Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ CHAPTER FUNDAMENTALS AMENTALS OF O ALGEBRA 25 25 Real numbers Section 1.1 Sets and Numerical Expressions Rational numbers Irrational numbers Integers Whole numbers 10 11 13 14 15 16 17 18 19 {−4, −1, 2, 3} {−5, −4, −3, 1, 3} {−250, −200, −150, −100, −50, 0, 50, 100, 150, 200} {−25, −20, −15, −10, −5, 0, 5, 10, 15, 20} {…, −3, −2, −1, 0, 1, 2, 3, …} {0, 1, 2, 3, …} prime 12 prime composite, since 147 = 3∙7∙7 composite, since 177 = 3∙59 No Whole numbers larger than are elements in set F, but are not elements of set G Yes Every element in set G is a whole number No The elements blue and red are part of set E, but are not part of set D Yes Both elements in set D are also part of set E Yes Every integer can be written as a ratio of two integers (For example, 17 can be written as 17 ) 24 0.2341 2341 10, 000 27 29 31 33 35 37 39 41 A, B, D A, B A, C A, B, D, E, F A, B H A, C 28 30 32 34 36 38 40 π −8 ± ± 13 ± 100 ± ± 0.43 ± ± ± ± A, B A, C A, B, D, E, G H A, B, D A, B A, B, D ± ± ± ± ± INSTRUCTOR R USE ONLY © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Whole numbers 399 1000 Whole numbers Integers 0.399 Integers Irrational numbers 23 Irrational numbers Rational numbers Yes Every irrational number is a real number Yes No irrational number is also a rational number No The set of integers has every one of its elements in common with the set of rational numbers Rational numbers Real numbers 20 21 22 26 ± Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 42 ± 0.67 π −2 ± ± (4, ∞) (−∞, 4) [4, ∞) (−∞, 4] ± ± ± ± ± ± Whole numbers ± Integers Rational numbers 36 Irrational numbers Real numbers 43 45 47 49 51.a 58 8a 58.a ± ± 58.b (−∞, −2] 59.a 59.b [25, 100) 60.a ± ± 44 46 48 50 60.b [30, 80) 61 92 81 ± (5, ∞) (−∞, 5) [5, ∞) (−∞, 5] 51.b (−∞, 7) 52.a 52.b (−∞, 8) 53.a 53.b [−4, ∞) 54.a 88 8 53 125 64 8 8 63 9 65 23 2 2 67 2 2 2 2 8 68 5 5 5 5 125 69 71 73 75 77 79 81 not a real number −5 83 36 has two square roots, and −6, but 36 is the principal square root of 36 and is equal to the positive square root of 36 Since any real number multiplied by itself always results in either or a positive number, there is no number that can be multiplied by itself to give −100 84 54.b [−6, ∞) 55.a 82 3 9 9 62 81 64 66 70 72 74 76 78 80 82 10 11 not a real number −1 85 86 So 100 cannot be a real number 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25 87 12 y 3 88 18 12 y 4 18 3 12 12 2 55.b (2, 10) 56.a 64 12 2 24 33 10 24 14 56.b (1, 8) 57.a 89 Đ 12 24 à ă â 5 Đ 12 à ă â 6 23 57.b (−∞, −15] 3 90 § 35 20 à ă â 4 Đ 55 à ă â 11 53 125 3 INSTRUCTOR O USE ONLY © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 91 y 3 92 81 93 992 92 y 10 8 42 3 30 16 62 22 x y 14 x y 8 x y 2 18 y 15 2 7 56 y x 21y x 6 x 35 y 4 28 32 28 y 17 8 12 48 36 95 14 72 y 32 11 15 14 72 y 4 14 72 y 4 14 4 22 4 26 11 36 y 32 21 11 36 y 19 11 19 34 97 257 16.031219 | 16.03 98 327 18.083141 | 18.08 99 212 100 21 21 21 441 101.a They used a ( on the left endpoint 101.b [−5, ∞) 102.a They have the numbers reversed 102.b (−∞, 6) 103.a They took the square root of 16 and made it negative 103.b Not a real number 104.a They listed as a prime number 104.b 2, 3, 5, 7, 11 Section 1.2 Algebraic Expressions 3w 11y 18w 20 y 1à Đ2 24 ă x x 6ạ â3 Đ2 à Đ5 Ã Đ 1à 24 ă x 24 ă x 24 ă â3 â8 ¹ © 6¹ 16 x 15 x 1à Đ5 30 