A Answers to All Eercises and Tests Appendi A Appendi A. (page A) Vocabulary Check (page A). rational. irrational. absolute value. composite. prime. variables; constants. terms. coefficient 9. Zero-Factor Property. (a),,,,, (c) 9,,,,,, (d),, 9,,,,,, (e). (a),,, (c),,,, (d),,,.,,,, (e). (a) (c),, (d).,,,,.... (e)..... (a) (c), (d),,.,. (e)....,. (a),, (c),,, (d),,.,,, (e),. (a), 9,,, 9,, (c),, 9,, (d), 9,.,,,.,.... 9..... <.. <.. >.. <..... < < > < 9. (a) denotes the set of all real numbers less than or equal to. (e) (c) Unbounded. (a) denotes the set of all real numbers greater than or equal to. (c) Unbounded. (a) < denotes the set of all real numbers less than. (c) Unbounded. (a) > denotes the set of all real numbers greater than. (c) Unbounded. (a), denotes the set of all real numbers greater than or equal to. (c) Unbounded. (a), denotes the set of all real numbers less than. (c) Unbounded. (a) < < denotes the set of all real numbers greater than and less than. (c) Bounded. (a) denotes the set of all real numbers greater than or equal to zero and less than or equal to. (c) Bounded. (a) < denotes the set of all real numbers greater than or equal to and less than. (c) Bounded. (a) < denotes the set of positive real numbers less than or equal to. (c) Bounded 9. (a), denotes the set of all real numbers greater than or equal to and less than. (c) Bounded. (a), denotes all real numbers greater than and less than or equal to. (c) Bounded. <. y <. y. y. t. k <. W >..% r % 9........ 9.. 9... >. <.
Answers to All Eercises and Tests A9. >.... 9...99. $, $, $ > $.$, $ Because the actual epenses differ from the budget by more than $, there is failure to meet the budget variance test.... $ < $.$9 $ Because the difference between the actual epenses and the budget is less than $ and less than % of the budgeted amount, there is compliance with the budget variance test. $, $, $ < $.$, $ Because the difference between the actual epenses and the budget is less than $ and less than % of the budgeted amount, there is compliance with the budget variance test. $ $ $ < $.$ $. Because the difference between the actual epenses and the budget is less than $ and less than % of the budgeted amount, there is compliance with the budget variance test.. (a) $9 $9 Surplus or deficit (in billions) Year Ependitures Surplus or deficit (in billions) (in billions) 9 $9. $. (s) 9 $9. $. (d) 9 $9.9 $. (d) 99 $. $. (d) $. $. (s) 9 9. (s). (d). (d) Year. (d). (s) 9 9 9 99. and older.9%.% Under.%.%.%.. 9.... miles. and are the terms; is the coefficient.. and are the terms; and are the coefficients..,, and are the terms; and are the coefficients.. and are the terms; is the coefficient..,, and are the terms; and are the coefficients.. and are the terms; and are the coefficients. 9. (a). (a). (a). (a). (a) Division by is undefined.. (a) Division by is undefined.. Commutative Property of Addition. Multiplicative Inverse Property. Multiplicative Inverse Property. Additive Inverse Property 9. Distributive Property 9. Additive Identity Property 9. Multiplicative Identity Property 9. Distributive Property 9. Associative Property of Addition 9. Associative and Commutative Properties of Multiplication 9. Distributive Property 9. Associative Property of Multiplication Multiplicative Inverse Property Multiplicative Identity Property 9. 9 9. 99....... (a) n.... n,,, APPENDIX A The value of n approaches infinity as n approaches.. (a) n,, n.... The value of n approaches as n increases without bound.
