Solution Manual for Beginning Algebra with Applications and Visualization, 3rd Edition

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’SRESOURCEMANUAL WITHTESTSANDMINI-LECTURESDEANARICHMONDBEGINNINGALGEBRA WITHAPPLICATIONS&VISUALIZATIONTHIRDEDITIONGary RockswoldMinnesota State University, MankatoTerry KriegerRochester Community and Technical College

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ContentsMini-Lectures (ML)Chapter 1 .......................................................................................................................................... 1Chapter 2 ........................................................................................................................................ 11Chapter 3 ........................................................................................................................................ 20Chapter 4 ........................................................................................................................................ 29Chapter 5 ........................................................................................................................................ 36Chapter 6 ........................................................................................................................................ 43Chapter 7 ........................................................................................................................................ 50Chapter 8 ........................................................................................................................................ 59Chapter 9 ........................................................................................................................................ 67Chapter 10 ...................................................................................................................................... 70Chapter 11 ...................................................................................................................................... 80Chapter 12 ...................................................................................................................................... 89Chapter 13 ...................................................................................................................................... 97Chapter 14 .................................................................................................................................... 103AdditionalResources (R)Teaching Tips Correlated to Textbook Sections .............................................................................. 1Outside Resources .......................................................................................................................... 85Available Supplements................................................................................................................... 86Graphing Calculator Resources...................................................................................................... 89Printable Test Forms (T)Chapter 1 .......................................................................................................................................... 1Chapter 2 ........................................................................................................................................ 11Chapter 3 ........................................................................................................................................ 21Chapter 4 ........................................................................................................................................ 39Chapter 5 ........................................................................................................................................ 57Chapter 6 ........................................................................................................................................ 67Chapter 7 ........................................................................................................................................ 77Chapter 8 ........................................................................................................................................ 89Chapter 9 ...................................................................................................................................... 107Chapter 10 .................................................................................................................................... 123Chapter 11 .................................................................................................................................... 133Chapter 12 .................................................................................................................................... 151Chapter 13 .................................................................................................................................... 169Chapter 14 .................................................................................................................................... 197Answers to Chapter Tests............................................................................................................. 213Cumulative Review Exercises and Answers ................................................................................ 165Final Exams and Answers ............................................................................................................ 319

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Mini-Lecture 1.1BEGINNING ALGEBRAML-1Numbers, Variables, and ExpressionsSection Topics:•Natural Numbers and Whole Numbers•Prime Numbers and Composite Numbers•Variables, Algebraic Expressions, and Equations•Translating Words to ExpressionsExamples:1.Classify each number asprimeorcomposite. If the number is composite, write it as a product of primenumbers.a)37b)65c)97d)2162.Evaluate the expressions when2a=,3b=, and6c=.a)23ab+b)5abc)cbd)()caa−3.Write each word phrase as an algebraic expression, usingxas the variable.a)The quotient of 6 less than a number and 3b)Fifty more than a numberc)The product of 8 and the total of a number and 5Teaching Notes:•When translating“the difference between a and b,”the order is kept the same:ab−.•Remind students that an expression evaluates to a number, while an equation can be translated into asentence (with an = symbol).Answers:1a) prime, b) composite, 5 13⋅, c) prime, d) composite, 2 2 2 3 3 3⋅⋅⋅⋅⋅; 2a) 13, b) 30, c) 2, d) 2; 3a)x63−,b) x50+, c)()8 x5+

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BEGINNING ALGEBRAMini-Lecture 1.2ML-2FractionsSection Topics:•Basic Concepts•Simplifying Fractions to Lowest Terms•Multiplication and Division of Fractions•Addition and Subtraction of Fractions•An ApplicationExamples:1.Simplify each fraction to lowest terms.a)510b)1664c)4277d)88902.Multiply and simplify to lowest terms.a)53915ib)5962ic)4189id)7524i3.Divide and simplify to lowest terms.a)53915÷b)5962÷c)5362424÷d)51534÷4.Find each sum or difference and write in lowest terms.a)341414+b)2133−c)2148+d)5768+e)791230+f)2213246++5.Rachel is jogging for exercise. This week, she ran121miles on Monday,142miles on Tuesday, and onThursday she ran122times the distance that she ran on Monday. How many miles did she run this week?

