. ( y ( y) z 81 8
More information( 4 p 3. ( 2 p 2. ( x 3 y 4. ( y. (2 p 2 ) 2 ( q 4 ) 2. ( x 2 ) POLYNOMIALS, PAGES CHECK IT OUT! PAGES
8. _ x 4 y 8 x 4-6 y 8-6 x 6 y 6 x - y y x 9. 5 m n 4 5 m - n 4-1 m n 5 m 0 n 3 30. ( 3 5) 3 3 3 31. _ ( 4 p 3 5 1 n 3 5 n 3 5 3 _ 7 15 4) p q ( 4 p 3-1 q -4 ) ( p q -4 ) ( p q 4 ) ( p ) ( q 4 ) _ ( p
More informationSolutions Key Exponential and Radical Functions
CHAPTER 11 Solutions Key Exponential and Radical Functions xzare YOU READY, PAGE 76 1. B; like terms: terms that contain the same variable raised to the same power. F; square root: one of two equal factors
More informationSolutions Key Exponents and Polynomials
CHAPTER 7 Solutions Key Exponents and Polynomials ARE YOU READY? PAGE 57. F. B. C. D 5. E 6. 7 7. 5 8. (-0 9. x 0. k 5. 9.. - -( - 5. 5. 5 6. 7. (- 6 (-(-(-(-(-(- 8 5 5 5 5 6 8. 0.06 9.,55 0. 5.6. 6 +
More informationChapter 4 Polynomial and Rational Functions
Chapter Polynomial and Rational Functions - Polynomial Functions Pages 09 0 Check for Understanding. A zero is the value of the variable for which a polynomial function in one variable equals zero. A root
More informationChapter 6 Rational Expressions and Equations. Section 6.1 Rational Expressions. Section 6.1 Page 317 Question 1 = = 5 5(6) = d) 77.
Chapter 6 Rational Epressions and Equations Section 6. Rational Epressions Section 6. Page 7 Question a) (6) 8 (7 ) 4, 0 5 5(6) 0 5 5(7 ) 5 c) 4 7 44 d) 77 e) (6) f) 8(6) 8 + 4( + ) 4 + 8 4( ) 4 y ( y
More informationReady To Go On? Skills Intervention 12-1 Inverse Variation
12A Find this vocabular word in Lesson 12-1 and the Multilingual Glossar. Identifing Inverse Variation Tell whether the relationship is an inverse variation. Eplain. A. Read To Go On? Skills Intervention
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types
More informationVILLA VICTORIA ACADEMY (2016) PREPARATION AND STUDY GUIDE ENTRANCE TO HONORS ALGEBRA 2 FROM ALGEBRA I. h) 2x. 18x
VILLA VICTORIA ACADEMY (06) PREPARATION AND STUDY GUIDE ENTRANCE TO HONORS ALGEBRA FROM ALGEBRA I ) Simplify. 8 43 ) Evaluate the expression if a ; b 3; c 6; d 3) Translate each statement into symbols,
More informationRational Expressions and Functions
Rational Expressions and Functions In the previous two chapters we discussed algebraic expressions, equations, and functions related to polynomials. In this chapter, we will examine a broader category
More informationVILLA VICTORIA ACADEMY (2017) PREPARATION AND STUDY GUIDE ENTRANCE TO HONORS PRECALCULUS PART 1 FROM HONORS ALGEBRA II. a) 2ab b) d a. h) 2x.
VILLA VICTORIA ACADEMY (07) PREPARATION AND STUDY GUIDE ENTRANCE TO HONORS PRECALCULUS PART FROM HONORS ALGEBRA II ) Simplify. 8 4 ) Evaluate the expression if a ; b ; c 6; d ) Translate each statement
More informationLesson 1: Writing Equations Using Symbols
Lesson 1 Lesson 1: Writing Equations Using Symbols Classwork Exercises Write each of the following statements using symbolic language. 1. The sum of four consecutive even integers is 28. 2. A number is
More informationRational Equations. You can use a rational function to model the intensity of sound.