A Answers to All Eercises and Tests. False. If a < b, then where a b. a > b,. False. The denominators cannot be added when adding fractions. 9. (a) No. If one variable is negative and the other is positive, the epressions are unequal. u v u v The epressions are equal when u and v have the same sign. If u and v differ in sign, u v is less than u v.. Yes. y is nonnegative if y. y is positive if y >.. The only even prime number is, because its only factors are itself and.. Real numbers. (a) Negative Negative. (a) Positive Positive. Yes. a if a <. a Irrational numbers Rational numbers Appendi A. (page A) Vocabulary Check (page A). eponent; base. scientific notation. square root. principle nth root. inde; radicand. simplest form. conjugates. rationalizing 9. power; inde....9.. (a). (a) 9. (a) 9. (a) 9. (a). (a). (a). (a)........ 9....... (a) z. (a) 9. y. (a) z 9. (a) y Integers...,,,,,,,,... Noninteger fractions (positive and negative) Negative integers...,,,, Whole numbers,,,... Natural numbers,,,... Zero y. (a). (a) r y. (a). (a) z. (a) y 9 b. (a) n y a. (a).. square miles. 9. kilometers 9..99 gram per cubic centimeter..9 inch.,,, ounces.,,c.. coulomb..9 meter. (a),,. (a),.. (a) 9... (a).9.9 9. (a),.9 9.9. (a)... (a). (a). (a). (a). (a). (a). (a)... (a).. 9. (a)... (a).9.9. (a). (a). (a). (a) z. (a) z. (a) a. (a) >. (a) z 9. (a). (a). (a) y. (a). (a). (a).. 9.. >......... 9. 9. 9. 9. y b y
Answers to All Eercises and Tests A 9. 9. 9. 9. 9. y 9., > 9. 99. (a). (a). (a). (a) a ab.. seconds.. inch. (a) h t.9.. 9... h 9 t.9....9.9 t..9.. minutes. True. When dividing variables, you subtract eponents.. False. When a power is raised to a power, you multiply the eponents: a n k a nk. a 9. a, a, using the property m a m a n amn : a m amm a.. (a) is also raised to the power, so. When two powers have the same base, the eponents are added: y y y. (c) When a power is raised to a power, the eponents are multiplied: a b a b. (d) The square of a binomial contains a cross-product term: a b a ab b. (e) If <, then > but < :. (f) Radicals can be added together only if they have the same radicand and inde:.. When any positive integer is squared, the units digit is,,,,, or 9. Therefore, is not an integer.. No. Rationalizing the denominator produces a number equivalent to the original fraction; squaring does not. Appendi A. (page A) Vocabulary Check (page A). n; a n ; a. descending. monomial; binomial; trinomial. like terms. First terms; Outer terms; Inner terms; Last terms. factoring. completely factored. d. e. b. a. f. c.. 9... (a) Degree: ; Leading coefficient: (c) Binomial. (a) Degree: ; Leading coefficient: (c) Trinomial. (a) Degree: ; Leading coefficient: (c) Trinomial. (a) Degree: ; Leading coefficient: (c) Monomial. (a) Degree: ; Leading coefficient: (c) Binomial. (a) y y Degree: ; Leading coefficient: (c) Trinomial. (a) Degree: ; Leading coefficient: (c) Monomial. (a) t 9 Degree: ; Leading coefficient: (c) Binomial 9. (a) Degree: ; Leading coefficient: (c) Trinomial. (a) Degree: ; Leading coefficient: (c) Binomial. (a) y Degree: ; Leading coefficient: (c) Monomial. (a) y y y Degree: ; Leading coefficient: (c) Trinomial. Polynomial:. Not a polynomial because of the negative eponent. Not a polynomial because it includes a term with a negative eponent. Polynomial:. Polynomial: y y y. Not a polynomial because of the square root 9...... 9...... z. y y y.. y y y 9. z z..... 9. y y. APPENDIX A
A Answers to All Eercises and Tests 9 r. y y.. 9.. 9..... y. 9y. 9. 9. y y.... y y y. y y y. 9. 9. m n m 9. y y 9.. y y y 9 y y y.. 9a b. 9. 9t.. t 9... 9..y 9y 9 9....y 9... u. y. y... 9. 9. y 9. 9. 9. 9. 9. 9. 9. 9. y 99.. yy y.. y y. 9 9.. y y. y y.. y y 9.. zz. u vu v. y y... t.. y. y 9 9. u v. y.. z.. 9.. y y y z z z. t t t. 9. 9 u vu uv 9v. y y y... s s. t t. y y. z z.. 9..... z z. u u.... 9.......... 9.......... yy y 9. 9. ).. 9.... u u.. 9. t t... t t t.... 9.. 9. 9. 9. 9. 9.,,, 9.,,,,, 9. 9.,,,,,,,,,,,, 9. Two possible answers:, 9. Two possible answers:, 99. Two possible answers:,. Two possible answers: 9,. (a) P, $,. (a) P $. (a) r r r % % %. (a) r r % % r $. $. $. $. $. (c) The amount increases with increasing r. r r r r % % r $. $. $. % r % r $9. $9. (c) The amount increases with increasing r. %
