美国algebra1 Chapter03

CHAPTER3CHAPTERTABLEOFCONTENTS3-1Using Letters to RepresentNumbers3-2Translating Verbal PhrasesInto Symbols3-3Algebraic Terms andVocabulary3-4Writing Algebraic

Expressions in Words3-5Evaluating AlgebraicExpressions3-6Open Sentences and

Solution Sets3-7Writing FormulasChapter SummaryVocabularyReview ExercisesCumulative ReviewALGEBRAICEXPRESSIONSAND OPENSENTENCESAn express delivery company will deliver a letteror package locally,within two company hasthe following schedule of addition to the basiccharge of $25,the cost is $3 per mile or part of a milefor the first 10 miles or less and $4.50 per mile or partof a mile for each additional mile over such as those described,that vary accordingto a schedule,are often shown by a formula or set as can be used to solve many this chapter,you will learn to write algebraicexpressions and formulas,to use algebraic expressionsand formulas to solve problems,and to determine thesolution set of an open sentence.88

Using Letters to Represent Numbers89Eggs are usually sold by the dozen,that is,12 in a ore,we knowthat:In 1 carton,there are 12 ϫ1 or 12 2 cartons,there are 12 ϫ2 or 24 3 cartons,there are 12 ϫ3 or 36 n cartons,there are 12 ϫn or ,n is called a variableor a placeholderthat can represent differentnumbers from the set of whole numbers,{1,2,3,...}.The set of numbers that canreplace a variable is called the domainor the replacement setof that that a numerical expression contains only algebraicexpression,such as 12n,however,is an expression or phrase that contains one ormore this section,you will see how verbal phrases are translated into algebraicexpressions,using letters as variables and using symbols to represent Phrases Involving AdditionThe algebraic expression aϩb may be used to represent several different ver-bal phrases,such as:aplus bthe sum of aand baadded to bbis added to aais increased by bbmore than aThe word exceeds means “is more than.”Thus,the number that exceeds 5 by2 can be written as “2 more than 5”or 5 ϩe the numerical and algebraic expressions shown below.A numerical expression:An algebraic expression:The number that exceeds 5 by 2 is 5 ϩ2,or number that exceeds aby b is aϩ Phrases Involving SubtractionThe algebraic expression a Ϫbmay be used to represent several different ver-bal phrases,such as:aminus badecreased by bbless than absubtracted from aadiminished by ba reduced by bthe difference between aand bVerbal Phrases Involving MultiplicationThe algebraic expressions aϫb,a ؒb,(a)(b) and abmay be used to representseveral different verbal phrases,such as:atimes bthe product of aand bb multiplied by a

90Algebraic Expressions and Open SentencesEXAMPLE 1The preferred form to indicate multiplication in algebra is ,a productis indicated by using no symbol between the variables being multiplication symbol ϫis avoided in algebra because it can be con-fused with the letter or variable raised dot,which is sometimes mistakenfor a decimal point,is also heses are used to write numericalexpressions:(3)(5)(2) or 3(5)(2).Note that all but the first number must be algebraic expressions,parentheses may be used but they are notneeded:3(b)(h) ϭ Phrases Involving DivisionThe algebraic expressions aϬb and

aent verbal phrases,such as:bmay be used to represent several differ-adivided by bthe quotient of aand bThe symbols aϬ4 and

a4mean one-fourth of aas well as adivided by s and CommasIn some verbal phrases,using a comma can prevent example,in“the product of xand y,decreased by 2,”the comma after ymakes it clear thatthe xand yare to be multiplied before subtracting 2 and can be written as(xy)Ϫ2 or xyϪt the comma,the phrase,“the product of x and ydecreased by 2,”would be written x(y Ϫ2).Use mathematical symbols to translate the following verbal phrases into alge-braic language: than 33 ϩwb.w less than 33 Ϫwc.r decreased by 2r Ϫ product of 5rand x,decreased by 102xϪ10f.25,diminished by 4 times n25 Ϫ sum of tand u,divided by 6t16uh.100 decreased by twice (x ϩ5)100 Ϫ2(x ϩ5)