UNIT Rational Equations You can use a rational function to model the intensit of sound. Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations,
More informationLesson 11: Classwork. Example 1 S.41
Classwork Example 1 Pauline mows a lawn at a constant rate. Suppose she mows a 35-square-foot lawn in 2.5 minutes. What area, in square feet, can she mow in 1 minutes? tt minutes? tt (time in minutes)
More informationAdvanced Algebra Scope and Sequence First Semester. Second Semester
Last update: April 03 Advanced Algebra Scope and Sequence 03-4 First Semester Unit Name Unit : Review of Basic Concepts and Polynomials Unit : Rational and Radical Epressions Sections in Book 0308 SLOs
More information5.1 Modelling Polynomials
5.1 Modelling Polynomials FOCUS Model, write, and classify polynomials. In arithmetic, we use Base Ten Blocks to model whole numbers. How would you model the number 234? In algebra, we use algebra tiles
More informationRational Functions and Equations
Rational Functions and Equations 12A Rational Functions and Epressions Lab Model Inverse Variation 12-1 Inverse Variation 12-2 Rational Functions 12-3 Simplifying Rational Epressions Lab Graph Rational
More informationFINAL REVIEW MATH 6 STUDENT NAME MATH TEACHER
FINAL REVIEW MATH 6 STUDENT NAME MATH TEACHER ** As you go through this review packet, be sure to show all work as you have done throughout the school year. Remember- NO WORK NO CREDIT ** REAL NUMBERS,
More informationUnit 5 RATIONAL FUNCTIONS. A function with a variable in the denominator Parent function 1 x Graph is a hyperbola
Unit 5 RATIONAL FUNCTIONS A function with a variable in the denominator Parent function 1 x Graph is a hyperbola I will be following the Alg 2 book in this Unit Ch 5 Sections 1-5 Use the Practice Packet
More informationCHAPTER 2 Solving Equations and Inequalities
CHAPTER Solving Equations and Inequalities Section. Linear Equations and Problem Solving........... 8 Section. Solving Equations Graphically............... 89 Section. Comple Numbers......................
More informationALGEBRA 2 Summer Review Assignments Graphing
ALGEBRA 2 Summer Review Assignments Graphing To be prepared for algebra two, and all subsequent math courses, you need to be able to accurately and efficiently find the slope of any line, be able to write
More informationReteach Multiplying and Dividing Rational Expressions
8-2 Multiplying and Dividing Rational Expressions Examples of rational expressions: 3 x, x 1, and x 3 x 2 2 x 2 Undefined at x 0 Undefined at x 0 Undefined at x 2 When simplifying a rational expression:
More information2. Which of the following expressions represents the product of four less than three times x and two more than x?
Algebra Topics COMPASS Review You will be allowed to use a calculator on the COMPASS test. Acceptable calculators are: basic calculators, scientific calculators, and graphing calculators up through the
More informationUnit 5 RATIONAL FUNCTIONS. A function with a variable in the denominator Parent function 1 x Graph is a hyperbola
Unit 5 RATIONAL FUNCTIONS A function with a variable in the denominator Parent function 1 x Graph is a hyperbola A direct variation is a relationship between two variables x and y that can be written in
More informationEureka Math. Algebra II Module 1 Student File_A. Student Workbook. This file contains Alg II-M1 Classwork Alg II-M1 Problem Sets
Eureka Math Algebra II Module 1 Student File_A Student Workbook This file contains Alg II- Classwork Alg II- Problem Sets Published by the non-profit GREAT MINDS. Copyright 2015 Great Minds. No part of
More informationModel Inverse Variation
. Model Inverse Variation Rational Equations and Functions. Graph Rational Functions.3 Divide Polynomials.4 Simplify Rational Epressions. Multiply and Divide Rational Epressions.6 Add and Subtract Rational
More informationPearson Learning Solutions
Answers to Selected Exercises CHAPTER REVIEW OF REAL NUMBERS Section.. a. b. c.. a. True b. False c. True d. True. a. b. Ú c.. -0. a. b. c., -, - d.,, -, -, -.,., - e. f.,, -, -,, -.,., -. a. b. c. =.
More information7.1 Rational Expressions and Their Simplification
7.1 Rational Epressions and Their Simplification Learning Objectives: 1. Find numbers for which a rational epression is undefined.. Simplify rational epressions. Eamples of rational epressions: 3 and 1
More informationAbout the Portfolio Activities. About the Chapter Project
Galileo is credited as the first person to notice that the motion of a pendulum depends only upon its length. About the Chapter Project Finding an average is something that most people can do almost instinctively.
More informationCHAPTER 8 Quadratic Equations, Functions, and Inequalities
CHAPTER Quadratic Equations, Functions, and Inequalities Section. Solving Quadratic Equations: Factoring and Special Forms..................... 7 Section. Completing the Square................... 9 Section.
