Solution Manual for Introductory Algebra, 6th Edition

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Solution Manual for Introductory Algebra, 6th Edition - Page 1 preview image

RESOURCEMANUALINTRODUCTORYALGEBRASIXTHEDITIONElayn Martin-GayUniversity of New Orleans

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Table of ContentsMini-Lectures (M)Chapter 1........................................................................................................................................... 5Chapter 2......................................................................................................................................... 13Chapter 3......................................................................................................................................... 21Chapter 4......................................................................................................................................... 29Chapter 5......................................................................................................................................... 37Chapter 6......................................................................................................................................... 45Chapter 7......................................................................................................................................... 53Chapter 8......................................................................................................................................... 57Chapter 9......................................................................................................................................... 65Answers........................................................................................................................................... 69Additional Exercises (E)Chapter R .......................................................................................................................................... 1Chapter 1......................................................................................................................................... 11Chapter 2......................................................................................................................................... 25Chapter 3......................................................................................................................................... 39Chapter 4......................................................................................................................................... 53Chapter 5......................................................................................................................................... 67Chapter 6......................................................................................................................................... 83Chapter 7....................................................................................................................................... 105Chapter 8....................................................................................................................................... 115Chapter 9....................................................................................................................................... 131Answers......................................................................................................................................... 141Group Activities (G)Chapter R .......................................................................................................................................... 1Chapter 1........................................................................................................................................... 3Chapter 2........................................................................................................................................... 5Chapter 3........................................................................................................................................... 7Chapter 4........................................................................................................................................... 9Chapter 5......................................................................................................................................... 11Chapter 6......................................................................................................................................... 13Chapter 7......................................................................................................................................... 15Chapter 8......................................................................................................................................... 17Chapter 9......................................................................................................................................... 19Answers........................................................................................................................................... 21Tests (T)Chapter R .......................................................................................................................................... 1Chapter 1......................................................................................................................................... 21Chapter 2......................................................................................................................................... 41Chapter 3......................................................................................................................................... 57Chapter 4......................................................................................................................................... 77Chapter 5......................................................................................................................................... 93Chapter 6....................................................................................................................................... 113Chapter 7....................................................................................................................................... 141Chapter 8....................................................................................................................................... 157Chapter 9....................................................................................................................................... 177Final Exam .................................................................................................................................... 205Test Answers................................................................................................................................. 215Final Exam Answers ..................................................................................................................... 245iii

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M-5Mini-Lecture 1.1Study Skill Tips for Success in MathematicsObjectives:A.Get ready for this course.B.Understand some general tips for success.C.Know how to use this text.D.Know how to use text resources.E.Get help as soon as you need it.F.Learn how to prepare for and take an exam.G.Develop good time management.Examples:1.Get ready for this course.a)Positive attitudeb)Be familiar with course structurec)Avoid schedule conflictsd)Allow adequate time for class arrivale)Bring all required materials2.Understand some general tips for success.a)Organize materialsb)Make contact with other studentsc)Choose to attend all classesd)Do your homeworke)Check your workf)Learn from mistakesg)Ask questionsh)Hand in assignments on time3.Know how to use this text.a)Each example in every section has a Practice exercise associated with it.b)At beginning of each section, a list of icons shows availability of support materials.c)Each chapter ends with Chapter Highlights, Reviews, and Practice Tests.4.Know how to use video and notebook organizer resources.a)Video resources include interactive lectures, test prep, and student success tips.b)Notebook organizer resources include video and student organizers5.Get help as soon as you need it.6.Learn how to prepare for and take an exam.a)Review previous homework assignments, class notes, quizzes, etc.b)Read Chapter Highlights to review concepts and definitions.c)Practice working out exercises in the end-of-the-chapter Review and Test.d)When taking a test, read directions and problems carefully.7.Develop good time management.a)Make a list of all weekly commitments with estimated time needed.b)Be sure to schedule study time. Don’t forget to eat, sleep, and relax!Teaching Notes:•Many developmental students are hesitant to ask questions and seek extra help.•Be sure to explain your expectations. Keep your expectations clear and concise.