Translating Verbal Phrases into Symbols91EXERCISESWriting About n why the sum of aand 4 can be written as aϩ4 or as 4 ϩn why 3 less than acan be written as aϪ3 but not as 3 Ϫping SkillsIn 3–20,use mathematical symbols to translate the verbal phrases into algebraic 79.c decreased by 612.5 less than than the difference of pand q19.5 times x,increased by 24.4 minus ed by -tenth of w13.8 divided by y5.7 times x8.10 divided by product of xand plied by -third of z18.a number that exceeds m by 420.10 decreased by twice aIn 21–30,using the letter n to represent “number,”write each verbal phrase in algebraic language.21.a number increased by 223.8 increased by a number25.2 less than a -fourths of a number29.3 less than twice a number22.20 more than a number24.a number decreased by 626.3 times a number28.4 times a number,increased by 330.10 times a number,decreased by 2In 31–34,use the given variable(s) to write an algebraic expression for each verbal number of baseball cards,if bcards are added to a collection of 100 ’s height,if he was hinches tall before he grew 2 total cost of nenvelopes that cost $0.39 cost of one pen,if 12 pens cost ddollarsA knowledge of arithmetic is important in the variables representnumbers that are familiar to you,it will be helpful to solve each problem by firstusing a simpler related problem;that is,relate similar arithmetic problems to thegiven algebraic one.

92Algebraic Expressions and Open SentencesProcedureTo write an algebraic expression involving variables: of a similar problem in an expression for the arithmetic problem,using a similar expression for the problem,using letters or E 1Represent each phrase by an algebraic expression.a.a distance that is 20 meters shorter than xmetersb.a bill for n baseball caps,each costing d dollarsc.a weight that is 40 pounds heavier than amount of money that is twice d to Proceed(1)Think of a similar problemin arithmetic:(2)Write an expression for this

arithmetic problem:(3)Write a similar expression using

the letter xin place of of a distance that is

20 meters shorter than 50 meters.50 Ϫ20xϪ to Proceed(1)Think of a similar problem

in arithmetic:(2)Write an expression for this

arithmetic ly

the number of caps by the cost

of one cap:(3)Write a similar expression

using nand d:Think of a bill for 6 caps,each

costing 5 dollars.6(5)ndAnswerNote:After some practice,you will be able to do steps (1) and (2) (pϩ40) pounds or (40 ϩp) poundsd.2d dollarsAnswersa.(x Ϫ20) metersc.(p ϩ40) or (40 ϩp)arsd.2ddollars

Translating Verbal Phrases into Symbols93EXAMPLE 2Brianna paid 17 dollars for batteries and film for her the batteriescost xdollars,express the cost of the film in terms of onIf Brianna had spent 5 dollars for the batteries,the amount that was left isfound by subtracting the 5 dollars from the 17 dollars,(17 Ϫ5) uld have been the cost of the Brianna spent xdollars for the batter-ies,then the difference,(17 Ϫx) dollars would have been the cost of the (17 Ϫx) dollarsNote:In general,if we know the sum of two quantities,then we can let xrep-resent one of these quantities and (the sum Ϫx) represent the SESWriting About ent the number of pounds of grapes you can buy with d dollars if each pound costsb the algebraic expression in part aalways represent a whole number? Explainyour answer by showing examples using xapples cost c cents,represent the cost of one x apples cost ccents,represent the cost of n the algebraic expressions in parts aand balways represent whole numbers?Explain your ping SkillsIn 3–18,represent each answer in algebraic language,using the variable mentioned in the number of kilometers traveled by a bus is represented by a train traveled 200 kilo-meters farther than the bus,represent the number of kilometers traveled by the invested $1,000 in he lost d dollars when he sold the stocks,represent theamount he received for cost of a mountain bike is 5 times the cost of a the skateboard costs xdollars,represent the cost of the mountain length of a rectangle is represented by the width of the rectangle is one-half of itslength,represent its 12 centimeters had been cut from a piece of lumber,c centimeters were ent the length of the original piece of lumber.