More informationMATH 110: FINAL EXAM REVIEW
MATH 0: FINAL EXAM REVIEW Can you solve linear equations algebraically and check your answer on a graphing calculator? (.) () y y= y + = 7 + 8 ( ) ( ) ( ) ( ) y+ 7 7 y = 9 (d) ( ) ( ) 6 = + + Can you set
More informationc) domain {x R, x 3}, range {y R}
Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..
More informationPOLYNOMIAL: A polynomial is a or the
MONOMIALS: CC Math I Standards: Unit 6 POLYNOMIALS: INTRODUCTION EXAMPLES: A number 4 y a 1 x y A variable NON-EXAMPLES: Variable as an exponent A sum x x 3 The product of variables 5a The product of numbers
More informationLesson 1: Successive Differences in Polynomials
Lesson 1 Lesson 1: Successive Differences in Polynomials Classwork Opening Exercise John noticed patterns in the arrangement of numbers in the table below. 2.4 3.4 4.4 5.4 6.4 5.76 11.56 19.36 29.16 40.96
More informationClassifying Polynomials. Simplifying Polynomials
1 Classifying Polynomials A polynomial is an algebraic expression with one or more unlike terms linked together by + or **Polynomials can be classified by the number of terms they have: A monomial has
More informationRational Equations and Graphs
RT.5 Rational Equations and Graphs Rational Equations In previous sections of this chapter, we worked with rational expressions. If two rational expressions are equated, a rational equation arises. Such
More informationClifton High School Mathematics Summer Workbook
Clifton High School Mathematics Summer Workbook Algebra II-H: 9 th grade Completion of this summer work is required on the first day of the school year. Date Received: Date Completed: Student Signature:
More informationMini-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.
More informationPolynomials. Lesson 6
Polynomials Lesson 6 MPMD Principles of Mathematics Unit Lesson 6 Lesson Six Concepts Introduction to polynomials Like terms Addition and subtraction of polynomials Distributive law Multiplication and
More informationName: Essential Skills Practice for students entering Geometry or Accelerated Geometry
Name: Essential Skills Practice for students entering Geometry or Accelerated Geometry Use this document to review the mathematics that you have learned previously. Completion of the Essential Skills Practice
More informationThe P/Q Mathematics Study Guide
The P/Q Mathematics Study Guide Copyright 007 by Lawrence Perez and Patrick Quigley All Rights Reserved Table of Contents Ch. Numerical Operations - Integers... - Fractions... - Proportion and Percent...
More informationSolutions Key Quadratic Functions
CHAPTER 5 Solutions Key Quadratic Functions ARE YOU READY? PAGE 11 1. E. C. A. B 5. (.) (.)(.) 10. 6. ( 5) ( 5 )( 5 ) 5 7. 11 11 8. 1 16 1 9. 7 6 6 11. 75 75 5 11 15 11 1. (x - )(x - 6) x - 6x - x + 1
More informationA2T. Rational Expressions/Equations. Name: Teacher: Pd:
AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest
More informationIOWA End-of-Course Assessment Programs. Released Items ALGEBRA II. Copyright 2010 by The University of Iowa.
IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA II 1 Which cubic equation has roots of 2, 1, and 3? A 3 6 = 0 INCORRECT: The student wrote a cubic
More informationFoundations for Algebra. Introduction to Algebra I
Foundations for Algebra Introduction to Algebra I Variables and Expressions Objective: To write algebraic expressions. Objectives 1. I can write an algebraic expression for addition, subtraction, multiplication,
More informationLesson 18: Word Problems Leading to Rational Equations
Opening Exercise 1 Anne and Maria play tennis almost every weekend So far, Anne has won 12 out of 20 matches A If Anne wants to improve her winning percentage to 75%, would winning the next 2 matches do
More informationCalculus - Chapter 2 Solutions
Calculus - Chapter Solutions. a. See graph at right. b. The velocity is decreasing over the entire interval. It is changing fastest at the beginning and slowest at the end. c. A = (95 + 85)(5) = 450 feet
More informationChapter 9 Rational Expressions and Equations Lesson 9-1 Multiplying and Dividing Rational Expressions Pages
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
More informationCollecting Like Terms
MPM1D Unit 2: Algebra Lesson 5 Learning goal: how to simplify algebraic expressions by collecting like terms. Date: Collecting Like Terms WARM-UP Example 1: Simplify each expression using exponent laws.