ă y y 15 2ạ â6 Đ5 Ã Đ Ã Đ1à 30 ă y 30 ă y 30 ă â6 â 15 ¹ ©2¹ 25 y 14 y 15 21w 31y 2 20 x 14 x 5 2 4 20 x 3 14 x 5 7 20 8x 8x 7 20 7 42 100 35 35 142 35 21 21 441 12 x 15 x 3 2 12 x 6 15 x 4 3 18 x 10 x x 3 27 x 6 35 x 96 y x 21y x y x 21y x 4 7 94 1.2 Algebraic Expressions 3x y 14 x y 10 16 16 81 3x y 14 x y 5h 14m 47h 11m 52h 25m INSTRUCTOR R USE ONLY © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 10 2Đ3 à ă a 1á 3â2 ¹ 21 21 16 2n 4 2Đ3 à ă a 1 3â2 a 2 a a 6 a a 4Đ5 à ă a 1á 5â4 10 16 2n 22 3x 28 7m 11 23 13 16 6n 12 24 12 28 35m 55 25 4x x 9 24 x 54 14 26 6x 4x 21x 28 x 17 x x x 9 21x3 28 x 24 x3 54 x p p 18 p p 1 2p 2 19 6x2 4x x 3x x 4 15 f 8 f 30 f 24 f 54 f a a 27 5a a 4a 20 Đ 2à 1ă g â 15 5 4g 15 25 4g 45 45 31 4 g 45 §3 · ă 2k 10 â Đ3à 1ă 2k 10 â ¹ 2k 10 28 15 2k 40 40 43 2 k 40 24 x 54 x 16 35m 24 § à ă 4g â 15 ¹ x x x 9 21x 28 x x 3x 6n 12 28 7m 11 28 7m 11 x x x 4 15 7m 36 16 2n 16 2n 4 x 4 21x 28 x 3x 2n 12 28 7m 11 28 m 11 4§5 · ă a 1 5â4 10 a 10 a 10 a 10 10 11 a 10 11 16 2n 0.25 4b 0.3 b 10 0.25 4b 0.25 8 0.3 b 0.3 10 b 0.3b 21 3d 9d 63d 36d 28 0.7b 0.75 12 z 24 0.5 z 10 0.75 12 z 0.75 24 0.5 z 0.5 10 99d z 18 0.5 z 8.5 z 13 INSTRUCTOR O USE ONLY © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 29 3.5 2n p 20 z 57 3.5 2n 3.5 3 p 3.5 20 z 4.5 3a 2b 30c 44 45 47 48 49 13.5a 9b 135c 32 25 z 52 y5 z 5y F 34 G A 36 H D 38 E J 40 E I 42 B Yes, since the sum of a real number and its additive inverse always equals 0, and is the additive identity of the real numbers Yes, since the product of a real number and its multiplicative inverse always equals 1, and is the multiplicative identity of the real numbers 46 52 7 33 5 7 If we consider 12 y , it is equal to since 12 Similarly y is equal to some number only if that number multiplied by is equal to But we know that multiplying zero by any number gives an answer of zero This means there is no number that can equal y , so y cannot equal 51 x8 y x y x 52 y 1 c d 54 x x a b d 65 x 20 y 2x y x y 60 67 69 1 71 a x a x12 a15 9 x y 8b c 9 x15 y 21 8b12 c 27 j jk 2 62 p 5vz j 32 j k 2 p 52 v z 32 j j k 2 52 p v z j 2 k 50 p v z §x· ă â2ạ x3 23 x3 64 h4 h9 9 h 66 73 p6 p10 p 4 p4 3x0 1 68 z2 z2 z Đ d v3 à ă 11 âd v Đ yÃ ă ©4¹ y3 43 y3 64 p 10 70 y0 1 p6 p6 p z0 d 2b c b c h 5 h5 10 y4 x y 4 6 5 63 c4 d 56 b6 x y x 45 j k c6 d c2 c u x3 4 17 21 u x3 u5 u 55 61 c d c17 d c x y 58 a 13k 4k 11 10 53 59 Following the text’s analogy, a model of the fraction is a pie cut into zero pieces It is impossible to conclude how many pieces are gone, and impossible that pieces could remain 20h 7h 5h3 13k 5k 9k 50 5h3 13h2 1.2 Algebraic Expressions a 24 b 4.5 3a 4.5 2b 4.5 30c 31 33 35 37 39 41 43 ab 6 7n 10.5 p 70 z 30 p0 72 x 3x x x x § x7 y8 à ă 20 âx y x 3x x 4 x x x 18 x3 24 x 18 x3 45 x 69 x INSTRUCTOR R USE ONLY © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 74 10 z z z z z 81 10 z z 10 z 9 z z z 1 z 96 z w7 z 20 wz13 76 w 1 z 20 13 w z 77 79 80 3 78 13 x y 15 11 x y 15 x 11 15 y b a4 3b a4 x3 y 12a 16a 12a 16a 3 36a 80a 15 15 44a 15 x y15 xy x 1 y 15 8 14 x9 y 30 x y13 82 12 3 a b 3a 4 b 20 z 90 z 15 z z 75 12a 3b9 4a b3 14u 9u 14u 9u 7 98u 27u 21 21 71u 21 83 3x x3 15 y11 y11 x 19 y 3 x 19 p k p 84 y11 27 p 12k 9k 15k 10k k k 10 5 k k 10 k k 10 10 17 k 10 85 x v8 Đ d v3 à ă 11 âd v d y Đ x7 y8 à ă 20 âx y 88 v 11 d v d v 8 4 7 20 x y 12 1 12 y y12 