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M-6Mini-Lecture 1.2Symbols and Sets of NumbersObjectives:A.Define the meaning of the symbols=,≠, <, >,≤, and >.B.Translate sentences into mathematical statements.C.Identify integers, rational numbers, irrational numbers, and real numbers.D.Find the absolute value of a real number.Key Vocabulary:set, member (element), variables, mathematical statements, integers, negativeintegers, positive integers, rational numbers, irrational numbers, real numbers, absolute valueExamples:1.Insert <, >, or = in the space between the paired numbers to make each statement true.a) 2 ____ 8b) 41 ____ 14c)37____921d)2.12____2.12−Determine whether each statement is true or false.e)1520≤f)3.0023.202≥g)147189≠h)611714≥2.Translate each sentence into a mathematical statement.a)Negative eleven is less than or equal to negative four.b)Fourteen is greater than one.3.Tell which set or sets each number belongs to: natural numbers, whole numbers, integers,rational numbers, irrational numbers, and real numbers.a) 5b)3−c) 83d)5e) 04.Find each absolute value.a)6.2b)14−c)29−d)0.03e)0Teaching Notes:•Encourage students to read an inequality statement from left to right.•Remind students that an inequality symbol < or > points toward the smaller value.•Plot values on a number line to visually illustrate less than or greater than.Answers: 1a) <; b) >; c) =; d) >; e) true; f) false; g) false; h) true; 2a) -11 < -4; b) 14 > 1; 3a) naturals, wholenumbers, integers, rationals, reals; b) integers, rationals, reals; c) rationals, reals; d) irrationals., reals; e) wholenumbers, integers, rationals, reals; 4a) 6.2; b) 14; c) 2/9; d) 0.03; e) 0

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M-7Mini-Lecture 1.3Exponents, Order of Operations, and Variable ExpressionsObjectives:A.Define and use exponents and the order of operations.B.Evaluate algebraic expressions given replacement values for variables.C.Determine whether a number is a solution of a given equation.D.Translate phrases into expressions and sentences into equations.Key Vocabulary:exponential notation, base, exponent, exponential expression, grouping symbols,algebraic expression, evaluate an algebraic expression, equation, solving, solutionExamples:1.Evaluate.a)32b)71c)267d)()30.32.Using the order of operations, simplify each expression.a)73 · 2+b)22532−⋅c)()65638−+−+d)()()()()2014321233−− −−−÷ −−3.Evaluate each expression whenx= 3,y= 2, andz= 6.a)xyz++b)3xz−c)25zx−d)253zyxz−4.Determine whether the given number is a solution of the given equation.a)1215; 27x−=b) 1229 : 7y+=c) 315 ; 5420x=d)32 : 0yy=+5.Write each phrase as an algebraic expression.a)The sum of a number and thirteenb) The quotient of forty-two and a numberWrite each sentence as an equation.c)The product of one-third and a number is nine.d)A number added to twelve is fourteen.Teaching Notes:•Be sure to identify base and exponent when working with exponential notation.•Some students may find order of operations challenging.•Many students will confuse expressions and equations. Be sure they understand thatthey will simplify an expression, but solve an equation.•Many students have problems translating sentences into equations.Answers: 1a) 8; b) 1; c) 36/49; d) 0.027; 2a) 13; b) 7; c) 150; d) -8; 3a) 11; b) 3; c) 3; d) 8; 4a) Solution;b) Not a solution; c) Not a solution; d) Not a solution; 5a) x + 13; b) 42/x; c) 1/3 x = 9; d) 12 + x = 14