More informationinequalities Solutions Key _ 6x 41 Holt McDougal Algebra 1 think and discuss 2-1 Check it out! b w 4-4 w
CHAPTER Inequalities Solutions Key Are You Ready?. B. E. F. D. C 6. b - a = 6 - = 7. ab = ()(6) = 9. a + b = + 6 = 8 8. b a = 6 =. .. % =..
More information3. There are many possible answers. Some examples are 1.3, 0.55, and
Chapter 2 Lesson 2.1 1. Absolute value represents the distance from zero when graphed on a number line. 2. proper fractions, improper fractions, equivalent 3. There are many possible answers. Some examples
More informationCycle 2: Why Does It Matter?
Lesson. It s All Relative 9 Part Cycle : Why Does It Matter? Lesson. It s All Relative. 5 5.. a. Negative; $0,000 Negative; 400 4. a. Loss of 0 yards Loss of 0.6 points for the day 5. 6. a. 6 6 4 4 c.
More informationWrite an equation for each relationship. Then make a table of input-output pairs and tell whether the function is proportional.
Functions Reteaching 41 Math Course, Lesson 41 A function is a rule that identifies a relationship between a set of input numbers and a set of output numbers. A function rule can be described in words,
More informationChapter 9 Prerequisite Skills
Name: Date: Chapter 9 Prerequisite Skills BLM 9. Consider the function f() 3. a) Show that 3 is a factor of f(). If f() ( 3)g(), what is g()?. Factor each epression fully. a) 30g 4g 6fg 8g c) 6 5 d) 5
More information150. a. Clear fractions in the following equation and write in. b. For the equation you wrote in part (a), compute. The Quadratic Formula
75 CHAPTER Quadratic Equations and Functions Preview Eercises Eercises 8 50 will help you prepare for the material covered in the net section. 8. a. Solve by factoring: 8 + - 0. b. The quadratic equation
More informationOne of your primary goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan.
PROBLEM SOLVING One of our primar goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan. Step Step Step Step Understand the problem. Read the problem
More information= The algebraic expression is 3x 2 x The algebraic expression is x 2 + x. 3. The algebraic expression is x 2 2x.
Chapter 7 Maintaining Mathematical Proficiency (p. 335) 1. 3x 7 x = 3x x 7 = (3 )x 7 = 5x 7. 4r 6 9r 1 = 4r 9r 6 1 = (4 9)r 6 1 = 5r 5 3. 5t 3 t 4 8t = 5t t 8t 3 4 = ( 5 1 8)t 3 4 = ()t ( 1) = t 1 4. 3(s
More informationChapter 1 Functions and Graphs. ( x x ) ( y y ) (1 7) ( 1 2) x x y y 100. ( 6) ( 3) x ( y 6) a. 101.
Chapter Functions and Graphs... ( ) ( y y ) ( 7) ( ) y y y ( 6) ( ) 6 9 5 5 6y 6y 6y9 9 ( y ) y y Solution set:. 5. a. h, k 6, r ; ( ) [ y( 6)] ( ) ( y6) ( y6) b. ( ) ( y) [ ( )] ( y) So in the standard
More information7 = 8 (Type a simplified fraction.)
