Chapter Exercise Set.. To prepare properly for this class, you need to do all the homework carefully and preview the new material that is to be covered in class.. At least hours of study and homework time for each hour of class time is generally recommended.. a. You need to do the homework in order to practice what was presented in class. b. When you miss class, you miss important information, Therefore it is important that you attend class regularly. 7. Answers will vary. Exercise Set.. Understand, translate, calculate, check, state answer. Substitute smaller or larger numbers so that the method becomes clear.. Rank the data. The median is the value in the middle. 6. We usually think of the mean as the average. 7. The mean is greater since it takes the value of 0 into account. 9. A mean average of 0 corresponds to a total of 00 points for the quizzes. Pat s mean average of 79 corresponds to a total of 790 points for the quizzes. Thus, he actually missed a B by points.. a. 7 + 97 + 9 + 7 + 7 = = 76. The mean grade is 76.. b. 9, 7, 7, 7, 97 The middle value is 7. The median grade is 7.. a. 0. +. +.0 + 9.76 +. The mean bill is $6.7. = 6. = 6.7 b. $9.76, $., $., $0., $.0 The middle value is $.. The median bill is $... a..6 +.67 +.6 +. +.6 +.76 +.7 +. +. +.9 =.9 ª.9 The mean inches for rainfall for the years is.9. b..6,.67,.6,.,.6,.76,.7,.,.,.9 The middle values are.6 and.76..6 +.76 =.6 =.6 The median inches for rainfall for the years is.6.
Chapter : Real Numbers 7. Barbara s earnings = % of sales Barbara s earnings = 0.0(900) = 70 Her week s earnings were $70. 9. a. sales tax = 7% of price sales tax = 0.07(6,700) = 69 The sales tax was $,69. b. Total cost = price + tax Total cost = 6,700 +,69 = 7,69 The total cost was $7,69.. a. total cost with payments = down payment + (number of months)(monthly payment) total cost with payments = 00 + () = 00 + 79 = 99 Making monthly payments, it costs $99. b. savings = total cost with payments total cost at purchase savings = 99-90 = He saves $ by paying the total at the time of purchase.. Women enlisted in the army = % of total enlisted women =.(9,600) =,0 Women enlisted in the navy = % of total enlisted women =.(9,600) = 9 Army Women Navy Women =,0-,9 = 9,6 There are approximately 9,6 more women enlisted in the army than the navy. number of miles. miles per gallon = number of gallons 6, 9. - 6, 7. =. = 9.. ª.9 His car gets about.9 miles per gallon. 7. savings = local cost mail order cost local cost = $ + ( 0.0) ( $) = $ + $ = $9 ( ) ( ) mail cost = $6.0 + $6.0 + $ = $76.0 = $06 savings = $9 - $06 = $ Eric saved $. 9.a. taxes = % of income taxes = 0.($,6) = $9.0 Their taxes were $9.0 b. taxes = $670 +7.% of (income - $,00) taxes ( ) ( ) = $670 +0.7 $,7 -$, 00 = $670 +0.7 $ = $670 +$0. = $90. Their taxes were $90..
Chapter : Real Numbers. a. gallons per year = 6(gallons per day) gallons per year = 6(. gallons) = 6. gallons additional money spent = (cost)(gallons wasted) additional money spent $.0 = 6. gallons 00 gallons ª $.. Cost = Flat Fee +.0(each quarter mile traveled) +.0(each 0 seconds stopped in traffic) =.00 +.0() =.0() = 6.0 His ride cost $6.0.. a. difference in cost = cost in Santa Monica cost in Austin = 0(number of weeks) + 9(number of weeks) = 0(0) 9(0) = 9600 0 =70 It costs $70 more for daycare in Santa Monica. b. hour of evening babysitting = total amount one can spend cost per hour of babysitter = 6 = You can obtain hours of babysitting. 7. A single green block should be placed on the on the right. 9. mean = total cost number of nights spent in a hotel mean = 70.7 = $. per night sum of grades. a. mean = number of exams 60 = 0 + 9 + 67 + 0 + 6 + last 6 60 = + last last = 60 - = Lamond needs at least a on the last exam.. Answers will vary. + last b. 70 = 6 0 = + last last = 0 - = Lamond would need points on the last exam, so he cannot get a C.