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M-8Mini-Lecture 1.4Adding Real NumbersObjectives:A.Add real numbers.B.Find the opposite of a number.C.Evaluate algebraic expressions using real numbers.D.Solve applications that involve addition of real numbers.Key Vocabulary:opposites (additive inverses)Examples:1.Add the following real numbers.a)811+b)()()315−+−c)()()1435−+−d)3152−+−e)()95−+f)()1625+ −g)()15.327.03−+h)1528+−i)()723−+ −j)4238−+k)()5322+ −l)53128−+2.Find the additive inverse (opposite).a)8b)9−c)0d)17−3.a)Evaluatexy+for2x= −and8.y=b)Evaluate 23xy+for4x=and5.y= −4.Solve each of the following.a)At the beginning of a chemistry experiment, Amy measured the temperatureof a liquid to be5 C.− °During the experiment, the temperature rose 14°C.What was the liquid’s temperature at the end of the experiment?b)A local restaurant reported net income of$1397,−$2042, and$809−forthe past three months. What was the total net income for the three months?Teaching Notes:•Some students will need to see addition performed on a number line.•Some students will need instruction with inputting negative numbers into a calculator.•Review the definition of absolute value.Answers: 1a) 19; b) -18; c) -49; d) -11/10; e) -4; f) -9; g) 11.73;h) -1/8; i) -30; j) -4; k) 31; l) -1/24; 2a) -8; b) 9;c) 0; d) -17; 3a) 6; b) -7; 4a) 9° C; b) -$164

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M-9Mini-Lecture 1.5Subtracting Real NumbersObjectives:A.Subtract real numbers.B.Evaluate algebraic expressions using real numbers.C.Determine whether a number is a solution of a given equation.D.Solve applications that involve subtraction of real numbers.E.Find complementary and supplementary angles.Key Vocabulary:complementary angles, supplementary anglesExamples:1.Subtract.a)84−−b)1118−c)()1510−− −d)1212−−e)()2213− −f)()132207−− −g)()1.33.8−h) 159714−−2.Evaluate each expression when3,x= −7,y= −and9.z=a)xy−b) 102xy−−c)xyz+−d)2xy−3.Determine whether the given number is a solution of the given equation.a)814;5x+=−b)– 817;9y= −−c)711;2xx−+=+−4.In a game of cards, Alicia won 11 chips, lost 6 chips, won 3 chips, lost 14 chips, andwon 1 chip. What was her final count of chips?5.Find the complementary or supplementary angle.a)b)x°x°42°53°Teaching Notes:•Remind students to always change subtraction to addition and “add the opposite”.•Some students forget to change the sign of the second value after changing to addition.•Encourage students to take the time to write the steps:()()3 –2325−=+ +=Answers: 1a) -12; b) -7;1c) -5; d) -24; e) 35; f) 75; g) -2.5; h) 39/14; 2a) 4; b) -13/9; c) 1; d) 16;3a) Not a solution; b) Solution; c) Solution; 4) -5; 5a) 138° ; b) 37°

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M-10Mini-Lecture 1.6Multiplying and Dividing Real NumbersObjectives:A.Multiply real numbers.B.Find the reciprocal of a real number.C.Divide real numbers.D.Evaluate expressions using real numbers.E.Determine whether a number is a solution of a given equation.F.Solve applications that involve multiplication or division of real numbers.Key Vocabulary:reciprocals (multiplicative inverses), undefinedExamples:1.Multiply the real numbers.a)( )6 5−b)() ()113−−c)()1050−d)() ()3.15.01−−2.Find the reciprocal of the real number.a) 37b) 5c)521−d) 0.33.Divide the real numbers.a) 273−b)()905−÷ −c)18215−÷−d)220−4.Evaluate each expression.a)29320−−b)()()2544 8464+ −+⋅⋅−c)()()()275239−− −+ −d)()()9417510−−−−e)Evaluate()642310xyz−−−+− −when5,x=1,y= −and0.z=5.a)Is –3 a solution of155?x−÷= −b)Is93−a solution of312 ?8x−= −6.The temperature falls three degrees per hour for eight hours. What is the total change intemperature?Teaching Notes:•Multiplying and dividing real numbers should be relatively easy for most students.•Remind students that0 50=and5 0is undefined.•Many students have difficulty with the fact that()2255.−≠−Answers: 1a) -30; b) 33; c) -500; d) 15.531; 2a) 7/3; b) 1/5; c) -21/5; d) 10/3; 3a) -9; b) 18; c) 15/16;d) undefined; 4a) 3/10; b) 53/8; c) undefined; d) 3/5; e) -2; 5a) Not a solution; b) Solutions; 6) –24 degrees