Student: Date: Assignment: Exponential and Radical Equations 1. Perform the indicated computation. Write the answer in scientific notation. 3. 10 6 10. 3. 4. 3. 10 6 10 = (Use the multiplication symbol
More informationCHAPTER 11: RATIONAL EQUATIONS AND APPLICATIONS
CHAPTER 11: RATIONAL EQUATIONS AND APPLICATIONS Chapter Objectives By the end of this chapter, students should be able to: Identify extraneous values Apply methods of solving rational equations to solve
More informationMath 10 - Unit 5 Final Review - Polynomials
Class: Date: Math 10 - Unit 5 Final Review - Polynomials Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Factor the binomial 44a + 99a 2. a. a(44 + 99a)
More informationUnit 4 Rational Functions
Unit 4 Rational Functions Test date: Name: By the end of this unit, you will be able to Simplify rational expressions Find the LCM for rational expressions Add and subtract rational expressions Solve rational
More informationAnswer Keys for Calvert Math
Answer Keys for Calvert Math Lessons CMAKF- Contents Math Textbook... Math Workbook... Math Manual... Answer Keys Math Textbook Lessons Math Textbook Answer Key Lessons. Area and Circumference of Circles
More informationLesson 24: True and False Number Sentences
NYS COMMON CE MATHEMATICS CURRICULUM Lesson 24 6 4 Student Outcomes Students identify values for the variables in equations and inequalities that result in true number sentences. Students identify values
More informationAlgebra II Notes Rational Functions Unit Rational Functions. Math Background
Algebra II Notes Rational Functions Unit 6. 6.6 Rational Functions Math Background Previously, you Simplified linear, quadratic, radical and polynomial functions Performed arithmetic operations with linear,
More information1 Chapter 1: Graphs, Functions, and Models
1 Chapter 1: Graphs, Functions, and Models 1.1 Introduction to Graphing 1.1.1 Know how to graph an equation Eample 1. Create a table of values and graph the equation y = 1. f() 6 1 0 1 f() 3 0 1 0 3 4
More informationFive-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal
Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Asymptote Example 2: Real-World Example: Use Graphs
More informationLesson 18: Recognizing Equations of Circles
Student Outcomes Students complete the square in order to write the equation of a circle in center-radius form. Students recognize when a quadratic in xx and yy is the equation for a circle. Lesson Notes
More informationMini-Lecture 5.1 Exponents and Scientific Notation
Mini-Lecture.1 Eponents and Scientific Notation Learning Objectives: 1. Use the product rule for eponents.. Evaluate epressions raised to the zero power.. Use the quotient rule for eponents.. Evaluate
More informationAlgebra. Robert Taggart
Algebra Robert Taggart Table of Contents To the Student.............................................. v Unit 1: Algebra Basics Lesson 1: Negative and Positive Numbers....................... Lesson 2: Operations
More informationSection 5.1 Model Inverse and Joint Variation
108 Section 5.1 Model Inverse and Joint Variation Remember a Direct Variation Equation y k has a y-intercept of (0, 0). Different Types of Variation Relationship Equation a) y varies directly with. y k
More informationName Class Date. You can use the properties of equality to solve equations. Subtraction is the inverse of addition.
2-1 Reteaching Solving One-Step Equations You can use the properties of equality to solve equations. Subtraction is the inverse of addition. What is the solution of + 5 =? In the equation, + 5 =, 5 is
More informationMath Review for Incoming Geometry Honors Students
Solve each equation. 1. 5x + 8 = 3 + 2(3x 4) 2. 5(2n 3) = 7(3 n) Math Review for Incoming Geometry Honors Students 3. Victoria goes to the mall with $60. She purchases a skirt for $12 and perfume for $35.99.
More informationVariables and Expressions
Variables and Expressions A variable is a letter that represents a value that can change. A constant is a value that does not change. A numerical expression contains only constants and operations. An algebraic
More informationLesson #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More informationMaths A Level Summer Assignment & Transition Work
Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first
More informationCHAPTER 5 RATIONAL FUNCTIONS
CHAPTER 5 RATIONAL FUNCTIONS Big IDEAS: ) Graphing rational functions ) Performing operations with rational epressions 3) Solving rational equations Section: 5- Model Inverse and Joint Variation Essential
More informationAlgebra SUMMER PACKET Ms. Bank
2016-17 SUMMER PACKET Ms. Bank Just so you know what to expect next year We will use the same text that was used this past year: published by McDougall Littell ISBN-13:978-0-6185-9402-3. Summer Packet
More informationClassifying Polynomials. Classifying Polynomials by Numbers of Terms
Lesson -2 Lesson -2 Classifying Polynomials BIG IDEA Polynomials are classifi ed by their number of terms and by their degree. Classifying Polynomials by Numbers of Terms Recall that a term can be a single
More informationWords to Review. Give an example of the vocabulary word. Numerical expression. Variable. Evaluate a variable expression. Variable expression
1 Words to Review Give an example of the vocabulary word. Numerical expression 5 12 Variable x Variable expression 3x 1 Verbal model Distance Rate p Time Evaluate a variable expression Evaluate the expression
More informationEureka Math. Grade, Module. Student _B Contains Sprint and Fluency, Exit Ticket, and Assessment Materials
A Story of Eureka Math Grade, Module Student _B Contains Sprint and Fluency,, and Assessment Materials Published by the non-profit Great Minds. Copyright 2015 Great Minds. All rights reserved. No part
More informationChapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)
Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J.. 7. 8 5. 6 6. 7. y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5
More informationUnit # 4 : Polynomials
Name: Block: Teacher: Miss Zukowski Date Submitted: / / 2018 Unit # 4 : Polynomials Submission Checklist: (make sure you have included all components for full marks) Cover page & Assignment Log Class Notes
More informationLinear Relations and Functions
Linear Relations and Functions Why? You analyzed relations and functions. (Lesson 2-1) Now Identify linear relations and functions. Write linear equations in standard form. New Vocabulary linear relations
More informationDivisibility Rules Algebra 9.0
Name Period Divisibility Rules Algebra 9.0 A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following eercise: 1. Cross
More informationMath-3. Lesson 3-1 Finding Zeroes of NOT nice 3rd Degree Polynomials
Math- Lesson - Finding Zeroes of NOT nice rd Degree Polynomials f ( ) 4 5 8 Is this one of the nice rd degree polynomials? a) Sum or difference of two cubes: y 8 5 y 7 b) rd degree with no constant term.