94Algebraic Expressions and Open and Martha saved 100 the amount saved by Paul is represented by x,repre-sent the amount saved by Martha.9.A ballpoint pen sells for 39 ent the cost of ent the cost of tfeet of lumber that sells for gcents a Hilda weighed 45 kilograms,represent her weight after she had lost ,who weighs c pounds,is d pounds ent the number of poundsRonald should weigh.13.A woman spent $250 for jeans and a ski she spent ydollars for the ski jacket,rep-resent the amount she spent for the jeans.14.A man bought an article for c dollars and sold it at a profit of $ent the amountfor which he sold width of a rectangle is represented by w ent the length of the rectangle ifit exceeds the width by 8 width of a rectangle is x ent the length of the rectangle if it exceedstwice the width by 3 a plane travels 550 kilometers per hour,represent the distance it will travel in h a car traveled for 5 hours at an average rate of r kilometers per hour,represent the dis-tance it ent the total number of days in wweeks and 5 ent the total number of days in wweeks and ng auditorium with m rows can seat a total of c each row in the auditorium hasthe same number of seats,represent the number of seats in one ent the total number of calories in x peanuts and ypotato chips if each peanut con-tains 6 calories and each potato chip contains 14 charges for a telephone call are $0.45 for the first 3 minutes and $0.09 for each addi-tional minute or part of a ent the cost of a telephone call that lasts m min-utes when m is greater than 3.23.A printing shop charges a 75-cent minimum for the first 8 photocopies of a onalcopies cost 6 cents ent the cost of c copies if c is greater than 8.24.A utility company measures gas consumption by the hundred cubic feet, companyhas a three-step rate schedule for gas ,there is a minimum charge of $5.00per month for up to 3 CCF of gas ,for the next 6 CCF,the charge is $0.75 y,after 9 CCF,the charge is $0.55 per ent the cost of gCCF of gas ifgis greater than 9.

Algebraic Terms and Vocabulary95TermsA termis a number,a variable,or any product or quotient of numbers and example:5,x,4y,8ab,kterms.c5,and are

2An algebraic expression that is written as a sum or a difference has morethan one example,4aϩ2b Ϫ5chas three terms,4a,2b,and 5c,are separated by ϩand Ϫs of a TermIf a term contains two or more numbers or variables,then each number,eachvariable,and each product of numbers and variables is called a factorof theterm,or factor of the example,the factors of 3xy are 1,3,x,y,3x,3y,xy,and we factor whole numbers,we write only factors that factor of an algebraic term is called the coefficientof the remainingfactor,or product of factors,of that example,consider the algebraicterm 3xy:3 is the coefficient of xy3yis the coefficient of x3xis the coefficient of yxyis the coefficient of 3When an algebraic term consists of a number and one or more variables,thenumber is called the numerical coefficientof the example:In 8y,the numerical coefficient is 4abc,the numerical coefficient is the word coefficient is used alone,it usually means a numerical ,since xnames the same term as 1x,the coefficient of x is under-stood to be is true of all terms that contain only example:7 is the coefficient of bin the

term 7b.2.25 is the coefficient of gtin

the term 2.25gt.1 is the coefficient of bin the term b.1 is the coefficient of gt in the term ,Exponents,and PowersYou learned in Chapter 2 that a poweris the product of equal factors.A powerhas a base and an baseis one of the equal factors of the power.

96Algebraic Expressions and Open SentencesThe exponentis the number of times the base is used as a factor.(If a termis written without an exponent,the exponent is understood to be 1.)42ϭ4(4):x3= x(x)(x):35m:25d= 5(d)(d):base ϭ4base ϭxbase ϭmbase ϭdexponent ϭ2exponent ϭ3exponent ϭ1exponent ϭ2power ϭ42ϭ16power ϭx3power ϭm1ormpower ϭd2An exponent refers only to the number or variable that is directly to its left,asseen in the last example,where 2 refers only to the base show the product5das a base (or to show any sum,difference,product,or quotient as a base),wemust enclose the base in parentheses.(5d)2ϭ(5d)(5d):(a + 4)2ϭ(a + 4)( a + 4):base ϭ5dbase ϭ(a + 4)exponent ϭ2exponent ϭ2power ϭ(5d)2power ϭ(aϩ4)2Note that –d4is not the same as (Ϫd)4.–d4ϭ–1(d)(d)(d)(d) is always a negative number.(Ϫd)4ϭ1(Ϫd)(Ϫd)(Ϫd)(Ϫd) is always a positive number since

the exponent is E 1For each term,name the coefficient,base,and sa.4x5b.Ϫw8c.2prcoefficient ϭ4coefficient ϭ–1coefficient ϭ2pbase ϭxbase ϭwbase ϭrexponent ϭ5exponent ϭ8exponent ϭ1Note:Remember that coefficient means numerical coefficient,and that 2pis a real SESWriting About squaring distribute over multiplication,that is,does (ab)2= (a2)(b2)? Write (ab)2as(ab)(ab) and use the associative and commutative properties of multiplication to justifyyour squaring distribute over addition,that is,does (aϩb)2ϭa2ϩb2? Substitute valuesfor aand bto justify your answer.