v8 87 k 21 x7 y8 x y 20 86 d v 8 20m8 35m12 30m6 50m10 10 12 10 m m 10 10 10 m m 10 10 20 21 m m 30 30 41 m 30 18 k2 14k 23 p15 d 5v3 d v11 d v 311 1 18 14 p 15 k 23 14 15 k 23 p 12 x 13 y15 12 13 y15 x 12 y15 x13 21 8 x 77 20 x y x y 12 5 d v 32 x y 60 v 32 v 32 1 1 v 12 y 60 y 60 INSTRUCTOR O USE ONLY © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken Full file at https://TestbankDirect.eu/ § 3x y · § x y à ă ă â © 89 R 1.3 Equations and an Inequalities in One Variable x7 98.b 10 x x 10 32 x y x y 52 9x y 4x y 25 y 94 x 25 36 x y 175 x 3 x3 99.a They didn’t distribute the negative sign correctly 99.b h h 11 h 5 h 11 3h 15 h 11 2h 100.a They did not write the answer with using positive exponents 90 § a b · § 2a b à ă ă â © 7 9 13 100.b x y x y 2 a b 2a 3b 52 4a b 2a 3b 25 42a b 25 11 8a b 175 91.a x 7 9 y 13 x 16 y 11 1 x16 y11 16 11 x y z8 z3 Section 1.3 Equations and Inequalities in One Variable 1.a zzzzz z z z zzz z5 91.b z8 z3 83 z z5 92.a z z 1.b 2.a [5, 5] 2.b 3.a [3, 3] 3.b 4.a (5, ∞) 4.b 5.a (3, ∞) 5.b 6.a (−∞, 5) z z z z z z z z z z9 92.b z z z z9 93 −32 94 −45 95 −264 96 −290 97.a They added exponents instead of multiplying them 97.b x x x 21 98.a They subtracted the exponent of the numerator from the exponent of the denominator INSTRUCTOR R USE ONLY 6.b (−∞, ( , 3) © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 7.a 18.a 18 8a w 12 34 12 12 7.b 8.a 8.b 9.a w 46 19.a 31 n 40 [5, ∞) 40 40 71 n 20.a 24 m 13 13 13 37 m [3, ∞) 21.a 9.b (−∞, 5] 10.a 10.b (, 3] 11.a 22.a ê3 11.b ô , ằ ¬7 7¼ 12.a 23.a 19.b 31 71 40 31 31 20.b 24 24 14 21.b 2 14 2 8b 2 16 16 14 8b b 16 2 6c 8 34 8 6c 6c c 42 42 7 x 15 14 14 22.b 7 24.a x 17 3x 3 3x 3x 17 3 20 20 20 x 14.b (−150, ∞) 15.a 15.b (−∞, 5) 16.a 25.a 12 z 34 34 23.b § 17 à ă 1á 15 â 17 Đ Ã ă 1 15 â 2ạ 17 15 15 15 24.b Đ 20 à ă 1á 17 â 20 Đ Ã ă 1 17 â ¹ 20 17 17 17 z 24 z 12 z 24 z 24 z z 12 z 12 2 z 6 17.b 29 21 21 21 34 42 34 15 2 17 17 17 x 13.b (100, ∞) 14.a 37 13 24 8b 2x 2 2x 2x ª4 4º 12.b ô , ằ ơ9 9ẳ 13.a 16.b (, 2) 17.a p 21 8 8 p 29 18.b 18 b 46 12 34 34 34 INSTRUCTOR O USE ONLY © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 25.b 12 6 6 18 36 26.a 10 y 29.a 36 14 y 80 y 84 y 80 y 80 y 2 y 2 y 2 y 4 2 2 26.b 10 2 14 2 12 16 96 27.a 30.a 96 x 15 x 14 x 28 15 x 31.a 14 x 19 15 x 19 19 14 x 15 x 22 15 x 15 x x 1 x x 22 1 22 32.a 22 27.b 22 15 22 48 327 336 327 327 327 x x 15 x 11 x 27 x 15 x 11 14 x 27 15 x 11 15 x 27 15 x 27 x 16 1 x x 1 16 16 35.a 28.b 16 16 15 16 11 32 128 240 11 41 128 251 123 128 251 h 15 4§3 · 15 ă há 3â4 60 h h 20 k 5Đ4 à ă ká 4â5 60 60 300 k k 75 33.a 0.8 w 1.9 1.9 1.9 1.1 w 34.a 0.29 y 0.81 0.81 0.81 0.52 y 44 330 28.a 1.3 Equations and an Inequalities in One Variable 17 17 5 c 29.b 4 4 17 3 5 4 4 20 17 20 c 4 4 17 17 c 4 27 27 d 30.b 4 8 8 5 27 4 8 8 32 27 32 d 8 8 27 27 d 8 36.a 251 251 a 15 0.3 § a · 0.3 ă 0.3 15 â 0.3 a 4.5 m 12 0.7 Đ m à 0.7 ă 0.7 12 © 0.7 ¹ m 8.4 31.b 32.b 20 15 60 15 15 15 75 300 60 60 60 60 33.b 0.8 1.1 1.9 0.8 0.8 34.b 0.29 0.29 35.b 36.b 0.52 0.81 0.29 4.5 15 0.3 15 15 8.4 12 0.7 12 12 INSTRUCTOR R USE ONLY © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 37.a x 3Đ2 à ă xá 2â3 x x x 38.a x 4Đ3 à ă xá 3â4 x x x 39.a k 0.35k 0.65k 0.65k 0.65 k 40.a z 0.48 z 0.52 z 0.52 z 0.52 z 41.a a a 8 a a Đ1 à 2ă a â2 a a a 3Đ8à ă 2â9ạ 24 18 4 3 16 4Đ Ã ă â 16 36 48 12 12 4 78 78 78 0.65 120 208 