More informationEvaluations with Positive and Negative Numbers (page 631)
LESSON Name 91 Evaluations with Positive and Negative Numbers (page 631) When evaluating expressions with negative numbers, use parentheses to help prevent making mistakes with signs. Example: Evaluate
More informationG.3 Forms of Linear Equations in Two Variables
section G 2 G. Forms of Linear Equations in Two Variables Forms of Linear Equations Linear equations in two variables can take different forms. Some forms are easier to use for graphing, while others are
More informationSolutions Key Exponential and Logarithmic Functions
CHAPTER 7 Solutions Key Exponential and Logarithmic Functions ARE YOU READY? PAGE 87 1. D. C. E. A. x ( x )(x) = x (x) = x 6 6. y -1 ( x y ) = ( y -1 y ) x = (y) x = 1 x y 7. a 8 = a (8 - ) a = a 6 9.
More informationAlgebra 2 Chapter 9 Page 1
Section 9.1A Introduction to Rational Functions Work Together How many pounds of peanuts do you think and average person consumed last year? Us the table at the right. What was the average peanut consumption
More informationSystems of Linear Equations
Systems of Linear Equations As stated in Section G, Definition., a linear equation in two variables is an equation of the form AAAA + BBBB = CC, where AA and BB are not both zero. Such an equation has
More informationSemester 1 Final Review. c. 7 d.
Solve the equation in questions 1-4. 1. 7 x + 5 = 8 a. 7 b. 1 7 c. 7 d. 7. 7 = d + 0 a. 10 b. 0 c. 1 d. 1. p 1 = 5(p 1) (7 p) a. b. 0 c. 9 d. 10 4. 5x 5 = x 9 a. b. 1 c. 1 d. 5. A customer went to a garden
More informationFinal Exam Review. Name: Class: Date: Short Answer
Name: Class: Date: ID: A Final Exam Review Short Answer. Use x, 2, 0,, 2 to graph the function f( x) 2 x. Then graph its inverse. Describe the domain and range of the inverse function. 2. Graph the inverse
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections.6 and.) 8. Equivalent Inequalities Definition 8. Two inequalities are equivalent
More information1 Linear and Absolute Value Equations
1 Linear and Absolute Value Equations 1. Solve the equation 11x + 6 = 7x + 15. Solution: Use properties of equality to bring the x s to one side and the numbers to the other: 11x (7x) + 6 = 7x (7x) + 15
More informationSolving and Graphing Polynomials
UNIT 9 Solving and Graphing Polynomials You can see laminar and turbulent fl ow in a fountain. Copyright 009, K1 Inc. All rights reserved. This material may not be reproduced in whole or in part, including
More informationChapter P Prerequisites
Section P. Real Numbers Chapter P Prerequisites Section P. Real Numbers Quick Review P.. {,,,,, 6}. {,, 0,,,,,, 6}. {,, }. {,,, }. (a) 87.7 (b).7 6. (a) 0.6 (b) 0.0 7. ( ) -( )+ ; (.) -(.)+.7 8. ( ) +(
More informationPolynomials. Lesson 6
Polynomials Lesson 6 MFMP Foundations of Mathematics Unit Lesson 6 Lesson Six Concepts Overall Expectations Simplify numerical and polynomial expressions in one variable, and solve simple first-degree
More informationLinear Equations in One Variable *
OpenStax-CNX module: m64441 1 Linear Equations in One Variable * Ramon Emilio Fernandez Based on Linear Equations in One Variable by OpenStax This work is produced by OpenStax-CNX and licensed under the
More informationInverse Variation. y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0.
Inverse Variation y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0. Inverse variation xy = k or y = k where k is not equal to 0. x Identify whether the following functions
More information