Chapter : Real Numbers Exercise Set.. a. Variables are letters that represent numbers. b. Letters often used to represent variables are x, y, and z.. ( x), ( )x, ( ) ( x), x, x. Divide out factors that are common to both the numerator and the denominator. 7. a. The least common denominator is the smallest number divisible by he two denominators. b. Answers will vary. 9. Part b) shows simplifying a fraction. In part a) common factors are divided out of two fractions.. and have no common factors other than. Therefore, the fraction is already simplified.. = + =. 0 + = = 7. = 6 + = 9 9. 76 + = = 9 9 9 9. 7 = because 7 = R. Part a) is incorrect because you cannot divide out common factors when adding.. = because = R. c); = =. Divide out the common factor,. This process can be used only when multiplying fractions and so cannot be used for a) or b). Part d) becomes so no common factor can be divided out.. Multiply numerators and multiply denominators. 7. Write fractions with a common denominator, add or subtract numerators, keep the common denominator. 9. Yes, it is simplified because the greatest common divisor of the numerator and denominator is.. The greatest common factor of and is. = =. The greatest common factor of and is. = =. The greatest common factor of 7 and 7 is 7. 7 7 7 = 7 7 7 = = 7. The greatest common factor of 6 and 76 is. 6 76 = 6 76 = 9 9 9. The greatest common factor of 0 and 6 is. 0 6 = 0 6 =. 7. 9.... 7. 9. 0 = 0 = because 0 = R 7 = because 7 = R 7 = = = = = = 9 = = = or = = = = = = = = 6 9 = 9 = 9 = = = 6 = 6 Ê 6. ˆ 7 = + = Ê ˆ 7 = Ê ˆ 7 = 7 = 77 0 or 7 0
Chapter : Real Numbers 6. = 0 + = = + = = 77. 7-9 7 = 7 = 6 9 = 9 = 6 7-9 = 6-6 = - = 6 6 6. 67. 69. 7. 7. 7. = = = = or + = + = = - = - = = 7 + 7 = 7 = 6 7 + = 6 + = 6 + = = 9 7 + 6 = = + 6 = + 6 = + 6 = = 6 or 6-6 = 6 = 6 - = - = - = = 9 - = = = = - = - = - = 7 79. 9-9 = 9 = 9 = 9 = = = 9 - = - - = =. 6-6 = + = 9 = 9 = 9 = + = 6 = 6 - = 9-9- = = 7 or. - = 0 + = = = 69 = = - = 69 - = 6 or. 6-6 6 = 0 + = 6 6 6 = + = = = 70 6 = 6 6-6 = 6-70 6 = 6 Kim has grown 6 inches.