208 208 0.52 400 37.b 38.b 2Đ4à ă 3â3ạ 9 3Đ3à ă 4â4ạ 16 16 16 1Đ 5à 3Đ5à ă á ă 8â3ạ 8â3ạ 15 24 24 20 24 5 6 5 6 7 42.a 42.b d d 48 60 60 10 10 10 10 60 420 d 48 10 10 10 480 10 Đ Ã 10 48 ă dá 10 © 10 ¹ 48 480 d d 60 1 k k 49 43.a · Đ1 12 49 12 ă k k ¹ ©4 41.b 39.b 120 0.35 120 78 120 42 78 78 78 40.b 400 0.48 400 208 400 192 208 208 208 48 48 48 48 Đ1 à Đ1 à 588 12 ă k 12 ă k â4 â3 588 3k 4k 588 588 84 7k 7k k 1 84 84 49 21 28 49 49 62 n n 44.a à Đ2 40 62 40 ă n n â 6 43.b 49 Đ5à 2ă â6ạ Đ2 à Đ3 à 40 ă n 40 ă n â5 â8 2480 16n 15n 2480 31n 2480 31n 31 31 80 n 10 5 3 2480 INSTRUCTOR O USE ONLY 10 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 44.b 62 62 62 45.a c c 2 c 2 c 2 c 21 21 21 2 21 2 p 2 § 13 · § 13 · 46.b ă ă â 2ạ © 2¹ 13 13 2 26 13 p 13 2 13 q 0.9q 45 10 q 0.9q 10 45 10q 9q 450 q 450 a 0.8a 48.a 54 10 a 0.8a 10 54 10a 8a 2a 2a a 49.a v 2v 2v 2v 3u 17 3u 27 3u 21 52.a 21 7c 18 5 9w 20 2w w 9w 20 10w 20 w 9w 20 w 20 9w 9w 20 20 Solution: The set of real numbers 52.b Check with w and w : 13 13 13 20 13 45 20 4 450 405 45 20 20 45 45 20 20 48.b 270 0.8 270 2 11 10 11 11 54 270 216 54 53.a 54 54 x 15 x x 14 x 28 15 x x 14 x 19 14 x 14 x 14 x 19 3 49.b No check possible NO SOLUTION 53.b No check possible 54.a x x 15 x x 16 x 27 x 15 x x 16 14 x 27 14 x 16 14 x 14 x 27 16 50.b No check possible 3u NO SOLUTION 54.b No check possible 17 27 NO SOLUTION 51.a 18 11 10 11 11 NO SOLUTION 3 u 18 18 21 2v 3u 17 9 18 21 9 50.a 18 47.b 450 0.9 450 540 540 540 270 2v 18 § 21 · § 21 · 45.b ă ă â 2ạ © 2¹ 21 21 2 42 21 46.a p p 13 2 p 13 47.a 1.3 Equations and an Inequalities in One Variable 51.b 51 b Check with c and c : 80 80 32 30 62 55.a 11n n 2n n 11n n 6n 4n 10n 10n 10n 10n 5 5 Solution: The set of real numbers 4c c 7c 18 8c 18 c 7c 18 7c 18 7c 7c 18 18 Solution: The set of real numbers INSTRUCTOR R USE ONLY 11 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 55.b Check with n 11 and n : 59.a 5 2 5 5 5 36 p 24 11 1 11 3 1 36 p 36 9u u 11 2u u p 9u u 11 6u 2u 14 8u 11 8u 11 8u 8u 11 11 Solution: The set of real numbers 56.b Check with u and u : p p 59.b 11 11 14 11 57.a 58.a 2z 9z 2 z z Đ 88 24 à ă 16 4â 3 57.b 0 Đ 64 à ă 16 4â 64 16 16 16 0 7z 7z z 12 y 2y 58.b 12 2 y y 10 y 10 y 10 y Đ Đ 22 à à ă 12 ă 16 4â â ¹ 3· § 88 16 ă 3ạ â 3 3 6 3 12 3 3 88 36 88 36 22 9 22 Đ 264 à 16 ă 4â ¹ 11 11 12 64 24 24 36 p 88 94 5 56.a 12 p 16 Đ3à ă 12 p 16 â4ạ 12 p 64 60.a 0 0 10 15k 18 §2· ă 15k 18 â3ạ 15k 54 30k 54 8 8 30k 62 30k 30 k k k 62 30 62 30 31 15 31 15 INSTRUCTOR O USE ONLY 12 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 1.3 Equations and an Inequalities in One Variable § § 31 à à 60.b ă15 ă 18 â â 15 ạ 31 27 54 3 18 18 h 61.a 12 Đhà 12 ă 12 â 12 h 108 62.a 63.a M M 64.a 18 18 a 18 a 18 15 n n 61.b 62.b 63.b Đ8à 75 ă â 15 600 15 40 P 11 128 16 Đ P Ã Đ 11 à 128 ă 128 ă â 128 â 16 1408 P 16 P 88 65.a 15 Đ8à 20 ă â 15 160 15 n 20 Đ n à 20 ă â 20 n 32 32 Đ a à 30 ă â 30 ¹ a 10 27 10 27 18 k 15 Đk à 15 ă 15 â 15 k 90 M 75 M Đ Ã 75 ă â 75 Đ Ã 66.a 30 ă â 20 270 20 64.b 67.a 6 40 75 8 15 15 15 15 15 88 128 11 16 11 16 11 16 11 16 11 16 65.b 15 15 15 15 15 27 30 27 30 27 60 9 20 20 5 x x 12 x 12 x 12 5 Ã Đ Ã Đ 12 ă x x 12 12 ă x 12 x 12 â4 â x 10 x 144 24 x 144 x 19 x 144 19 x 144 19 x 19 x 144 144 Solution: The set of real numbers 67.b Check with x and x : 108 12 9 90 15 66.b 20 20 20 20 20 5 12 12 12 12 12 