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M-11Mini-Lecture 1.7Properties of Real NumbersObjectives:A.Use the commutative and associative properties.B.Use the distributive property.C.Use the identity and inverse properties.Key Vocabulary:commutative property of addition, commutative property of multiplication,associative property of addition, associative property of multiplication, distributive propertyof multiplication over addition, identity propertiesExamples:1.Use the commutative property of addition or multiplication to complete each statement.a)3_____y+=b)()9_____a+−=c)10_____x−⋅=d)_____s t⋅=Use the associative property of addition or multiplication to complete each statement.e)(3)_______xy++=f)()25________x−⋅=Use the commutative and associative properties to simplify each expression.g)()124x++h)()7 5x−i)15312x−++j)()0.131.2y−2.Use the distributive property to write each expression without parentheses.Then simplify the result, if possible.a)()8xy+b)()3 7– 9x−c)()26– 10y−−d)()6 4– 3– 9xyUse the distributive property to write each sum as a product.e) 66xy⋅+⋅f)1313 4x⋅+⋅g)()()22xy−+ −h) 11 633a+⋅3.Name the property that is illustrated by each true statement.a)01111+=b)1313⋅=c)()550+ −=d)12 112⋅=Teaching Notes:•Many students use the Properties of Real Numbers without realizing that they are usingthese properties.•Some students, when using the Distributive Property, forget to multiply the second term.Answers: 1a) y+3; b) -9+a; c) x(-10); d) t·s; e) 3 + (x + y); f) (-2 · 5) x; g) 16 + x; h) -35x; i) x + 1/12;j) -0.156y; 2a) 8x + 8y; b) -21x + 27; c) 12y + 20; d) 24x – 18y – 54; e) 6(x + y); f) 13(x + 4); g) -2(x + y);h) 1/3(a + 6); 3a) Identity Element for Addition; b) Multiplicative Inverse Property; c) Additive Inverse Property;d) Identity Element for Multiplication

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M-12Mini-Lecture 1.8Simplifying ExpressionsObjectives:A.Identify terms, like terms, and unlike terms.B.Combine like terms.C.Simplify expressions containing parentheses.D.Write word phrases as algebraic expressions.Key Vocabulary:term, numerical coefficient, like terms, unlike terms, combine like termsExamples:1.Identify the numerical coefficient of each term.a)9xb)3y−c)x−d)22.7xyIndicate whether the terms in each list are like or unlike.e)6 ,3xx−f)22,xyx y−−g)15,2abba−h)32332,xyzx yz−2.Simplify each expression by combining any like terms.a)724xx−+b)92 – 16– 7yy−+++c)5251.60.9– 0.3xxx+3.Simplify each expression. Use the distributive property to remove any parentheses.a)()36x+b)()56– 2mnp− −+c)()1 693x−Remove parentheses and simplify each expression.d)()14 26 – 4x+e)()10– 5 – 2– 3aaf)()()3 2– 57xx−+4.Write each phrase as an algebraic expression. Simplify if possible.a)Add43y−+to 69.y−b)Subtract21x−from37.x+c)Triple a number, decreased by sixd)Six times the sum of a number and twoTeaching Notes:•Students will need repeated practice with identifying terms and like terms.•Remind students that a variable without a written numerical coefficient actually has acoefficient of 1.•Some students will forget to distribute the negative symbol in 3b), 3e), and 3f).It may be helpful to write a 1 in front of the parentheses in 3b) and 3f).Answers: 1a) 9; b) -3; c) -1; d) 2.7; e) like; f) unlike; g) like; h) unlike; 2a) 5x+4; b) -8y; c) 1.3x5+0.9x2;3a) 3x+18; b) 5m - 6n+2p; c) 2x - 3; d) 28x+80; e) 8a+1; f) 5x - 22; 4a) (-4y+3) + (6y - 9) = 2y - 6;b) (3x+7) - (2x - 1) = x + 8; c) 3x - 6; d) 6(x + 2)