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Mini-Lecture 1.2BEGINNING ALGEBRAML-36.In a recent survey, 1000 people were asked about their favorite breed of dog. The table shows the approximatefractions of survey respondents who listed each breed as his or her favorite.Dog BreedFractionPug3/40Labrador Retriever91/200Golden Retriever27/100Yorkshire Terrier1/5a)What fraction of people chose either Golden Retriever or Labrador Retriever as their favorite breed ofdog?b)Of the 1000 people surveyed, how many people chose the Yorkshire Terrier as their favorite breed of dog?c)Of the 1000 people surveyed, how many peopledid notchoose the Pug as their favorite breed of dog?Teaching Notes:•Encourage students to simplify fractions by dividing numerator and denominator by the same number andby factoring into primes and dividing out common factors.•Remind students that “fraction of” phrases generally indicate multiplication.•Some students add/subtract the denominators when adding/subtracting fractions.Answers: 1a)12, b)14, c)611, d)4445; 2a)19, b)154, c) 8, d)358; 3a)259, b)527, c)536, d)49; 4a)12, b)13, c)58,d)4124, e)5360, f)1112; 5)127miles ; 6a) 29/40, b) 200, c) 925

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BEGINNING ALGEBRAMini-Lecture 1.3ML-4Exponents and Order of OperationsSection Topics:•Natural Number Exponents•Order of Operations•Translating Words to ExpressionsExamples:1.Find the value of each exponential expression.a)26b)43c)434⎛⎞⎜⎟⎝⎠2.Find the value of each expression.a)482−÷b)2233⋅−c)21032−+d)316322−÷e)21646222−+−+f)2324596÷+−g)()()2 313 424++ ⎡−+⎤⎣⎦3.Translate each phrase into a mathematical expression and then evaluate it.a)Seven squared decreased by fifteenb)Four times twenty increased by elevenc)Five times the quantity of twelve plus nineTeaching Notes:•Some students do not know how to say23, or32, or45, etc., in words and need to see the words written.•Students will often compute42as2 4⋅. Illustrate the difference between the expressions with examples.•Students should be reminded to work from inside out when evaluating expressions with grouping symbols.Answers: 1a) 36, b) 81, c) 81/256; 2a) 0, b) 9, c) 3, d) 12, e) 6, f) 13, g) 18; 3a) 72−15, 34, b) 4⋅20 + 11, 91,c) 5(12+9), 105

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Mini-Lecture 1.4BEGINNING ALGEBRAML-5Real Numbers and the Number LineSection Topics:•Signed Numbers•Integers and Rational Numbers•Square Roots•Real and Irrational Numbers•The Number Line•Absolute Value•InequalityExamples:1.List all numbers from the set310,5,2, 0,5, 104⎧⎫−−−⎨⎬⎩⎭that are:a)natural numbersb)whole numbersc)integersd)rational numberse)irrational numbersf)real numbers2.Evaluate the square root. Approximate your answer to three decimal places when appropriate.a)121b)40c)273.Plot each of the numbers on a number line:112, 5,4, 132−−.4.Select the lesser number in each pair.a)–3, 8b)41,53−−c)10, –125.Find the opposite of each number.a)6b)–(–(–5))c)56−6.Simplify.a)3−b)6− −c)1316−7.Decide whether the statement is true or false.a)102−>b)35−≤ −c)71− −>

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BEGINNING ALGEBRAMini-Lecture 1.4ML-6.Teaching Notes:•Some students find it helpful to see each type of number in Example 1 on a number line.•Remind students that integers are rational numbers; any integer can be written as the ratio of itself and 1.•Decimal numbers that terminate or repeat in a fixed block are rational numbers – ask students to giveexamples of both.•The decimal form of an irrational number neither terminates nor repeats.•The number line is a good way to illustrate that opposite numbers are equidistant from 0 but on oppositesides of 0.•Some students have never seen absolute value before and will need examples.Answers: 1a) 10, b) 0, 10, c) –10, –5, 0, 10, d) –10, –5,32 4−, 0, 10, e)5 , f) –10, –5,32 4−, 0,5 , 10;2a) 11, b) 6.325, c) 5.196; 3); 4a) –3, b)45−, c) –12; 5a) –6, b) 5,c) 56 ; 6a) 3, b) –6, c) 3; 7a) true, b) true, c) false;