12 12 5 12 12 12 5 12 12 12 10 144 24 144 12 12 12 12 12 12 125 125 12 12 32 68.a x x 18 x 18 x 35 32 à Đ4 Ã Đ 35 ă x x 18 35 ă x 18 x 35 â5 â 28 x 10 x 630 70 x 630 32 x 38 x 630 38 x 630 38 x 38 x 630 630 Solution: The set of real numbers 32 20 32 20 32 60 8 15 15 INSTRUCTOR R USE ONLY 13 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 68.b Check with x and x : 70.b 32 18 18 35 18 18 32 18 18 35 32 18 18 35 35 35 32 18 16 7 35 35 35 28 10 630 560 32 35 35 35 35 35 592 592 35 35 69.a a 5à Đ3 24 ă a 6ạ â4 18a 20 20 18a 18a 18 a 69.b 70.a 3Đ Ã ă á â 18 15 72 15 12 72 12 15 60 72 72 45 72 c 7à Đ5 24 ă c 8ạ â6 20c 21 21 20c 20c 20 71.a 72.a Đ5à 24 ă â8ạ 15 20 5 5 18 18 8 45 72 45 72 5Đ Ã ă á â 20 ¹ 15 120 15 15 120 15 15 105 120 120 90 120 4 30 30 90 120 90 120 v Đ Ã 2 ă v 2 â v 16 71.b 16 8 q Đ Ã 4 ă q â ¹ q 72.b 24 6 4 6 24 73.a x x 84 x x 84 x 84 6 6 x 90 x 90 9 x 10 73.b 10 10 84 50 40 84 50 34 84 84 84 74.a x 10 x 44 x 10 x 44 x 10 44 10 10 §3· 24 ă â4ạ 18 21 3 3 20 c 20 9x 9x x 54 54 74.b 10 24 10 30 44 44 24 20 44 44 44 INSTRUCTOR O USE ONLY 14 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 75.a z z 77.a 3z z 3z 8z 3z 3z 11z 11 z x Đ1à 8ă â 11 11 11 11 78.a 7x z Đ4 à 9ă z ©9 ¹ 4z 4x z z 78.b 0 9z 4z 4z 5z 11 11 11 11 11 11 11 11 8 11 11 76.a p 12 p 1 79 80 p 12 p p 12 p 81 5z 5 z The variable terms are the same and will cancel, leaving constants that are not equal Both sides of the equation are identical, so any x value placed in the left hand side of the equation will produce the same result when placed in the right hand side of the equation Answers will vary One possible answer is: x 3x 5p 5p 17 p 82 17 p 17 17 Answers will vary One possible answer is: x x 83 Solving an equation in one variable: Apply the distributive property to clear all parentheses (and other grouping symbols) on both sides of the equation Combine like terms on each side of the equation Use the addition and/or subtraction properties of equality to get all variable terms on one side of the equals sign and all constant terms on the other side Use the multiplication and/or division properties of equality to make the coefficient of the variable equal to Clearing fraction from an equation in one variable: Find the least common multiple of all denominators of all fractions in the problem Multiply both sides of the equation by the least common multiple 17 p § § 2· Ã Đ 2à 76.b ă ă 12 ă â 17 â â 17 ạ Đ 10 à ă 2á 17 â 24 17 10 2 17 10 2 17 17 10 2 17 17 34 10 17 17 24 17 24 17 24 17 24 17 24 17 24 17 4 Equations and an Inequalities in One Variable 77.b x 0 0 x 3x 3x 4x 11z 11 11 § §1· · 75.b ă ă 1á â â 11 ạ Đ3 à ă 1á â 11 ¹ 1 11 1 11 1.3 x Đ3 à 7ă xá â7 3x 84 INSTRUCTOR R USE ONLY 15 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra Since x can be any number ber larger than 6, the interval interva begins at and goes to positive infinity Since is not part of the solution, use a ( on the left side of the interval When infinity is a right endpoint of an interval, that side of the interval always uses a ) So the interval is written (6, ∞) 86 Since x can be any number smaller than or equal to 2, the interval begins at negative infinity and continues to Since is part of the solution, use a ] on the right side of the interval When negative infinity is a left endpoint of an interval, that side of the interval always uses a ( So