Answers to All Eercises and Tests A. (a). (a).. 9. (a). (a) T..99. mihr.. (c) The stopping distance increases at an accelerating rate as speed increases. V (cm) V cm V (cm) V cm T (feet).9.9.. r. r.. 9. (a) hr rr r. kq. False.. False. 9. True. a b a ba b. False. A perfect square trinomial can be factored as the binomial sum squared.. m n. n.. 9. If either or y is zero, then y y. 9... n y n n y n n y n n n y n y n n y n is completely factored.. Answers will vary. Sample answer: The roots of the equation can be found when a polynomial is in factored form.. Answers will vary. Sample answer: Appendi A. (page A) Vocabulary Check (page A) V R r R r h. domain. rational epression. comple. smaller. equivalent. difference quotient APPENDIX A... All real numbers. All real numbers. All nonnegative real numbers. All positive real numbers. All real numbers such that. All real numbers such that. All real numbers such that. All real numbers such that 9.,., y..., y y,. y.., y.,., 9. y, y.,.,. y., y, y.., 9,
A Answers to All Eercises and Tests.. z, ± yy. y y, y 9. The epressions are equivalent ecept at.. The epressions are equivalent ecept at.. The epression cannot be simplified.. The polynomial,, can not be factored.., r.,..,,. r, r r., y, 9. t y. t t, t y, y.,., ±...... 9....,. The error was incorrect subtraction in the numerator.. The error was an incorrect epansion of in the numerator...,,,.,,., 9..., > t t Undef. Undef.......,. 9. h, h. h h, h. h, h. h. h,.. z z h, h. h, h.., 9. (a) minute minute(s) (c) minutes. t MN P. (a) 9.9% 9.9% NMN P ;. (a).% MN P.% NMN P ;. (a) t T.9... t T....9.. The model is approaching a T-value of.. (a) Year, t Banking Paying bills (millions) (millions).9.9.......... The estimates are fairly close to the actual numbers of households. (c).t Rate.t.t..t.t.9t.t.
Answers to All Eercises and Tests A (d) Eplanation will vary.. False. In order for the simplified epression to be equivalent to the original epression, the domain of the simplified epression needs to be restricted. If n is even,,. If n is odd,.. False. The two epressions are equivalent for all values of such that.. Completely factor each polynomial in the numerator and in the denominator. Then conclude that there are no common factors. Appendi A. (page A) Vocabulary Check (page A). equation. solve. identities; conditional. a b. etraneous. quadratic equation. factoring; etracting square roots; completing the square; Quadratic Formula. Identity. Conditional equation. Conditional equation. Identity. Identity. Identity. Identity. Identity 9. Conditional equation. Conditional equation... 9.... 9. 9. No solution. All real numbers..... 9.. No solution. The -terms sum to zero.. No solution. The -terms sum to zero. 9. 9........ No solution. The variable is divided out.. 9. No solution. The solution is etraneous. Year Ratio..9. Year Ratio....... No solution. The solution is etraneous.. No solution. The solution is etraneous.... All real numbers. No solution. The -terms sum to zero. 9..... 9..,.,.,. 9, 9....,, ±.,.,.,.,. a. a b, a b 9. ±. ±. ±. ±. ±. ±.,.,. ±. ± 9. ±. ±.. 9.,.,.,. ±. ±. ± 9. ± 9. No solution 9. ± 9 9., 9 9. 9., 9. 9.,,, 9. ± 9. ± 99. ±. ±. ±. ±.. ± ±.... ± ± ± 9... ± ±. ±. ±. ±. ± 9. ±..9,...,. 9..,...,...,...,.9..9,...99,.9. ±.,.,. 9. ±. ±.. b. ± 9 a, b a. ±., ±., ±. ±. ± 9...,.,.,,.. ±.. ±, ±. ± 9. ±, ±. ±.,... 9.... 9.,.,.. No solution. 9.. ±. 9,. ±. ± 9..,,.,.,. ±.,., APPENDIX A