Chapter : Real Numbers 7. - 6 = 6 6-6 - = = 6 6 6 About of all U.S. employees 6 were not online. 9. - 9 0 = 0 0-9 0-9 = = 0 0 0 About of sales were for imported 0 vehicles in 00. 9. = + 6 = 6 + 6 = + = 9 = 9 6 6 = = 7 6 = 7 6 = = 7 = 7 = + 6 + = + + = 9 = 7 The total weight is 7 tons. 9. feet in. - feet in. = feet in. - feet in. = feet in. or () + = + = in. or 7 + = = 7 ª.9 ft. 9. = + = = = = 6 or 9 6 Each piece is 6 or 9 inches long. 6 97. = + = = = = or cups of chopped onions are needed. 99. = = = = 0 = 0 Tierra can wash her hair 0 times.. + + = + + = 6 = or The total thickness is inches.. = + = = = = 6 = 6 There will be 6 whole strips of wood.. a. Total height of computer + monitor 7. a. = 7 in.+ in. 7 = = = 60 = + = 7 + = 60 + = 7 or 7 Total height of computer and monitor is 7 or 7 inches, so there is sufficient room. b. = + = = = 0-7 = 0-7 = There will be inch of extra height. c. = + = = = 90 6 = + = = = 6 = + = = = = + = + 6 + + = 90 + 6 + + = or The height of the desk is in. * a +? a = *+? a 6
Chapter : Real Numbers b. c. d. e.? -? = -? D + = D + x - = x - 9. number of pills x - x = - = x x ( mg per day) ( days per month) number of months = ( ) mg per pill number of pills = ( 0) ( 0) ( 6) 00 = 70 Dr. Highland should prescribe 70 pills. Answers will vary.. 9 + + + + 6 The mean is 6. = 0 = 6. In order, the values are:, 9,, 6,. The median is.. Variables are letters used to represent numbers. Exercise Set.. A set is collection of elements.. Answers will vary. One possible answer is the set of all natural numbers less than 0.. The set of whole numbers contains the natural numbers and zero which is not a natural number. 7. a. A rational number is any number that can be expressed as a quotient of two integers, denominator not 0. b. Any integer can be expressed as a quotient of two integers by writing it with a denominator of. Therefore every integer is a rational number. 9. a. yes b. no b. no c. yes. The integers are {,,,, 0,,,, }.. The whole numbers are {0,,, }.. The negative integers are {,,, }. 7. True; the whole numbers are {0,,, }. 9. True; any number that can be represented on a real number line is a real number.. False; the integers are {,,, 0,,, }.. False; cannot be expressed as the quotient of two integers.. True; - is a quotient of two integers, -. 7. True; 0 can be expressed as a quotient of two integers, 0. 9. False; is rational since it can be expressed as a quotient of two integers.. False; - is rational since it is a quotient of two integers.. True, either or { } is used.. False; irrational numbers are real but not rational. 7. True; any rational number can be represented on a real number line and is therefore real. 9. True; irrational numbers are real numbers which are not rational.. True; the counting numbers are {,,, }, the whole numbers are {0,,, }.. True; the symbol R represents the set of real numbers.. False; every number greater than zero is positive but not necessarily an integer. 7. True; the integers are Ï Ô K, -, -, { 0,,, K Ô Ì Ó Ô negative integers zero positive integers Ô. 9. a. 0 is an integer. 7