Algebraic Terms and Vocabulary97Developing SkillsIn 3–6,name the factors (other than 1) of each 4.3a5.7mn6.1stIn 7–14,name,in each case,the numerical coefficient of x.7.8x11.Ϫ1.4x8.(5 ϩ2)x12.2 ϩ7x9.12x13.3.4x10.x14.ϪxIn 15–22,name,in each case,the base and exponent of the power.15.m219.10616.Ϫs320.(5y)417.t21.(xϩy)518.(Ϫa)422.12c3In 23–29,write each expression,using exponents.23.b ؒb ؒb ؒb ؒb26.7 ؒr ؒr ؒr ؒs ؒs24.pؒr ؒr27.(6a)(6a)(6a)25.aؒa ؒa ؒaؒb ؒb28.(aϪb)(a Ϫb)(a Ϫb) fourth power of (m +2n)In 30–33,write each term as a product without using exponents.30.r631.5x432.4a4b233.(3y)5In 34–41,name,for each given term,the coefficient,base,and exponent.34.Ϫ3k38."2y35.Ϫ2440.32a37.(ax)541.(Ϫb)3Applying xrepresents the cost of a can of soda,what could 5xrepresent? rrepresents the speed of a car in miles per hour,what could 3rrepresent? nrepresents the number of CDs that Alice has,what could nϪ5 represent? drepresents the number of days until the end of the year,what could

d7represent? srepresents the length of a side of a square,what could 4srepresent? rrepresents the measure of the radius of a circle,what could 2rrepresent? w represents the number of weeks in a school year,what could

w4represent? drepresents the cost of one dozen bottles of water,what could

12represent? qrepresents the point value of one field goal,what could 7q represent?

98Algebraic Expressions and Open SentencesIn Section 1 of this chapter,we listed the words that can be represented by eachof the four basic can use these same lists to write algebraicexpressions in words and to write problems that can be represented by a givenalgebraic an algebraic expression such as 2nϪ3,ncould be any real ,associated with any real number n,there is exactly one real number that is thevalue of 2nϪr,if nand 2nϪ3 represent the number of cans of tunathat two customers buy,then nmust be a whole number greater than or equalto 2 in order for both nand 2nϪ3 to be whole this situation,the domain or replacement set would be the set of E 1If nrepresents the number of points that Hradish scored in a basketball gameand 2n Ϫ3 represents the number of points that his friend Brad scored,describein words the number of points that Brad is a possible domain forthe variable n?SolutionThe number of points scored is always a whole order that 2nϪ3be a whole number,nmust be at least The number of points that Brad scored is 3 less than twice the number thatHradish scored.A possible domain for n is the set of whole numbers

greater than or equal to E 2Molly earned ddollars in July and

12d 1 10dollars in be in wordsthe number of dollars that Molly earned in In August,Molly earned 10 more than half the number of dollars that sheearned in E 3Describe a situation in which xand 12 Ϫxcan be used to represent the domain or replacement set for the onIf xeggs are used from a full dozen of eggs,there will be 12 Ϫxeggs The domain or replacement set is the set of whole numbers less than or equalto 12.

Writing Algebraic Expressions in Words99Another SolutionThe distance from my home to school is 12 my way to school,after Ihave traveled xmiles,I have 12 Ϫxmiles left to The domain or replacement set is the set of non-negative real numbers thatare less than or equal to other answers are SESWriting About 4 ϩnrepresents the number of books Ken read in September and 4 Ϫnrepresentsthe number of books he read in October,how many books did he read in these twomonths? is the domain of the variable n? said that the replacement set for the amount that we pay for any item is the set ofrational numbers of the form 0.01xwhere xis a whole you agree with Pedro?Explain why or why ping SkillsIn 3–14: in words each of the given algebraic be a possible domain foreach one route,the distance that Ian walks to school is a different route,the dis-tance is dϪ0.2 pays ncents for a can of soda at the grocery he buys soda from a machine,he pays n ϩ15 day Alexander spent aminutes on leisurely reading and 3aϩ10 minutes width of a rectangle is wmeters and the length is 2wϩ8 a school day,Abby spends hhours in class,h6hours at lunch and

3hours on spends dhours at work and

12hours driving to and from ’s score for 18 holes of golf was gand her son’s score was 10 ϩ paid dcents for a notebook and 5dϩ30 cents for a ’s essay for English class had wwords and Dominic’s had

34w 1 ia read rbooks last month and Anna read 3rϪ5 and Pete are playing a card game where it is possible to have a negative ’sscore is sand Mario’s score is sϪ220.