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M-13Mini-Lecture 2.1The Addition Property of EqualityObjectives:A.Use the addition property of equality to solve linear equations.B.Simplify an equation and then use the addition property of equality.C.Write word phrases as algebraic expressions.Key Vocabulary:solving, linear equation in one variable, equivalent equations, additionproperty of equalityExamples:1.Solve each equation. Check each solution.a)618y−=b)185t−=−c) 8.113.9y+=d)2334a+= −2.Solve each equation. If possible, be sure to first simplify each side of the equation.Check each solution.a)()()526– 3yy+=b)10495xxx=++c)856310zzz−++= −+d)5461528xx−++=−e)11516362xx−−=+f)14.94– 2.725.171.5aaa−++= −+−3.Write each algebraic expression described.a)Two numbers have a sum of 72. If one number isz, express the other number interms ofz.b)During a recent marathon, Tom ran 8 more miles than Judy ran. If Judy ranxmiles,how many miles did Tom run?c)On a recent car trip, Raymond drovexmiles on day one. On day two, he drove 170miles more than he did on day one. How many miles, in terms ofx,did Raymonddrive for both days combined?Teaching Notes:•Some students need a quick review of “like terms.”•Avoid shortcuts! Write out each step until they have mastered this concept.•Teach students to work a problem in sequential order, showing each step.•Encourage students to take their time and organize their work. This will help when theproblems become more complex.Answers: 1a) y=24; b) t=-13; c) y=5.8; d) a=-17/12; 2a) y=28; b) x=9; c) z=5; d) x=-17; e) x=-5/6; f) a=-11;3a) 72 – z; b) x+8; c) 2x+ 170

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M-14Mini-Lecture 2.2The Multiplication Property of EqualityObjectives:A.Use the multiplication property of equality to solve linear equations.B.Use both the addition and multiplication properties of equality to solve linear equations.C.Write word phrases as algebraic expressions.Key Vocabulary:multiplication property of equalityExamples:1.Use the multiplication property of equality to solve each linear equation.Check each solution.a)824x−= −b)70x=c)19z−=d)322x= −e) 2125a=f)2.511y=−g)308b−=h)10.23.4c−= −2.Use the addition property of equality and the multiplication property of equality to solveeach linear equation. Check each solution.a)5646x+=b)7119a−=c)243– 9x−= −d) 11633y−= −e)5.81.932.5 – 1.5zz−+= −f) 876 – 2– 10yyy+=g)()()()4 4– 18 –24x=−−3.Write each algebraic expression described. Simplify if possible.a)Ifzrepresents the first of two consecutive even integers, express the sum of the twointegers in terms ofz.b)Ifxrepresents the first of three consecutive odd integers, express the sum of the firstand third integer in terms ofx.c)Houses on one side of a street are all numbered using consecutive odd integers.If the first house on the street is numberedx,write an expression in terms ofxfor the sum of the numbers of the first five houses.Teaching Notes:•Review “like terms” with students.•Many students do not combine like terms before using one of the properties.•Encourage students to always take the time to check their solution.Answers: a) x=3; b) x=0; c) z=-19; d) x=-22/3; e) a=30; f) y=-27.5; g) b=0; h) c=3; 2a) x=8; b) a=162; c) x=5;d) y=-17; e) z=8; f) y=-1/20; g) x=5/4; 3a) 2z+2; b) 2x+4; c) 5x+20

