THÔNG TIN TÀI LIỆU
Preface This manual contains detailed solutions to all of the exercises of the text College Algebra, eleventh edition, by R David Gustafson and Jeff Hughes Many of the exercises in the text may be solved using more than one method, but it is not feasible to list all possible solutions in this manual Also, some of the exercises may have been solved in this manual using a method that differs slightly from that presented in the text There are a few exercises in the text whose solutions may vary from person to person Some of these solutions may not have been included in this manual For the solution to an exercise like this, the notation "answers may vary" has been included If you are a student using this manual, please remember that only reading a solution does not teach you how to solve a problem To repeat a commonly used phrase, mathematics is not a spectator sport You MUST make an honest attempt to solve each exercise in the text without using this manual first This manual should be viewed more or less as a last resort Above all, DO NOT simply copy the solution from this manual onto your own paper Doing so will not help you learn how to the exercise, nor will it help you to better on quizzes or tests I would like to thank Paul McCombs from Rock Valley College and Cynthia Ashton of Brooks/Cole Publishing Company for their help and support This solutions manual was prepared using EXP 5.1 This book is dedicated to John, who helps me to realize that mathematics cannot describe everything in life May your study of this material be successful and rewarding Michael G Welden Contents Chapter Sets of Real Numbers Chapter Equations and Inequalities 52 Chapter The Rectangular Coordinate System and Graphs of Equations 174 Chapter Functions 236 Chapter Exponential and Logarithmic Functions 309 Chapter Solving Polynomial Equations 359 Chapter Linear Systems 410 Chapter Conic Sections and Quadratic Systems 505 Chapter Sequence, Series and Probability 551 Chapter The Mathematics of Finance 585 SECTION 0.1 Exercises 0.1 (page 13) set subset union intersection decimal variable # even composite 10 rational 11 decimals 12 Ÿ 13 negative 14 15 B 16 CB 17 &7 18 commutative, multiplication 19 interval 20 no 21 two 22 half-open 23 positive 24 distance 25 Every natural number is a whole number, so N § W TRUE 26 Every rational number is a real number, so Q § R TRUE 27 The rational number "# is not a natural ¸ N FALSE number, so Q § 28 Every integer is a rational number, so Z § Q TRUE 29 Every whole number is an integer, so W § Z TRUE 30 31 &†# E ∪ F œ ea, b, c, d, e, f, gf 32 C D The real number È# is not an integer, Z FALSE so R Đ E ∩ F œ ed, ef 33 E ∩ G œ ea, c, ef 34 F ∪ G œ ea, c, d, e, f, gf 35 * œ !