100Algebraic Expressions and Open the past month,Agatha has increased the time that she walks each day from mminutesto 3mϪ10 in has 1 more tape than 3 times the number of tapes that Julia a has n tapes,then Benjamin has 3nϩ1 algebraic expression 3nϩ1 represents an unspecified en the variable n is replaced by a specific number does 3nϩ1 become a spe-cific example:If n ϭ10,then 3nϩ1 ϭ3(10) ϩ1 ϭ30 ϩ1 ϭ nϭ15,then 3nϩ1 ϭ3(15) ϩ1 ϭ45 ϩ1 ϭ in this example,nrepresents the number of tapes that Julia has,onlywhole numbers are reasonable replacements for ore,the replacementset is the set of whole numbers or some subset of the set of whole we substitute specific values for the variables in an algebraic expres-sion and then determine the value of the resulting expression,we are evaluatingthe algebraic we determine the number that an algebraic expression represents forspecific values of its variables,we are evaluating the algebraic ureTo evaluate an algebraic expression,replace the variables by the givenvalues,and then follow the rules for the order of e the variables by the given te the expression within the grouping symbols such as parentheses,always simplifying the expressions in the innermost groupings fy all powers and ly and divide,from left to and subtract,from left to E 1Evaluate 50 Ϫ3xwhen xϭonHow to Proceed(1)Write the expression:50 Ϫ3x

Evaluating Algebraic Expressions101(2)Replace the variable by its given value:(3)Multiply:(4)Subtract:50 Ϫ3(7)50 Ϫ2129Answer29EXAMPLE 2Evaluate 2x2Ϫ5x ϩ4 when:a.x = Ϫ7 b.x = 1.2SolutionHow to Proceeda.(1) Write the expression:(2)Replace the variable by the value Ϫ7:(3)Evaluate the power:(4)Multiply:(5)Add:b.(1) Write the expression:(2) Replace the variable by the value 1.2:(3) Evaluate the power:(4) Multiply:(5) Add and subtract:2x2Ϫ5xϩ42(Ϫ7)2Ϫ5(Ϫ7) ϩ42(49) Ϫ5(Ϫ7) ϩ498 ϩ35ϩ41372x2Ϫ5xϩ42(1.2)2Ϫ5(1.2) ϩ42(1.44) Ϫ5(1.2) ϩ42.88Ϫ6ϩ40.88Answersa.137EXAMPLE 3b.0.882aEvaluate when aϭϪ4,nϭ10,and dϭ3.5 1 (n 2 1)dSolutionHow to Proceed(1)Write the expression:(2)Replace the variables with their

given values:(3)Simplify the expressions grouped by

parentheses or fraction bar:(4)Multiply and divide:(5)Add:2a5 1 (n 2 1)d2(24)5 1 (10 – 1)(3)285 1(9)(3)2135 1 2752135 1 2652525Answer

102Algebraic Expressions and Open SentencesCalculatorThe values given for the variables can be stored in the onENTER:(-)4

STOŁALPHAAENTER10

STOŁ3

STOŁDISPLAY:ALPHAALPHANDENTERENTER-4—>A-410—>N103—>D3Now enter the algebraic expression to be evaluated.(؉ALPHAA،ALPHAENTER:2 5

ALPHADISPLAY:DENTERN؊)1

2A/5+(N–1)D25.4Answer2525525.4EXAMPLE 4Evaluate (2x)3Ϫ2x3when xϭ ϪonHow to Proceed(1)Write the expression:(2)Replace the variable by its given value:(3)Simplify the expression within brackets:(4)Evaluate the powers:(5)Multiply:(6)Subtract:(2x)3Ϫ2x3[2(Ϫ0.40)]3Ϫ2(Ϫ0.40)3[Ϫ0.80]3Ϫ2(Ϫ0.40)3Ϫ0.512Ϫ2(Ϫ0.064)Ϫ0.512 ϩ0.128Ϫ0.384AnswerϪ0.384

Evaluating Algebraic Expressions103EXERCISESWriting About n why,in an algebraic expression such as 12ab,12 is called a constant and aand barecalled variables?n why,in step 2 of Example 1,parentheses were needed when xwas replaced by its ping SkillsTo understand this topic,you should first evaluate the expressions in Exercises 3 to 27 without a ,store the values of the variables in the calculator and enter the given algebraic expres-sions to check your 3–27,find the numerical value of each a ϭ8,b ϭϪ6,d ϭ3,x ϭϪ4,and y ϭ0.5.3.5a7.b Ϫ211.7xy3315.34x1x4.25.0.3y9.3bd9113.25a 1