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M-15Mini-Lecture 2.3Further Solving Linear EquationsObjectives:A.Apply the general strategy for solving a linear equation.B.Solve equations containing fractions or decimals.C.Recognize identities and equations with no solution.Key Vocabulary: linear equation in one variable, no solution, identity, all real numbersExamples:1.Solve each linear equation.a)()6– 5– 14aa=b)()4 3– 116b=c)()48 29zz=+d)()()283– 5xx+=e)()()3 2– 354aa=+f)()12 4– 23– 4cc=2.Solve each equation containing fractions.a)416y−=b) 13548xx−=c)655144xx−++= −Solve each equation containing decimals.d)()0.050.061500570xx+−=e)()()0.47 – 0.1 360.8xx++= −3.Solve each equation. Indicate if it is an identity or an equation with no solution.a)()67642zz+=+b)312– 184– 1xxx+=+c)23136xx−=+Teaching Notes:•Refer students to the beginning of this section in the textbook for steps:To Solve Linear Equations in One Variable.•Most students find it difficult to solve equations with fractions or decimals.•Common error: When multiplying equations with fractions by the LCD, some studentsmultiply only the terms with fractions instead of all terms.•Common error: When solving equations with decimals and parentheses, some studentsmultiply terms both inside parentheses and outside of parentheses by a power of 10.Answers: 1a) a=3; b) b=5/3; c) z=-6; d) x=31; e) a=29; f) c=4/9; 2a) y=30; b) x=-40; c) x=9; d) x=6000;e) x=-30; 3a) identity; all real numbers; b) no solution; c) no solution

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M-16Mini-Lecture 2.4An Introduction to Problem SolvingObjectives:A.Solve problems involving direct translations.B.Solve problems involving relationships among unknown quantities.C.Solve problems involving consecutive integers.Key Vocabulary:general strategy for problem solving (understand, translate, solve, interpret)Examples:Solve.1.a)The sum of a number and eight is doubled. The result is 12 more than the number.Find the number.b)The difference between two positive integers is 42. One integer is three times asgreat as the other. Find the integers.2.a)A graduating class is made up of 450 students. There are 206 more girls than boys.How many boys are in the class?b)A 22-ft pipe is cut into two pieces. One piece is 7 feet shorter than the other piece.What is the length of the longer piece?c)A triangle has three angles,A,B, andC. AngleCis 18° greater than angleB.AngleAis 4 times angleB. What is the measure of each angle?(Hint: The sum of the angles of a triangle is 180°).3.a)The room numbers of two adjacent hotel rooms are two consecutive odd numbers.If their sum is 1380, find the hotel room numbers.b)In an open book, the left and right page numbers are consecutive natural numbers.The sum of the page numbers is 349. What is the number of the page on the left?Teaching Notes:•Many students find application problems challenging.•Encourage students, whenever possible, to draw diagrams, charts, etc.•Encourage students to use algebra to solve a problem even though they may be able tosolve without it.•Refer students toGeneral Strategy for Problem Solving.Answers: 1a) -4; b) 21, 63; 2a) 122 boys; b) 14.5 feet; c) A=108°, B=27°, C=45°; 3a) 689, 691; b) 174

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M-17Mini-Lecture 2.5Formulas and Problem SolvingObjectives:A.Use formulas to solve problems.B.Solve a formula or equation for one of its variables.Key Vocabulary:formulaExamples:1.Substitute the given values into each given formula and solve for the unknown variable.If necessary, round to one decimal place.a)Distance Formulab)Perimeter of a rectangle;9,63drttd===22;32,7PlwPw=+==c)Volume of a pyramidd)Simple interest1;40,83VBhVh===;345,2300,0.03IPRTIPR====Solve.e)Convert the record high temperature of 102°F to Celsius. (9325FC=+)f)You have decided to fence an area of your backyard for your dog. The length of thearea is 1 meter less than twice the width. If the perimeter of the area is 70 meters,find the length and width of the rectangular area.g)For the holidays, Chris and Alicia drove 516 miles. They left their house at 7 a.m.and arrived at their destination at 4 p.m. They stopped for 1 hour to rest and re-fuel.What was their average rate of speed?2.Solve each formula for the specified variable.a)Area of a triangleb)Perimeter of a triangle12Abh=forb1233forPssss=++Teaching Notes:•Most students may need algebra reminders when working with a formula given values.•Refer students toSolving Equations for a Specified Variablechart in the textbook.•Many have problems with applications. SeeGeneral Strategy for Problem Solving.Answers: 1a) r=7; b) l=9; c) B=15; d) t=5; e) 38.9°C; f) l=23, w=12;g) 64.5 mph; 2a)2/;bAh=b)312sPss=−−

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