A Answers to All Eercises and Tests. ±.,. ± 9.,.,.,.,,.,.,. (a). inches Yes. The estimated height of a male with a 9-inch femur is 9. inches. (c) Height, Female Male femur length femur length..9 9. 9... 9...... inches (d).9; There would not be a problem because it is not likely for either a male or a female to be inches tall (which is feet inches tall)..,. miles. y.t ; after about hours. (a) ww (c) w feet w l feet w + 9. 9. inches inches inches. centimeters 9.. inches 9. miles per hour and miles per hour 9. (a) 99 During 9., passengers 9. units 9., units 9. False. The equation cannot be written in the form a b. 9. False. The product must equal zero for the Zero-Factor Property to be used. 99. False. See Eample on page A.. False. has only one solution to check,.. Equivalent equations have the same solution set, and one is derived from the other by steps for generating equivalent equations.,. Remove symbols of grouping, combine like terms, reduce fractions. Add (or subtract) the same quantity to (from) each side of the equation. Multiply (or divide) each side of the equation by the same nonzero quantity. Interchange the two sides of the equation.. Yes. The student should have subtracted from both sides to make the right side of the equation equal to zero. Factoring out an shows that there are two solutions, and.. (a) and, (c) The method used in part (a) reduces the number of algebraic steps..... 9... a 9, b 9. Isolate the absolute value by subtracting from both sides of the equation. The epression inside the absolute value signs can be positive or negative, so two separate equations must be solved.. (a), b, a Appendi A. (page A) Vocabulary Check (page A). solution set. graph. negative. solution set. double. union.. Bounded. <. Bounded. >. Unbounded.. Unbounded. <. Unbounded.. Unbounded. b. f 9. d. c. e. a. (a) Yes No (c) Yes (d) No. (a) No No (c) Yes (d) No. (a) Yes No (c) No (d) Yes. (a) No No (c) Yes (d) No. (a) Yes Yes (c) Yes (d) No. (a) No Yes (c) No (d) Yes 9. <. <. <. <..
Answers to All Eercises and Tests A. >.. < <. <.. 9. 9 < <. < 9. < <. < < 9.... No solution. 9. <. <.. <.. >.. > 9. No solution. No solution.. All real numbers.. < <. <, >. <, >. <, > 9 9.,. < <. < <. <, > 9. 9,. 9.. >.. <.. <, >.., 9.. 9 (a) (a).. 9 APPENDIX A.,. < < 9 9 (a) (a)
A Answers to All Eercises and Tests.. (a) (a),,.,.,.,., 9.,.,. All real numbers within eight units of. All real numbers more than four units from...... > 9. > 9. 9. > 9. More than, copies 9. r >.% 9. r > % 9. 9.,9 9. 9. weeks 99. (a) 9. (a) and (c) (d). (e) An athlete s weight is not a particularly good indicator of the athlete s maimum bench-press weight. Other factors, such as muscle tone and eercise habits, influence maimum bench-press weight.. (a) t t >. (a) t t... square inches area 9. square inches. square centimeters area 9. square centimeters. Might be undercharged or overcharged by $.9.. $... < t <... h... 9. 9 9. h. (a) vibrationssecond. millimeters (c). millimeters t. millimeters (d) v < vibrationssecond t. h. False. c has to be greater than zero.. False. If, then and.. b. One set (the solution is not unique): a, b, c Appendi A. (page A) Vocabulary Check (page A). numerator. reciprocal. Change all signs when distributing the minus sign. y y. The is distributed to both terms. z z. Change all signs when distributing the minus sign.. The epression on the right should be negated.. z occurs twice as a factor. zz z. yz is one term, not two. yz yz. The fraction as a whole is multiplied by a, not the numerator and denominator separately. a a y y. The eponent also applies to the coefficient. 9. 9 cannot be simplified.. Do not apply radicals term-by-term.. Divide out common factors, not common terms. cannot be simplified. y. cannot be simplified. y. To get rid of negative eponents: a b ab a b ab ab b a.. The negative eponent is on a term of the denominator, not a factor. y y y, y. Factor within grouping symbols before applying eponent to each factor.. Eponents are applied before multiplying.
Answers to All Eercises and Tests A9. To add fractions, first find a common denominator. y y y. Be careful when using a slash to denote division.. (a) Answers will vary. y..9..9.. y y y 9.......... 9.., 9, 9 9.,.,...... t 9..... 9..... 9 9.... 9...... 9... (a)..... True. y y y y. False. Cannot move term-by-term from denominator to numerator.. True.. False. 9 does not factor into.. Add eponents when multiplying powers with like bases. n n n. There is no error. 9. When a binomial is squared, there is also a middle term. n y n n n y n y n n y n. There is no error.. The two answers are equivalent and can be obtained by factoring. (a) y..9..9.. APPENDIX A t....9 (c).... t..... mile