5b217.14xy6.a ϩ310.5x Ϫ2y14.0.2d ϩ0.3b18.a2ϩ3d2222.12x(y 1 0.1) Ϫdx16.(3y)220.x(yϪ2)19.(ay)323.3yϪ(x Ϫd)26.(2aϪ5d)221.4(2xϩ3y)24.2(xϩy) Ϫ527.(2a)2Ϫ(5d)225.(xϪd)5Applying one car rental agency,the cost of a car for one day can be determined by using the alge-braic expression $32.00 ϩ$0.10mwhere mrepresents the number of miles ine the cost of rental for each of the following: Baier drove the car he rented for 35 Morse drove the car he rented for 435 Szalach drove the car he rented for 102 local pottery co-op charges $40.00 a year for membership and $0.75 per pound for fir-ing pottery pieces made by the algebraic expression 40 ϩ0.75prepresentsthe yearly cost to a member who brings ppounds of pottery to be ine theyearly cost for each of the following:y is an amateur potter who fired 35 pounds of work this sells her pottery in a local craft shop and fired 485 pounds of work this a stone is thrown down into a deep gully with an initial velocity of 30 feet per second,thedistance it has fallen,in feet,after tseconds can be found by using the algebraic expression16t2ϩ the distance the stone has fallen: 1 2 3 seconds.

104Algebraic Expressions and Open Parkside Bread Company sells cookies and scones as well as (b) costs$4.50 a loaf,cookies (c) cost $1.10 each,and scones (s) cost $1.50 cost of a bakeryorder can be represented by 4.50bϩ1.10cϩine the cost of each of the fol-lowing orders: cookies and two loaves of bread and one loaf of bread,a dozen cookies,and a half-dozen scones32.A Green Thumb volunteer can plant shrubbery at a rate of 6 shrubs per hour and aFriendly Garden volunteer can plant shrubbery at a rate of 8 shrubs per totalnumber of shrubs that gGreen Thumb volunteers and fFriendly Garden volunteers canplant in hhours is given by the algebraic expression 6ghϩine the number ofshrubs planted: 3 hours by 2 Green Thumb and 1 Friendly Garden 2 hours by 4 Green Thumb and 4 Friendly Garden this chapter,you learned how to translate words into algebraic value of an algebraic expression depends on the value of the the values of the variables change,the value of the algebraic example,xϩ6 is an algebraic value of xϩ6depends on the value of one value is assigned to an algebraic expression,an algebraic sentence sentences may be formulas,equations,or exam-ple,when the value 9 is assigned to the algebraic expression xϩ6,we can writethe sentence “Six more than xis 9.”This sentence can be written in symbols asxϩ6 ϭ sentence that contains a variable is called an open sentence.xϩ6 ϭ93yϭ122nϾ0xϩ5 Յ8An open sentence is neither true nor sentence will be true or false only when the variables are replaced bynumbers from a domain or a replacement set,such as {0,1,2,3}.The numbers from the domain that make the sentence true are the elementsof the solution setof the open sentence.A solution set,as seen below,cancontain one or more numbers or,at times,no numbers at all,from the replace-ment E 1Using the domain {0,1,2,3},find the solution set of each open sentence:a.xϩ6 ϭ9 b.2nϾ0

Open Sentences and Solution ure:Replace xin the open

sentence with numbers from thedomain {0,1,2,3}.x + 6 ϭ9Let xϭ 0 ϩ6 ϭ9 is xϭ l ϩ6 ϭ9 is xϭ 2 ϩ6 ϭ9 is xϭ 3 ϩ6 ϭ9 is ,only when xϭ3 does the

open sentence become a true

ure:Replace nin theopen sentence with numbersfrom the domain {0,1,2,3}.2n Ͼ0Let nϭ 2(0) Ͼ0 or 0 Ͼ0 is nϭ 2(l) Ͼ0 or 2 Ͼ0 is nϭ 2(2) Ͼ0 or 4 Ͼ0 is nϭ 2(3) Ͼ0 or 6 Ͼ0 is ,three elements of the

domain make the open :on set ϭ{3}.Answer:on set ϭ{1,2,3}EXAMPLE 2Find the solution set for the open sentence 3yϭ12 using: domain ϭ{3,5,7} domain ϭ{whole numbers}ure:Replace ywith 3,5,and yϭ3,then 3(3) ϭ12 is yϭ5,then 3(5) ϭ12 is yϭ7,then 3(7) ϭ12 is yis replaced by each ofthe numbers from the domain,no

true statement is

solution set for this domainis the

empty set or the null set,written

in symbols as {} or as

л.ure:Of course,you

cannot replace ywith everywhole number,but you canuse multiplication factslearned know that 3(4) ϭ

yϭ 3(4) ϭ12 is other whole number wouldmake the open sentence 3yϭ12 a true : solution set is {4}.Answer: solution set is {} or л.