Þ&'#&; terminates "' 36 $ œ !Þ$(&; terminates ) 37 $ œ !Þ#(#(#(ÞÞÞ; repeats "" 38 & œ !Þ%"''''ÞÞÞ; repeats "# 39 natural: ", #, ', ( 40 whole: !, ", #, ', ( 41 integers: 42 rational: 43 irrational: È# 46 even: 49 oqñqñqñqp # $ % &, %, !, ", #, ', ( %, !, #, ' 44 prime: #, ( 47 odd: &, ", ( 50 &, %, # $, !, ", #, #Þ(&, ', ( 45 composite: ' 48 negative: &, oñqqñqqñqñp ' ) * % %, # $ SECTION 0.1 51 oñqñqqñqñp "" "$ "( "* 52 oñqñqñqñqñqñqñp # " ! " # $ % 53 oñqñqñqñp $ # " % 54 oñqñqñqñp ) ' # % 55 qđqđqđqđp & $ " " $ 56 qqqđqqqqđqp !Þ( "Þ(& $ 57 B # p #, ∞ 58 %p ∞, % íïïĐqqp # % # B $p #, $ 61 oqÐïïĐqp # #ŸB 63 # % 67 oqqỊïïỵ & BŸ!p & # Ÿ B Ÿ $ p c #, $d ! 70 oqỊïïĨqp # 71 B & and B 72 & &, !d oqỊïïĐqp % Ÿ B Ÿ % p c %, %d $B # %p ∞, % ∞, % % p c $, % $ŸB 68 &, ∞ ∩ ∞, % oqÐïïïỵ & íïïïĐqp % oqÐïïĐqp & % #p $ % # Ÿ B Ÿ $ p c #, $d oqỊïïĨqp ' &, ∞ &, ∞ ∩ íïïĨqqp % oqỊïïĨqp 73 ∞, & d oqỊïïĨqp $ ' B # p # Ÿ B Ÿ ' p c#, 'd # ∞, $ BŸ&p 65 oqÐïïÓqp " 69 " % $p $ %, "d BŸ"p & íïïĐqqp oqÐïïĨqp # " p c ", ∞ B 62 oqqÐïïỵ 64 & p !, & oqÐïïÑqp %, ∞ % # p c #, # B ! %p $ oqỊïïĐqp B B ! 59 oqqÐïïỵ 60 66 B ( ) % $ SECTION 0.1 74 B 'p c $, ∞ ∩ $ and B c $, ∞ oqỊïïïỵ $ íïïïĐqp ' oqỊïïĐqp ∞, ' c $, ∞ ∩ 75 B ∞, ' $ ) and B Ÿ c ), ∞ ∞, B $p c ), ∞ ∩ $d c ), ∞ ∩ 76 ∞, ∞, ( d ", ∞ ∩ B # or B $d ∞, ) ∞, ( d " and B Ÿ (p ", ∞ ∩ ∞, ( d ' oqỊïïïỵ ) íïïïĨqp $ oqỊïïĨqp $d ", ∞ 77 ∞, ' $ oqÐïïïỵ " íïïïĨqp ( oqÐïïĨqp #p " ∞, ( # ∪ #, ∞ 78 íïĐqqÐïỵ # 79 BŸ # " or B $ p ∞, íïĨqqỊïỵ " 81 83 85 "d ∪ c$, ∞ & 80 B !p ∞, $ or B # p $ Since ! !, k!k œ ! ) œ ) ! ∞, íïĐqqỊïỵ $ !, k )k œ ) œ ) & or B íïĨqqÐïỵ Since "$ !, k"$k œ "$ Since ) k )k œ BŸ "( !, k "(k œ "( œ "( !, k #&k œ #& œ #& Since 84 Since '$ !, k'$k œ '$ k'$k œ '$ œ '$ 86 Since $ ∪ c#, ∞ # 82 #& & d ∪ !, ∞ SECTION 0.1 87 89 91 Since $# !, k$#k œ $# k$#k œ $# œ $# Since & !, k1 &k œ & œ k1 &œ& Since ' k 'k œ 90 Since ) 92 Since #1 !, k#1k œ #1 1k œ k!k œ ! !, k 'k œ ' œ ' 88 !, k) 1k œ ) 93 If B #, then B " ! Then kB "k œ B " 94 If B Ÿ #, then B " kB "k œ B " 95 If B !, then B % kB %k œ B % 96 If B "!, then B kB (k œ B ( 97 99 distance œ k) distance œ k $ ! Then $k œ k&k œ & ) k œ k& k œ & 98 distance œ k"# 100 distance œ k #! ' œ ' ! Then ( ! Then & k œ k"(k œ "( 'k œ k #'k œ #' 101 Since population must be positive and never has a fractional part, the set of natural numbers should be used 102 Since the subdivisions on a ruler are measured in fractions of an inch, the set of rational numbers should