the interval is written (−∞, 2] 87 Since x can be any number smaller than or equal to 2, the number line is shaded beginning at negative infinity and continuing to Since is part of the solution, use a ] on the right side of the shaded line 88 Since x can be any number larger than 6, the number line is shaded beginning at and continuing to positive infinity Since is not part of the solution, use a ( on the left side of the shaded line 89.a x d 15 9 9 x d 24 x 24 d 3 xd8 89.b Check with any number less than or equal to Letting x = 7: 85 911 a 91.a 3 x 15 9 9 3 x 24 3 x 24 ! 3 3 x ! 8 91.b Check with any number greater than −8 Letting x 5 : 3 5 15 15 15 15 91.c (−8, ∞) 92.a 4 x 12 16 12 12 4 x 28 4 x 28 ! 4 x ! 7 92.b Check with any number greater than −7 Letting x 6 : 4 6 12 16 24 12 16 12 16 92.c (–7, ∞) 93.a 3 x ! 15 9 9 3 x ! 3 x 6 3 x 2 93.b Check with any number less than Letting x : d 15 21 d 15 12 d 15 89.c (−∞, 8] 90.a x 12 d 16 12 12 3 ! 15 x d 28 x 28 d 4 xd7 90.b Check with any number less than or equal to Letting x : 3 ! 15 12 ! 15 93.c (−∞, 2) 94.a 4 x 12 ! 16 12 12 4 x ! 4 x 4 4 x 1 94.b Check with any number less than Letting x : 12 d 16 12 d 16 90.c (−∞, 7] 4 12 ! 16 12 ! 16 INSTRUCTOR O USE ONLY 16 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 1.3 Equations and an Inequalities in One Variable 98.b 98 b Check with any number lless than −27 Letting r 30 : 94.c (−∞, 1) 95.a x t 15 9 9 x t 6 x 6 t 3 x t 2 95.b Check with any number greater than or equal to −2 Letting x : 30 30 10 35 29 20 35 58 20 35 38 35 98.c (–∞, –27) 99.a n 50 ! 2n 4n 200 ! 6n 18 6n 200 6n 200 2n ! 218 2n 218 2 2 n 109 99.b Check with any number less than −109 Letting n 110 : t 15 9 t 15 95.c [−2, ∞) 96.a x 12 t 16 12 12 x t 4 x 4 t 4 x t 1 96.b Check with any number greater than or equal to −1 Letting x : 110 50 ! 110 160 ! 214 640 ! 642 99.c (−∞, −109) 12 t 16 12 t 16 96.c [−1, ∞) 100.a w 15 ! 5w 14 8w 120 ! 10 w 28 10 w 120 10 w 120 2 w ! 148 97.a d d 15 2d d 15 2 w 148 2 2 w 74 100.b Check with any number less than −74 Letting w 75 : d 15 6 6 d 9 97.b Check with any number less than −9 Letting d 10 : 75 15 ! 75 14 10 10 15 90 ! 361 9 2 15 720 ! 722 100.c (−∞, −74) 101.a x 15 x 18 15 16 15 97.c (–∞, –9) 98.a r r 10 35 Ã Đ Ã Đ 12 ă x 12 ă15 x © ¹ © 9 x 96 180 x 2r r 10 35 r 35 8 8 r 27 8x 8x x 96 180 96 96 x 84 1 x ! 1 84 x ! 84 INSTRUCTOR R USE ONLY 17 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 105 05 a 12 y t 2 y 105.a 2y 2y 101.b Check with any number greater than −84 84 Letting x 72 : 72 15 72 54 15 48 62 63 101.c (−84, ∞) 102.a x 18 ! 34 x 10 y t 10 y d 10 10 yd0 105.b Check with any number less than or equal to Letting y 1 : 12 1 t 2 1 Ã Đ Ã Đ ă x 18 ! ă 34 x ¹ © ¹ © 3 x 144 ! 272 10 x 10 x 144 +144 10 x 13x ! 416 12 t 105.c (−∞, 0] 106.a 24h t 2h 2h 2h 22h t 22h d 22 22 hd0 106.b Check with any number less than or equal to Letting h 1 : 13 x 416 13 13 x 32 102.b Check with any number less than −32 Letting x 40 : 40 18 ! 34 40 15 18 ! 34 50 3 ! 16 102.c (−∞, −32) 103.a x ! 36 § · 2 ă x 2 36 â x 72 103.b Check with any number less than −72 Letting x 74 : 24 1 t 2 1 24 t 106.c (−∞, 0] x 107.a d0 Đ x 5à 18 ă d 18 â2 9ạ x 10 d 10 10 x d 10 x 10 d 9 10 xd 74 ! 36 37 ! 36 103.c (−∞, −72) x 52 104.a § à 3 ă x ! 3 52 â ¹ x ! 