106Algebraic Expressions and Open SentencesEXERCISESWriting About the open sentence xϩ7 Ͼ12,write a domain for which the solution set is the the open sentence xϩ7 Ͼ12,write a domain for which the solution set has only the open sentence xϩ7 Ͼ12,write a domain for which the solution set is an ping SkillsIn 4–11,tell whether each is an open sentence,a true sentence,or an algebraic expression.4.2 ϩ3 ϭ5ϩ08.nϾ75.xϩ10 ϭ149.3 ϩ2 Ͻ106.yϪ410.3rϩ27.3 ϩ7 ϭ2(5)11.2xϪ7 ϭ15In 12–15,name the variable in each open sentence.12.xϩ5 ϭ913.4yϭ2014.rϪ6 Ͼ215.7 Ͻ3 ϩaIn 16–23,using the domain {Ϫ5,Ϫ4,Ϫ3,Ϫ2,Ϫ1,0,1,2,3,4,5},find the solution set for each opensentence.16.nϩ3 ϭ720.2nϩ1 Ͻ817.xϪxϭ01 15421.2n

318.5 Ϫnϭ222.3x2,x19.nϩ3 Ͼ923.2xϽϪ4Applying s sell for $0.19 wants to buy at least one but not more than 10 pencils andhas $1.50 in his the number of pencils that Torry wants to buy to write a domain for this number of pencils that Torry might buy,x,can be found using the open sentence0.19x Յ the solution set of this open sentence using the domain from part many pencils can Torry buy? local grocery store has frozen orange juice on sale for $0.99 a can but limits the numberof cans that a customer may buy at the sale price to no more than domain for this problem is the number of cans of juice that a customer may buy atthe sale the does not want to spend more than $10,the number of cans that shemight buy at the sale price,y,is given by the equation 0.99yՅ the solution setof this equation using the domain from part many cans can buy if she does not want to spend more than $10?

Writing ion to a recreation park is $ includes all rides except for a ride called TheBroncothat costs $1.50 for each has $25 to the domain for this problem,the number of times a person might ride number of times Ian might ride The Bronco,z,can be found using the open sen-tence 17.50 ϩ1.50zՅ the solution set of this open sentence using the domainfrom part many times can Ian ride The Bronco?A formulauses mathematical language to express the relationship between twoor more formulas are found by the strategy of looking for example,how many square units are shown in the rectangle on theleft? This rectangle,measuring 4 units in length and 3 units in width,contains atotal of 12 square units of such examples led to the conclusion that the area of a rectangle isequal to the product of its length and relationship is expressed by theformula Aϭlw where A,l,and w are variables that represent,respectively,thearea,the length,and the width of a rectangle.A formula is an open sentence that states that two algebraic expressions formulas,the word is is translated into the symbol ϭ.EXAMPLE 1Write a formula for each of the following perimeter Pof a square is equal to 4 times the length of one total cost Cof an article is equal to its price pplus an 8% tax on sum Sof the measures of the interior angles of an n-sided polygon is180 times 2 less than the number of srepresent the length of each side of a square.Pϭ4sAnswer8b.8% (or 8 percent) means 8 hundredths,written as 0.08 or

100.8Cϭp ϩ0.08por C5p 1

100pAnswerc.“2 less than the number of sides”means (nϪ2).Sϭ180(nϪ2)

Answer

108Algebraic Expressions and Open SentencesEXAMPLE 2Write a formula that expresses the number of months m that are in onLook for a 1 year,there are 12 2 years,there are 12(2) or 24 3 years,there are 12(3) or 36 yyears,there are 12(y) or equals m,the number mϭ12yEXAMPLE 3The Short Stop Diner pays employees $6.00 an hour for working 40 hours aweek or working overtime,an employee is paid $9.00 for each hourover 40 a formula for the wages,W,of an employee who works hhours in a onTwo formulas are needed,one for hՅ40 and the other for hϾ hՅ40,the wage is 6.00 times the number of hours,h.Wϭ6.00hIf hϾ40,the employee has worked 40 hours at $6.00 an hour and the remain-ing hours,hϪ40,at $9.00 an hour.Wϭ6.00(40) ϩ9.00(hϪ40).AnswerWϭ6hif hՅ40 and Wϭ6(40) ϩ9(hϪ40) if hϾ40.(Note that the formula for hϾ40 may also be given as W ϭ240 ϩ9(hϪ40).)EXERCISESWriting About said that a recipe is a type of you agree or disagree with Fran? Explainyour an algebraic expression a formula? Explain why or why a formula an open sentence? Explain why or why not.