Chapter : Real Numbers b. 0 and c. 0 and are rational numbers. are real numbers. 7. a. There are an infinite number of fractions between any numbers. b. There are an infinite number of fractions between any numbers.. a. and 77 are positive integers. b. 0,, and 77 are whole numbers c. 0,,, and 77 are integers. d. - 7, 0, -,, 6,.6, and 77 are rational numbers. e. 7 and - are irrational numbers. f. - 7, 0, -,, 6,, - 7,.6, and 77 are real numbers. For Exercises 6, answers will vary. One possible answer is given.. 0,,., 7, p 7., 6.,.6 9., 7, 6. 0,, 6. 7,, 6. {, 9,,,, 9} 9-+=6+=7 The set has 7 elements. 67. a. A = {,,,, } b. B = {,, 6, 7, } c. A and B = {, } d. A or B = {,,,,, 6, 7, } 69. a. Set B continues beyond. b. Set A has elements. c. Set B has an infinite number of elements. d. Set B is an infinite set. 7. 7. = + = + = 7 6 = because 6 = R 7. + = = 0 = 0 + = 0 + 0 = 9 0 or 9 0 76. Ê ˆ 9 Ê ˆ = + = Ê ˆ 9 Ê ˆ = 9 = 70 7 Exercise Set.. a. b. or 6 7 c. is greater than because it is farther to the right on the number line. d. < e. >. a. is units from 0 on a number line. b. is units from 0 on a number line. c. 0 is 0 units from 0 on a number line.. Yes; for example, > and <. Also, > and <. 7. No, > but <. 9. No, > but <.. 7 = 7
Chapter : Real Numbers. - =. 0 = 0 7. - - = -() = - 9. - = -() = -. > ; is to the right of on a number line.. 6 < 0; 6 is to the left of 0 on a number line.. > - ; is to the right of - on a number line. 7. 0.7 < 0.; 0.7 is to the left of 0. on a number line. 9. - > -; - is to the right of on a number line.. > ; is to the right of on a number line... < ;. is to the left of on a number line.. > - ; is to the right of - line. 7. - < ; - is to the left of line. on a number on a number 9. 0.9 > 0.; 0.9 is to the right of 0. on a number line.. > 7; is to the right of 7 on a number line.. 0.006 > 0.007; 0.006 is to the right of 0.007 on a number line.. > 0.6 because = 0.6 and 0.6 is to the right of 0.6 on a number line. 7. - < - ; - is to the left of - line. on a number. < 9 ; 9 is to the left of 0 on a number line.. -(-6) > - (-); 6 is to the right of on a number line. 7. > since = 9. - > since - = 6. 0 < - since 0 = 0 and - = 6. < - 9 since - 9 = 9 or 6. - < - since - = = 6 0 and - = = 0 67. -.6 = - since -.6 =.6 and - = =.6 69. 7. 7. + + + = since + + + = + + + = and = = < since = = and = = = - < since - = - = and = = 9. - > - ; - is to the right of - line.. 0. < ; 0. is to the left of... on a number line. on a number 7. - -,., 9,, -6 because - = -, 9 = 0..., = 0.6 and -6 = 6. 9
Chapter : Real Numbers 77., 0.6,, 9, -.6 because =.66..., 9 =.666..., =.76 and -.6 =.6. 79. and since = - = For Exercises -7, answers will vary. One possible answer is given.. Three numbers greater than and less than 6 are,,... Three numbers less than and greater than 6 are,,.. Three numbers greater than and greater than are,, 6. 7. Three numbers greater than - and less than -6 are,,. 9. a. Between does not include endpoints. b. Three real numbers between and 6 are.,, and. c. No, is an endpoint. d. Yes, is greater than and less than 6. e. True 9. a. The taxes were less than 0 cents all the time b. In March 000 the gasoline was greater than $.0 for the firs time. c. Gas was less than $.00 from January to March, 999. d. March 000 May 00, except for June 000 9. The result of dividing a number by itself is. Thus, the result of dividing a number between 0 and by itself is a number,, which is greater than the original number. 9. Yes, 0 since 0 = 0 and - 0 = -(0) = 0 9. - = + = = = 0 = = 9 - = 0-9 = or 7 99. The set of whole numbers is {0,,,, }. 0. The set of counting numbers is {,,,, }.. a. is a natural number. b. and 0 are whole numbers. c.,, and 0 are integers. d.,, 0,, -, and. are rational numbers. 9 e. is an irrational number. f.,, 0,,, -, and. are real numbers. 9 Exercise Set.6. The basic operations of arithmetic are addition, subtraction, multiplication, and division. b. One example is and.. a. No; - + does not equal 0. b. The opposite of - is because - + = 0.. The sum of a positive number and a negative number can be either positive or negative. The sum of a positive number and a negative number has the same sign as the number that has larger absolute value. 7. Answers will vary. 9. a. He owed 7, a negative, and then paid a positive amount, 9, toward his debt. b. 7 + 9 The numbers have different signs so find the difference between the absolute values. -7-9 =