be used 103 Since temperatures are usually reported without fractional parts and may be either positive or negative (or zero), the set of integers should be used 104 Since the financial condition of a business is usually described in terms of dollars and cents (fractional parts of a dollar), the set of rational numbers should be used 105 B will represent a positive number if B itself is negative For instance, if B œ $, $ œ $, which is a then B œ positive number 106 Every integer is a rational number because every integer is equal to itself over 107 The statement is always true 108 The statement is always true 109 The statement is not always true For example, let + œ & and , œ # 110 The statement will be true if + ! and , !, or if + Ÿ ! and , Ÿ ! 111 The statement + , - could be interpreted to mean that + -, when this is not necessarily true Exercises 0.2 factor 112 k, +k œ k " + , k œ k " k † k+ œ k+ , k ,k (page 24) natural $, #B exponential SECTION 0.2 13 15 17 scientific, integer 23 œ B8 C8 10 Answers may vary B7 œ B7 B8 "$# œ "$ † "$ œ "'* &# œ "†&†& œ #& %B$ œ % † B † B † B % &B 19 21 BC œ )B% œ &B &B &B &B B 27 $> $> )†B†B†B†B 11 B! œ " 14 "!$ œ "! † "! † "! œ ",!!! 16 & 18 %B $> œ $ $ $ >$ œ BBBCC œ B$ C# 30 32 (Þ"% œ #&%"Þ"')" 33 !Þ&% œ 35 B# B$ œ B# 36 C$ C% œ C$ œ B& & % œ )B 31 40 +$ +' +% œ +* +% œ +"$ $ 42 >$ % >& 43 +# $ +% # œ +' +) œ +"% 44 +# % +$ 45 $B $ œ $$ B$ œ #(B$ 46 #C % œ 47 B# C 48 B$ D % ' œ B$ 51 B ! œ" œ +' ,$ œ C( 50 52 % D# œ D ' D #! œ D #' 49 #Þ#$ œ "!Þ'%) 37 œ C( & $ " † # † # † # † # † ,% "', % % D% $ )B 34 $ +# +# Œ œ $ , , )B !Þ!'#& D# $ )B #, #, #, #, œ œ 41 œ B# C$ œ B' C$ " B8 )C% ( $ $ œ '†B†B +++,,,, œ +$ , % C& C# œ B78 œ %B %B %B >' 39 B & œ #& 38 œ >'†( œ >%# B7 #+ #+ #+ œ # † # † # † +$ œ )+$ 28 29 œ )CCCC œ 26 #(>$ $ )B 22 " B# œ B# " # 12 'B# œ 24 B œ $ B7 B8 œ B7 20 (BBB œ (B$ 25 %B! œ % † " œ % œ C#" Œ # !Þ# $ œ !Þ!!"' œ D #†$ œ D' œ >"# >"! œ >## $ œ +) +* œ +"( # % C% œ "'C% ' D% ' œ B") D #% B B% B% œ œ % C$ C"# C$ % 53 %B ! œ" SECTION 0.2 #B! œ 54 #†" œ 57 C #C $ œC 59 B$ B % # 61 B( œ B( B$ 63 +#" œ +#" +"( & # D 55 % œ " D% " C& œ # 67 69 64 >"$ œ >"$ >% 66 =* =$ ="# œ % œ ="# # = =# 68 >% % $ $ " $ $ Œ $ œ ˆ> ‰ œ ˆ> ‰ œ > > 70 > œ œ " B# #Œ " &œ + " + " & œ +# #+ " & œ #+ " &œ+ " %œ+ # B $ œ B &B ' œ B# &B ' œ (B "#B ' œ "# "#B œ ' Bœ 53 + œ $ B % (B "# "# ' œ "# " # % SECTION 1.1 54 $ #BŒ B $ B " % œ # B " % œ #B † # B ' Bœ) Bœ# $ 56 B $ $B " $ œ B B # 57 " $ œB B # † # B B # " B # œ $B $B B # œ $B %B # œ $B Bœ# The answer does not check Ê no solution BB 58 #Œ # # 55 B $B "Œ B # " " œ $ B " " " œ$ B