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Mini-Lecture 1.5BEGINNING ALGEBRAML-7Addition and Subtraction of Real NumbersSection Topics:•Addition of Real Numbers•Subtraction of Real Numbers•ApplicationsExamples:1.Find each sum or difference.a)912+b)()1610−+ −c)()1426−+ −d)6.35.2−+e)()12617−+ −+f)9 – 11g)()2617−+ ⎡+ −+⎤⎣⎦h)72105⎛⎞+−⎜⎟⎝⎠i)111324−−2.Write a numerical expression for each phrase, and simplify the expression.a)The sum of –6 and 3 and –1b)The sum of –10 and –15, increased by 12c)0.29 more than the sum of 3.56 and –2.41Solve each problem.3.A scuba diver is at a depth of 16 feet below the surface. He descends another 8 feet. What is his new depth?4.On January 14, in New Market, Indiana, the temperature rose 17º F in three hours. If the starting temperaturewas –5º F, what was the temperature three hours later?Teaching Notes:•Some students need to see addition problems done on a number line first.•Caution students about the difference between the subtraction key and the change-of-sign key on acalculator.•Refer students to the summary boxes for adding and subtracting real numbers.•Emphasize the uses of the – symbol with students.Answers: 1a) 21, b) –26, c) – 40, d) –1.1, e) –1, f) –2, g) 10, h)310, i)512−; 2a)();6314−++ −−,b)();10151213⎡−+ −⎤ +−⎣⎦, c)()...;.3 562 410 29 1 44⎡+ −⎤ +⎣⎦; 3) 24 feet below; 4) 12ºF

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BEGINNING ALGEBRAMini-Lecture 1.6ML-8Multiplication and Division of Real NumbersSection Topics:•Multiplication of Real Numbers•Division of Real Numbers•ApplicationsExamples:1.Find each product.a)()2 0−b)()415−c)()()305−−d)()()2.23.3−e)3849⎛⎞ ⎛⎞−−⎜⎟ ⎜⎟⎝⎠ ⎝⎠2.Find each quotient.a)168−÷b)93−−c)4.60−d)315520−÷e)04−3.Write each number as a decimal.a)418b)43500c)112 54.Write each decimal as a fraction in lowest terms.a)0.72b)0.375c)0.0055.In a box of 50 premium colored pencils, the fraction of pencils that are shades of blue is425 . Write thisfraction as a decimal.Teaching Notes:•Refer students to the rules for signs of products and quotients.•Give examples to show why division by zero is undefined but zero can be divided by any number exceptzero.•,,xxyy÷andy x(where0y≠) all indicate division.Answers: 1a) 0, b) –60, c) 150, d) –7.26, e)23; 2a) –2, b) 3, c) undefined, d)45−, e) 0; 3a)0.2, b) 0.086, c) 12.2;4a) 18/25, b) 3/8, c) 1/200; 5) 0.16

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Mini-Lecture 1.7BEGINNING ALGEBRAML-9Properties of Real NumbersSection Topics:•Commutative Properties•Associative Properties•Distributive Properties•Identity and Inverse Properties•Mental CalculationsExamples:1.Name the property (commutative, associative, identity property, inverse property, or distributive property)illustrated by each statement.a)()3773+ −= −+b)23132⎛⎞ ⎛⎞ =⎜⎟ ⎜⎟⎝⎠ ⎝⎠c)()()()362362−+ ⎡+ −⎤ = −++ −⎣⎦d)11055−+=e)()555cdcd−+= −−f)()()57101057−++=+ −+g)337777⎛⎞⎛⎞−=−⎜⎟⎜⎟⎝⎠⎝⎠h)56 156−⋅= −2.Use a distributive property to rewrite each expression. Simplify if possible.a)()36k+b)()52h−−c)33yz⋅+⋅d)()2 52abc−−−e)()2 723−⋅+ −⋅3.Use properties of real numbers to calculate the expression mentally: 22 + 36 + 58 + 44Teaching Notes:•Remind students that the commutative property deals with the order of addition (or multiplication), whereasthe associative property deals with grouping.•Students often confuse the additive identity (zero) and the multiplicative identity (one).•Have students provide examples to show whether or not the commutative / associative properties hold forsubtraction and division.Answers: 1a) commutative (add.), b) inverse (mult.), c) associative (add.), d) inverse (add.), e) distributive,f) commutative (add.), g) commutative (mult.), h) identity (mult.); 2a)3k18+, b)5h10−+, c)()3 yz+,d)10a4b2c−++, e)();2 7320−+−; 3) 160