Chapter : Real Numbers -7 is greater so sum is negative. 7 + 9 = c. Answers will vary.. Yes, it is correct.. The opposite of 9 is 9 since 9 + ( 9) = 0.. The opposite of is since + = 0. 7. The opposite of 0 is 0 since 0 + 0 = 0. 9. The opposite of is - since + Ê - ˆ = 0.. The opposite of is - since + Ê - ˆ = 0.. The opposite of.7 is.7 since.7 + (-.7) = 0.. Numbers have same sign, so add absolute values. + 6 = + 6 = Numbers are positive so sum is positive. + 6 = 7. Numbers have different signs so find difference - - = - =. is greater than - so the sum is positive. + ( ) = 9. Numbers have same sign, so add absolute values. - + - = + = 6. Numbers are negative, so sum is negative. + ( ) = 6. Numbers have different signs, so find difference between absolute values. 6 - -6 = 6-6 = 0 6 + (-6) = 0. Numbers have different signs, so find difference between absolute values. - - = - = 0 + = 0. Numbers have same sign, so add absolute values. - + - = + =. Numbers are negative, so sum is negative. + ( ) = 7. Numbers have different signs, so find difference between absolute values. -6-6 = 6-6 = 0 6 + 6 = 0 9. Numbers have same sign, so add absolute values. - + - = + = Numbers are negative, so sum is negative. + ( ) =. 0 + 0 = 0. 6 + 0 = 6. Numbers have different signs, so find difference - -9 = - 9 = 9. is greater than -9 so sum is positive. + (-9) = 9 7. Numbers have same sign, so add absolute values. - + - = + = 6 Numbers are negative, so sum is negative. -+ (-) = -6 9. Numbers have same sign, so add absolute values. - + -9 = + 9 =. Numbers are negative, so sum is negative. - + (-9) = -. Numbers have different signs, so find difference -0 - = 0 - = 6. -0 is greater than so sum is negative. + (-0) = -6. Numbers have different signs, so find difference 0 - - = 0 - =. 0 is greater than - so sum is positive. + 0 =. Numbers have different signs, so take difference 6. -. = 6.. =.. 6. is greater than. so sum is positive.. + 6. =. 7. Numbers have same sign, so add absolute values. 9.7 +. = 9.7 +. =.. Numbers are negative, so sum is negative.. 9.7 + (.) =. 9. Numbers have different signs, so find difference
Chapter : Real Numbers -00-0 = 00-0 = 0. -00 is greater than 0 so sum is negative. 0 + (-00) = -0 6. Numbers have different signs, so find difference -67 - = 67 - = 9. -67 is greater than so sum is negative. 67 + = 9 6. Numbers have different signs, so find difference - -9 = - 9 = 9. is greater than -9 so sum is positive. + (-9) = 9 6. Numbers have different signs, so find difference - - = - = 0. - is greater than so sum is negative. - + = -0 67. Numbers have same sign, so add absolute values. -6 + -79 = 6 + 79 =. Numbers are negative so sum is negative. -6 + (-79) = - 69. Numbers have same sign, so add absolute values..6 +.9 =.6 +.9 =.. Numbers are negative so sum is negative..6 + (.9) =. 7. Numbers have different signs, so find difference.9 6. =.9 6. =.6..9 is greater than 6. so sum is negative. 6. + (.9) =.6 7. 