156 104.b Check with any number greater than −156 Letting x 153 : 107.b Check with any number less than or equal to Letting x 2 : 2 d0 1 d 9 1 d 9 d0 9 d0 153 52 51 52 104.c (−156, ∞) 10 INSTRUCTOR O USE ONLY 18 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 10 Đ 107.c ă f, ằ 9ẳ © 108.a 1.3 Equations and an Inequalities in One Variable 111.b 111 b Check with any number lless than Letting k 1 : 2 1 z d0 Đ z 5à 24 ă d 24 â3 8ạ z 15 d 0 2 111.c (−∞, 0) 112.a d 8c 8c t 8 8 0tc 112.b Check with any number less than or equal to Letting c 1 : 15 15 z d 15 z 15 d 8 15 zd d 8 1 108.b Check with any number less than or equal to 0d8 112.c (−∞, 0] 113.a x t x 15 6 x x Letting z 2 : 2 d0 8 d0 8 16 15 d0 24 24 d0 24 15 Đ 108.c ă f, ằ 8ẳ â 109.a 2x t 2x t 2 xt 113.b Check with any number greater than or equal to Letting x 2: t 16 t 12 f !3 7t6 1 f 1 ê3 à 113.c ô , f ơ2 114.a 12 x t 10 x f 3 109.b Check with any number less than −3 Letting f 4 : 10 x 10 x 4 ! x t 1 x t 1 xt 4!3 109.c (−∞, −3) p ! 110.a 1 p 1 114.b Check with any number greater than or equal to Letting x : p 4 110.b Check with any number less than −4 Letting p 5 : 12 t 10 3 t 4 5 ! ª · 114.c ô , f 115.a x 12 ! x 12 6 x 12 x 12 x!0 5!4 110.c (−∞, −4) 111.a 2k 2 k ! 2 2 0!k INSTRUCTOR R USE ONLY 19 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 115.b Check with any number greater than 00 Letting x : x t 340 20 § x · § à 340 ă t 340 ă â 340 â 20 x t 119 119.b Check with any number greater than or equal to 119 Letting x 120 : 119.a 12 ! 12 12 ! 12 5 ! 6 115.c (0, ∞) 116.a x 34 ! x 34 4 x 34 x 34 x!0 116.b Check with any number greater than Letting x : 120 120 OR t t 340 20 340 20 0.3529 t 0.35 120 17 t 340 20 17 120 119 t 340 340 119.c [119, ∞) x t 120.a 945 15 § x · §7· 945 ă t 945 ă â 945 â 15 ¹ x t 441 120.b Check with any number greater than or equal to 441 Letting x 442 : 34 ! 34 34 ! 34 29 ! 30 116.c (0, ∞) 117.a x d x x 24 d x 45 x 24 x 24 x d 21 x 21 d 3 x d 7 117.b Check with any number less than or equal to −7 Letting x 8 : 442 442 OR t t 945 15 945 15 0.4677 t 0.4666 442 63 t 945 15 63 442 441 t 945 945 120.c [441, ∞) 15 x 121.a 16 336 15 § Ã Đ x à 336 ă 336 ă â 16 â 336 315 x 121.b Check with any number greater than 315 Letting x = 316: 15 316 15 316 16 336 16 336 OR 0.9375 0.94047 15 21 316 16 21 336 315 316 336 336 121.c (315, ∞) 11 x 122.a 12 276 § 11 Ã Đ x à 276 ă 276 ă â 12 â 276 253 x 8 d 8 22 d 17 88 d 85 117.c (−∞, −7] 118.a x d x 15 12 x d x 135 9x 9x 3 x d 132 x 132 d 3 x d 44 118.b Check with any number less than or equal to −44 Letting x 45 : 45 d 45 15 181 d 60 543 d 540 118.c (−∞, −44] INSTRUCTOR O USE ONLY 20 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 1.3 Equations and an Inequalities in One Variable 125.a 125 a 8p 3p 122.b Check with any number 253 mber greater than 25 253 Letting x = 254: 11 254 11 254 12 276 12 276 OR 0.91666 0.920289 11 23 254 12 23 276 253 254 276 276 122.c (253, ∞) 3 p p 5p 5p 5 p 0 125.b Check with any number less than Letting p = −1: 1 1 123.a x x x 8 3 125.c (−∞, 0) 126.a k 3k 3k 3k 4k 4k 4 k 0 126.b Check with any number less than Letting k = −1: x 18 x x x 18 x x 18 x 18 3 x 17 3 x 17 ! 3 3 17 x! 123.b Check with any number greater than 17 1 1 Letting x = 6: 7 3 126.c (−∞, 0) 127.a 8 p p 3p 3p 3 6 12 30 36 31 30 31 11 p § 17 à 123.c ă , f â 11 p ! 11 11 p!0 127.b Check with any number greater than Letting p = 1: 124.a x x x 2x x 7x x 7x 1 8 7x 7x 8 127.c (0, ∞) 128.a 7 k 3k 3k 3k 6 x x 7 ! 