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BEGINNING ALGEBRAMini-Lecture 1.8ML-10Simplifying and Writing Algebraic ExpressionsSection Topics:•Terms•Combining Like Terms•Simplifying Expressions•Writing ExpressionsExamples:1.Simplify each expression.a)835xx−+b)()3 27xx−+c)()435m−−+2.Give the numerical coefficient of each term.a)4mb)216x−c)23p q−d)45r−3.Simplify each expression.a)()2 364mm−++b)36bb+c)23bb−d)()758cc−+e)()()22342 51eee−−−4.Translate each phrase into a mathematical expression. Usexas the variable. Simplify.a)A number increased by the difference between 3 and the numberb)Six plus the product of 4 more than a number and 2c)A number plus 7 added to the difference between 6 and twice the numberTeaching Notes:•Students have difficulty distinguishing betweentermsandfactors– terms are separated by a + or – sign,while factors are multiplied.•Like terms not only have the same variables but also the same exponents.•Remind students that expressions can be simplified, whereas equations are solved.Answers: 1a) 13x3−, b) 7 x21−, c)3m9−−; 2a) 4, b) –16, c) –1, d)45−; 3a)2m12−−, b) 9b , c)13b−,d) 2c8−, e)27e12e2−−+; 4a)();x3x3+−, b)();62 x42x14+++, c)()();62xx7x13−++−+

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Mini-Lecture 2.1BEGINNING ALGEBRAML-11Introduction to EquationsSection Topics:•Basic Concepts•Equations and Solutions•The Addition Property of Equality•The Multiplication Property of EqualityExamples:1.Which pairs of equations are equivalent equations?a)58x+=and3x=b)1210x−=−and2x= −c)155x−=and6x=d)2330x+=and9x= −2.Solve each equation, and check your solution.a)1412x=−b)1518x+=c)1916x−=+d)()618x− −=e)155x=−f)1536x+=3.By what number is it necessary to multiply each side of each equation in order to isolatexon the left side? Donot solve.a)135x=b)26x= −c)3214x−=d)41x−=4.Tell whether you would use the addition or multiplication property of equality to solve each equation. Do notsolve.a)45x+= −b)525x= −5.Solve each equation, and check your solution.a)1254x= −b)512x=c)756x= −d)112x−=e)1.188x=f)16x−= −g)3.2116.8x−= −h)538x= −i)196x= −j)434x−=k)9.8210.994x−= −l)88x−=

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BEGINNING ALGEBRAMini-Lecture 2.1ML-12.Teaching Notes:•Encourage students to write all of the addition property steps and to avoid using shortcuts until they havemastered these types of equations.•Encourage students to write the steps for solving the equations in a neat and organized manner. This habitwill help immensely when the equations become more complex.•Refer students to the addition property of equality definition in the text.•Mention that letters besidesxcan be used for variables in equations.•Some students need to be shown that1111212112 112xxxx⋅=⋅==⋅.•Refer to the multiplication property of equality definition in the text.•Ask students what should be done to the right side of an equation to balance it if the left side of theequation is doubled.Answers: 1) choices a and b; 2a) 26, b) 3, c)35−, d) 12, e) 20, f) 1/2; 3a) 5, b) 6, c)43−, d) –1;4a) addition property of equality, b) multiplication property of equality; 5a)100−, b) 60, c)8−,d)112−or15 2−, e) 80, f) 16, g) 36.5, h)538−, i)54−, j)434−, k) 21.53, l)64−

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Mini-Lecture 2.2BEGINNING ALGEBRAML-13Linear EquationsSection Topics:•Basic Concepts•Solving Linear Equations•Applying the Distributive Property•Clearing Fractions and Decimals•Equations with No Solutions or Infinitely Many SolutionsExamples:1.Which of the following are linear equations in one variable?a)4732xx−=+b)225xx=+c)280x−=d)38xx=2.Determine whether the given solution is correct.a)Is 12 a solution for 5x+ 4 – 2x= 3x– 5 +x?b)Is 9 a solution for 5x+ 4 – 2x= 3x– 5 +x?3.Solve each equation, and check your solution.a)107107x+=b)7827x−=c)3363x=−d)1641514x=+e)1822x−= −f)28323x−−= −4.Solve each equation, and check your solution.a)4x= –2x+ 60b)8x– 6 = 3 + 9xc)–6x– 10 = –7 + 10xd)0.6y– 0.3 = 0.7 – 0.3ye)x– 12 = 10 –xf)–9x+ 4 + 7x= –3x+ 95.Determine whether the equation has no solutions, one solution, or infinitely many solutions.a)6(21)1230xx−−=b)–1(x+ 11) = 20c)()68648xx−=−d)()369915xxx++=−e)0.4x– 0.2(3 –x) = 7.6f)73(8)324xxx−−=+

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