7. + 7 = + = + = 6 + 6 7 = + 7 + 7 = = 7 77. Numbers have different signs, so find difference - + = - 0 + = - - 0 = - 0 = 0 is greater than - so sum is positive. - + = 79. Numbers have different signs, so find difference - 7 + 90 = - 6 90-90 = 6 90-6 - = = 90 90 90 = 6-6 is greater than so sum is negative. 90 90-7 + 90 = - 6. Numbers have different signs, so find difference between larger and smaller absolute values 9 + Ê - ˆ = 0 0 - - 0 = 0-0 = - = 0 0 = is greater than - so sum is positive. 0 0 9 + Ê - ˆ = 0. Numbers have same sign, so add absolute values. - 7 0 + Ê - ˆ 6 = - 7 0 + - 0 = 7 0 + 0 = 7 + = 0 0 = 6 Numbers are negative so sum is negative. - 7 0 + Ê - ˆ 6 = - 6. Numbers have same sign, so add absolute values. - + Ê - ˆ 7 = - 60 7 + - 7 = 60 7 + 7 = 60 + = 7 6 7 = Numbers are negative so sum is negative. - + Ê - ˆ 7 = - 7. Numbers have different signs, so find difference 6 + Ê - ˆ = 7 + Ê - ˆ = - 7 7-7 = 7-7 = 7 - is greater than so sum is negative. 7 7 6 + Ê - ˆ = - 7
Chapter : Real Numbers 9. Numbers have same sign, so add absolute values. - + Ê - ˆ = - 60 + Ê - ˆ 60 = - 60 + - 60 = 60 + 60 = 60 Numbers are negative so sum is negative. - + Ê - ˆ = - 60 9. Numbers have same sign, so add absolute values. - + Ê - 7 ˆ = - 9 + Ê - 7 ˆ = - 9 + - 7 = 9 + 7 = 9 + 7 = 6 = Numbers are negative so sum is negative. - + Ê - 7 ˆ = - 9. a. Positive; 7 is greater than -97 so sum will be positive. b. 7 + (-97) = 90 c. Yes; By part a) we expect a positive sum. The magnitude of the sum is the difference between the larger and smaller absolute values. 9. a. Negative; the sum of negative numbers is always negative. b. - + (-9) = -7 c. Yes; the sum of negative numbers should be (and is) a larger negative number. 97. a. Negative; -97 is greater than 9 so sum will be negative. b. -97 + 9 = - c. Yes; by part a) we expect a negative sum. Magnitude of the sum is the difference between the larger and smaller absolute values. 99. a. Negative; the sum of negative numbers is always negative. b. -96 + (-0) = -00 c. Yes; the sum of negative numbers should be (and is) a larger negative number.. a. Negative; -7 is greater than 6 so sum will be negative. b. -7 + 6= - c. Yes; by part a) we expect a negative sum. The magnitude of the sum is the difference between the larger and smaller absolute values.. a. Negative; the sum of negative numbers is always negative. b. - + (-07) = -0 c. Yes; The sum of negative numbers should be (and is) a larger negative number.. a. Positive; is greater than -0 so sum will be positive. b. + (-0) = c. Yes; by part a) we expect a positive sum. Magnitude of sum is difference between larger and smaller absolute values. 7. a. Negative; the sum of negative numbers is always negative. 9. True b. - + (-) = -00 c. Yes; the sum of negative numbers should be (and is) a larger negative number.. True; the sum of two positive numbers is always positive.. False; the sum has the sign of the number with the larger absolute value.. Mr. Peter s balance was $9. His new balance can be found by adding. -9 + (-) = -77 Mr. Peter owes the bank $77. 7. Total loss can be represented as + ( ). - + - = + =. The total loss in yardage is yards. 9. The depth of the well can be found by adding 7 + ( ) = 6. The well is 6 feet deep.. Distance from sea level to place on mountain can be represented by 67 + 9. 67 9 = 67 9 = 69