B " $ ' " B " œ$ " ' B "œ$ B (œ$ Bœ % # 59 + + ( + # + $ † + # œ ( + # + # œ + ( + # # + $ œ% + #+ ' œ %+ #+ œ $% + œ "( % $ + # ( + # + ( #) 57 " B " *> ' ( œ > $ >> $ *> ' ( >> $” •œ> > $ † >> $ > $ *> ' œ (> #> ' œ ! #> œ ' >œ $ The answer does not check Ê no solution #B " $B# œ $B & $B & # #B " $B# $B & ”B • œ $B & † $B & $B & # B $B & # #B " œ $B # $B &B %B # œ $B# B #œ! Bœ # B " † $ † % + $ + # SECTION 1.1 # 60 " # # 61 62 # "Œ # " " # " " œ 8# œ $B B# # # " # " † # &B ' B ' % #B $ $B # &B # œ B $ B # B $ B # B # B B # #B $ B # $B # œ B $ &B #B# B ' $B# %B % œ &B# "$B &B# $B "! œ &B# "$B $B "! œ "$B ' "!B œ % % # Bœ œ "! & B# $ # " œ 8 # " #8 " " # œ" #8 # # œ " $8 œ " " 8œ $ #B " œ " # " &B B# # # ' ' emultiply by common denominatorf $B #B B # œ # B# B B# &B B 'B & $B #B B # œ BB " BB & B & B " $ # B # œ B " B & B & B " emultiply by common denominatorf $B & # B " œB # $B "& #B # œ B # B "$ œ B # "$ Á # Ê no solution 58 SECTION 1.1 63 $B B$ # $B % B # B# $B & $ # $B œ B # B# #B % B # B # B # B# ˆB# #B %‰ $ œ # B # $B B # $B & $B# 64 " ) " 65 ) &8 " "" " "" " "" 66 "! $ B# œ $B# 'B "# œ 'B# )B ) 'B# (B # œ 'B# )B ) "&B œ "! "! # Bœ œ "& $ # #B % # #B % # emultiply by common denominatorf $8 % " œ %#8 "' &8 # $8 % " œ &8 # ) &8 # # " $8 % œ ) emultiply by common denominatorf &8 # $8 % œ ) #8 ' œ ) 8œ# &8# # $8 " " œ (8# (%8 $$ (8 $ # $8 " " œ # (8 (%8 $$ (8 $ '8 # " œ (8 $ "" (8 $ (8 $ '8 # œ "" " (8 $ '8 # œ "" & œ "" #8 œ ' 8œ$ % # # "$+ %) + ")+ % # "' + $ + "' + %+ # #+ %+ ) #+ #+ +# + B & ) $# œ +# emultiply by common denominatorf " + ' " œ # + $ + $ œ " + "' ' œ + "' "% œ + "' +œ # 59 # emultiply by common denominatorf SECTION 1.1 67 & C # % C ' œ # & C % œ C % C & C # œ% C &C "! œ %C &C "! œ %C Cœ& ' 68 #+ ' $ ' $ " 'C ) " # C # C % % " emultiply by common denominatorf "' " "& C# # $+ œ " %+ $ " $ + " " $ + " emultiply by common denominatorf +# $ œ + + $ " œ + $ + " + $+ " " + $ œ" $+ $ + $ œ " %+ ' œ " %+ œ ( ( +œ % #+ 69 70 $ $" $C #C ' $C #C % $C #C $# C #C # C C # C # C C # C C # C #C C# #C C# %C C $ œ œ œ ) % # C# ) C # C ) C # C emultiply by common denominatorf # œ) œ) œ) œ # Ê The solution does not check, so the equation has no solution # $+ &+ # œ # +# ' + + % #+ $ $+ # &+ # œ + # + $ + $ + # + # + + # #+ $ + # $+ # œ + $ &+ #+# + ' $+# %+ % œ &+# "$+ &+# $+ "! œ &+# "$+ "!+ œ % % # +œ œ "! & +# #+ ' &+ 60 # # ' ' emultiply by common denominatorf SECTION 1.1 + 71 + # + + #” # + ++ +# # #+ +# #+ + + + # + # + ˆ+# %+ +# 72 B 73 76 $ B " B# %B œ #Þ#: #Þ#: œ #Þ# #Þ# œ: #Þ# " # 1< $ " œ $ †
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