6 x! 124.b Check with any number greater than 10k 10k ! 10 10 k !0 128.b Check with any number greater than Letting k = 1: Letting x = 2: 2 14 7 12 14 7 128.c (0, ) 10 13 Đ7 à 124.c ă , f â6 INSTRUCTOR R USE ONLY 21 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 129.a p 3 p 3 p 3p 133.a 11 p 11 p 11 11 p 0 129.b Check with any number less than Letting p = –1: m !1 Đ Ã 12 ă m ! 12 6ạ â4 9m ! 12 2 2 9m ! 10 9m 10 ! 9 10 m! 1 3 1 8 129.c (−∞, 0) 130.a 7k 3k k 3k 133.b Check with any number greater than Letting m = 2: ! 6 !1 6 !1 18 !1 12 12 20 !1 12 § 10 · 133.c ă , f â9 3k 3k 10k 10k 10 10 k 0 130.b Check with any number less than Letting k = –1: 1 3 1 7 130.c (−∞, 0) 131.a 8 p 3 p 3p 3p 5 p 134.a 5 p ! 5 5 p!0 131.b Check with any number greater than Letting p = 1: OR 10 ! 6 !1 1.666 ! n !1 Đ Ã 12 ă n ! 12 4ạ â6 10n ! 12 3 3 8 3 10n ! 8 3 131.c (0, ∞) 132.a 7 k 3k 10n ! 10 10 n! 10 3k 3k 4 k 4 k ! 4 4 k !0 132.b Check with any number greater than Letting k = 1: 7 3 7 3 132.c (0, ∞) INSTRUCTOR O USE ONLY 22 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 1.3 Equations and an Inequalities in One Variable 136.a h6 d h2 §1 à Đ5 à 4ă h 6á d 4ă h 2á â2 â4 2h 24 d 5h 134.b Check with any number greater than 10 Letting n = 1: 5 1 ! 1 ! 6 OR 5 !1 !1 6 1.08333 ! !1 10 !1 12 12 13 !1 12 Đ9 à 134.c ă , f 10 â ¹ 5h 24 5h 24 3h d 32 3h 32 t 3 3 32 ht 136.b Check with any number greater than or equal to Letting h = –10: 5 10 d 10 10 d 10 4 OR 5 d 12.5 50 5 d 11 d 10.5 50 11 d 4 44 50 d 4 44 42 d 4 135.a k 8 d k §1 · §3 · 15 ă k d 15 ă k â3 â5 5k 120 d 9k 30 9k 120 9k 120 4k d 150 4k 150 t 4 4 150 kt 75 kt 2 75 kt ê 32 à 136.c ô , f 137.a 135.b Check with any number greater than or equal to 75 2 w5 §2 · § 4· 9ă w 5á 9ă â ¹ © 9¹ w 45 4 45 45 w 41 Letting k = –30: 30 d 30 10 d 18 w 41 6 41 w 18 d 16 ê 75 à 135.c ô , f ¬ ¹ 137.b Check with any number less than 41 Letting w = 6: 45 1 INSTRUCTOR R USE ONLY 23 © Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ 32 Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 41 Ã Đ 137.c ă f, 6ạ â 138.a 139.b Check with any number greater than Letting h = 1: m 8 10 Đ4 Ã Đ 3à 10 ă m 10 ă â5 © 10 ¹ 8m 80 3 80 80 8m 77 8m 77 8 77 m 5 5 2 20 4 22 ! 2 OR ! 2 4 ! 2 4 ! 4 ! ! 2 ! 2 5.5 ! 1.25 § à 139.c ă , f â 28 77 138.b Check with any number less than Letting m = 5: 10 48 10 4 10 140.a 77 Ã Đ 138.c ă f, â 139.a 28 ! 2h 1· 3· § § ¨ 5h ¸ ! ¨ 2h ¸ â â 20h ! 8h 8h 8h 28h ! 28h ! 28 28 h! 28 5h ! 5c 2à 5Ã Đ Đ ă 9c ! ă 5c 6ạ â © 54c ! 30c 30c 30c 84c ! 84c ! 84 84 c! 84 9c 140.b Check with any number greater than 84 Letting c = 1: 9 2 9 54 6 58 ! 5 ! 5 6 ! 5 6 30 ! 6 25 ! OR ! 5 ! 5 9.666 ! 4.1666 Đ Ã 140.c ă , f â 84 INSTRUCTOR O USE ONLY 24 â Cengage Learning All Rights Reserved Full file at https://TestbankDirect.eu/ .. .Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR O SALE Full file 1atFundamentals https://TestbankDirect.eu/ Chapter of Algebra 42 ± 0.67 π −2... Learning All Rights Reserved Full file at https://TestbankDirect.eu/ Solution Manual for Intermediate Algebra 1st Edition by Bracken NOT FOR R SALE Full file at https://TestbankDirect.eu/ 91 y 3... square root of 36 Since any real number multiplied by itself always results in either or a positive number, there is no number that can be multiplied by itself to give −100 84 54.b [−6, ∞) 55.a 82