Chapter : Real Numbers 67 > 9 so 67 + 9 = 69 Their vertical distance is 69 feet below sea level.. a. In 00, the Postal Service will have a net loss of $ billion. b. 999: 0, billion dollars 000: - 0. billion dollars 00: - billion dollars 0.+(-0.)+(-)= -.9 billion dollars From 999 through 00, the Postal Service had a net loss of $.9 billion.. (-) + (-6) + (-) = (-) + (-) = - 7. 9 + (-6) + 7 = (-7) + 7 = 0 9. (-) + (-) + + (-) = (-) + + (-) = + (-) = 0. + Ê - ˆ + = Ê 6 - ˆ 6 + = 6 + = 0 + 6 0 = 0. + + +L + = (+ ) + ( + 9)L + ( + 6) = ()() =. = + = Ê ˆ Ê ˆ = Ê ˆ Ê ˆ = = 6. = 6 6 = 6-6 = 6-6 = - = or 6 6 6 7. {,,,,... }. - > since - = 9. > -7 since -7 = 7 Exercise Set.7. - 7. * -. a. to subtract b from a, add the opposite of b to a. b. + (-) c. + (-) = -9 7. a. a - ( +b ) = a - b b. 7 - ( +9) = 7-9 c. 7-9 = 7 + (-9) = - 9. a. - (-6) + (-) = + 6 - b. + 6 - = 9 - =. Yes it is correct.. = + (-) = 7. - 9 = + (-9) = - 7. - - = - + (-) = -6 9. - - (-) = - + = -. - - = - + (-) = -. 0 7 = 0 + (-7) = -7. = + (-) = 0 7. - - = - + (-) = - 9. - = + (-) =. 6 - (-) = 6 + = 9. 0 - (-) = 0 + =. -9 - = -9 + (-) = -0 7. (-) - (-) = - + = 0 9. - - (-) = - + =. -6 - (-) = -6 + = -. -9 - = -9 + (- ) = -. - (-) = - + = -7 7. -90 60 = -90 + (-60) = -
Chapter : Real Numbers 9. - - 7 = - + (-7) = -. 70 - (-70) = 70 + 70 = 0.. 9.7 =. + ( 9.7) = -7.. 7. (.9) = 7. +.9 = -.9 7. 0 9 = 0 + ( 9) = 9 9. - - = - + (-) = -6 6. - - (-) = - + = - 6. 9 - (-) = 9 + = 6. - = + (-) = - 67. 6. (.) = 6. +. = 6. 69. 7. - 6 = 0 - - = = - 0 0 0-7 = - 7 = - 7 = 7 7. - - = - - = -- = - 7. - - 0 = - 6 60-9 60 = -6-9 = 60-60 = - 77. - 6 = - 6 = - 6 = = 79. - 7-0 = - 70 0 - -70 - = 0 0 = - 0 = - 7. 6 - Ê - ˆ = 6 + 6 = + 6 = 6. - - Ê - ˆ = - + 9 = - + 9 = -. 7-7 9 = 6 6-9 6-9 = = - 6 6 6 7. - - Ê - ˆ = - 60 + 60 = -+ 60 = - 7 60 9. a. Positive; 7 79 = 7 + (-79) 7 is greater than -79 so the sum will be positive. b. 7 + (-79) = 99 c. Yes; by part a) we expect a positive sum. The size of the sum is the difference between the absolute values of the numbers. 9. a. Negative; - 7 = - + (-7) The sum of negative numbers is always negative. b. - + (-7) = -69 c. Yes; the sum of two negative numbers should be (and is) a larger negative number. 9. a. Positive; - (-7) = + 7. The sum of positive numbers is always positive. b. + 7 = c. Yes; by part a) we expect a positive answer. The size of the sum is the sum of the absolute values of the numbers. 9. a. Positive; -0 - (-60) = -0 + 60. 60 is greater than -0 so the sum will be positive. b. -0 + 60 = 96 c. Yes; by part a) we expect a positive answer. The size of the answer is the difference between the larger and smaller absolute values. 97. a. Negative; - - (-76) = - + 76. - is greater than 76 so the sum will be negative. b. - + 76 = - c. Yes; by part a) we expect a negative answer. The size of the answer is the difference between the larger and the smaller absolute values. 99. a. Positive; 6.7-9.6 = 6.7 + (-9.6). 6.7 is greater than -9.6 so the sum will be positive. b. 6.7 + (-9.6) = 6.
Chapter : Real Numbers c. Yes; by part a) we expect a negative answer. The size of the answer is the difference between the larger and the smaller absolute values.. a. Negative; 9-6 = 9 + (-6). Since -6 is greater than 9 the answer will be negative. b. 9 + (-6) = -69 c. Yes; by part a) we expect a negative answer. The size of the answer is the difference between the larger and the smaller absolute values.. a. Negative; 67 = + ( 67). The sum of two negative numbers is always negative. b. - + (-67) = -670 c. Yes; the sum of two negative numbers should be (and is) a larger negative number.. a. Zero; -7.6 - (-7.6) = -7.6 + 7.6. The sum of two opposite numbers is always zero. b. -7.6 + 7.6 = 0 c. Yes; by part a) we expect zero. 7. 7 + (+) = 7 + +(-) = + (-) = 9. -6 + (-6) + 6 = - + 6 = -6. - - ( +) + = - + (-) + = - + = -. -9 - (-) + = -9 + + = -6 + = -. - (-9 ) + (-) = + 9 + (-) = + (-) = 7. 7 + (-) - ( +) = 7 + (-) + (-) = 9 + (-) = - 9. -6 - + 9 = -6 + (-) + 9 = - + 9 = -. - + 7-9 = - + 7 + (-9) = + (-9) = -. - 9 + = + (-9) + = 6 + = 9. (-) + (-6) + - 7 = (-) + (-6) + + (-7) = - + + (-7) = - + (-7) = - 7. 7 + (-) - 9 - (-7) = 7+ (-) + (-9) + 7 = + (-9) + 7 = + 7 = 9. -9 + (-7) + (-) - (-) = -9 + (-7) + (-) + = -6 + (-)+ = -+ = -. a. 00 - = 00 + (-) = - They had sweaters on back order. b. + 0 = They would need to order sweaters.. - = + (-) = -6 The bottom of the well is 6 feet below sea level.. - (-6) = + 6 = 0 Thus the temperature dropped 0 F. 7. a. + (-) = 76 In 00, his score was 76. b. - (-) = - + = - He had strokes less in 00 than in 000. c. 000 stroke score: + (-) = 00 stroke score: + (-6) = 7 00 stroke score: + (-) = 76 + 7 + 76 Average = = ª 77. His average stroke score was 77.. 9. - + - + - 6 + 7 - + 9 - = ( - ) + ( - ) + ( - 6) + ( 7 - ) + ( 9 - ) = (-) + (-) + (-) + (-) + (-) = -. a. 7 units b. - (- ) = 7. a. + + + + = 9 The ball travels 9 feet vertically. b. - + + (-) + + (-) = - The net distance is feet.. The integers are { K, -, -, 0,,, K }.. The set of rational numbers together with the set of irrational numbers forms the set of real numbers. 6
Chapter : Real Numbers 6. - > - since - = 7. -6 < -7 since -6 = 6 and -7 = 7. - = 0 - - = = 7 0 0 0 Exercise Set.. Like signs: product is positive. Unlike signs: product is negative.. The product of 0 and any real number is 0. A fraction of the form or -a b. a -b is generally written - a b 7. a. With you subtract, but with ( ) you multiply. b. - = + (-) = - ( ) = 9. a. With x y you subtract, but with x( y) you multiply. b. x - y = - (-) = + = 7 c. x( y) = [ ( )] = () = d. - x - y = - - (-) = - + = -. The product ()()( ) is negative since there is an odd number () of negatives. The product ( )( 6)()(9) is positive since there is an even number of negatives.. The product ( 0)( 6)(0)(0)( ) is negative since there is an odd number () of negatives. 7. Since the numbers have like signs, the product is positive. ( )( ) = 0 9. Since the number have unlike signs, the product is negative. 6( ) =. Since the numbers have unlike signs, the product is negative. ( )() =. Zero multiplied by any real number equals zero. 0( ) = 0. Since the numbers have like signs, the product is positive. 6(7) = 7. Since the numbers have unlike signs, the product is negative. (7)( ) = -6 9. Since the numbers have like signs, the product is positive. ( )( 6) = 0. Zero multiplied by any real number equals zero. 0()() = 0() = 0. Since there is one negative number (an odd number), the product will be negative. ( ) (-) ( ) = (-)( ) = -. Since there are three negative numbers (an odd number), the product will be negative. -( -)( ) (-) = ( ) (-) = 9( -) = -7 7. Since there are two negative numbers (an even number), the product will be positive. (-)( ) (-7)( ) = (-0)( -7) ( ) = ( 0) ( ) = 0 9. Zero multiplied by any real number equals zero. ( )()(0)( 7) = ( )(0)( 7) = 0( 7) = 0... 7. Ê - ˆ Ê ˆ = (-)( ) = - = - Ê -ˆ Ê -7 ˆ = (-)( -7) = (-)( -7) 9 9 9 = 7 7 Ê 6 ˆ Ê ˆ - - = Ê ˆ - Ê ˆ ( ) ( ) = - (-)(-) = = Ê - ˆ Ê ˆ 6 Ê = - ˆ Ê ˆ = (-) ( ) ( ) ( ) = - 6 = - 6 9. Since the numbers have like signs, the quotient is positive. =. Since the numbers have like signs, the quotient is positive. -6 (-) = -6 - =. Since the numbers have like signs, the quotient is positive. -6-9 =. Since the numbers have unlike signs, the quotient is negative. 6 - = - 7