Writing Formulas109Developing SkillsIn 3–17,write a formula that expresses each total length lof 10 pieces of lumber,each m meters in length,is 10 times the length ofeach piece of article’s selling price Sequals its cost c plus the margin of profit perimeter P of a rectangle is equal to the sum of twice its length land twice its

width average mof three numbers,a,b,and cis their sum divided by area A of a triangle is equal to one-half the length of the base b multiplied by thelength of the altitude area A of a square is equal to the square of the length of a side volume V of a cube is equal to the cube of the length of an edge surface area S of a cube is equal to 6 times the square of the length of an edge surface area Sof a sphere is equal to the product of 4pand the square of the

radius average rate of speed r is equal to the distance that is traveled d divided by the timespent on the trip Fahrenheit temperature Fis 32°more than nine-fifths of the Celsius tempera-ture Celsius temperature C is equal to five-ninths of the difference between the Fahrenheittemperature Fand 32°. dividend D equals the product of the divisor d and the quotient qplus the

remainder r.16.A sales tax T that must be paid when an article is purchased is equal to 8% of the price ofthe article v.17.A salesman’s weekly earnings Fis equal to his weekly salary s increased by 2% of his totalvolume of sales ng Skills18.A ferry takes cars,drivers,and passengers across a body of total ferry charge Cin dollars is $20.00 for the car and driver,plus d dollars for each a formula for Cin terms of dand the number of passengers, the cost of the ferry for a car if dϭ$15 and there are 5 persons in the the cost of the ferry for a car with only the driver.

110Algebraic Expressions and Open cost Cin cents of an internet telephone call lasting mminutes is xcents for the first 3minutes and ycents for each additional two formulas for C,one for the cost of calls lasting 3 minutes or less (mՅ3),and another for the cost of calls lasting more than 3 minutes (mϾ3). the cost of a 2.5 minute telephone call if xϭ$0.25 and yϭ$ the cost of a 10 minute telephone call if xϭ$0.25 and yϭ$ cost Din dollars of sending a fax of ppages is adollars for sending the first page and bdollars for each additional two formulas for D,one for the cost of faxing 1 page (pϭ1),and another forthe cost of faxing more than 1 page (pϾ1). the cost of faxing 1 page if aϭ$1.00 and bϭ$ the cost of faxing 5 pages if aϭ$1.00 and bϭ$0.60.21.A gasoline dealer is allowed a profit of 12 cents a gallon for each gallon more than25,000 gallons are sold in a month,an additional profit of 3 cents for every gallon over thatnumber is two formulas for the gasoline dealer’s profit,P,one for when the number of gal-lons sold,n,is not more than 25,000 (nՅ25,000),and another for when more than25,000 gallons are sold (nϾ25,000). Pwhen 21,000 gallons of gasoline are sold in one Pwhen 30,000 gallons of gasoline are sold in one l earns a bonus of $25 for each sale that he makes if the number of sales,s,in amonth is 20 or earns an extra $40 for each additional sale if he makes more than 20sales in a a formula for Gabriel’s bonus,B,when sՅ a formula for Bwhen sϾ August,Gabriel made 18 his bonus for September,Gabriel made 25 his bonus for is selling cookies at a local bake she sells exactly 3 dozen cookies,the costof ingredients will equal her she sells more than 3 dozen cookies, willmake a profit of 25 cents for each cookie a formula for ’s earnings,E,when the number of cookies sold,c,isequal to a formula for Ewhen cϾ the number of cookies sold if she makes a profit of $2.00.

Chapter Summary111CHAPTER SUMMARYAn algebraic expression,such as xϩ6,is an expression or a phrase that con-tains one or more variables,such as variableis a placeholder for evaluate an expression,replace each variable with a number and follow theorder of operations.A termis a number,a variable,or any product or quotient of numbers the term 6by,6 is the numerical the term n3,the baseis n,the exponentis 3,and the poweris power n3means that base nisused as a factor 3 open sentence,which can be an equation or an inequality,contains the variable is replaced by numbers from a domain,the numbersthat make the open sentence true are the elements of the solution setof the sen-tence.A formulais a sentence that shows the relationship between two or LARY3-1Variable • Placeholder • Domain • Replacement set •

Algebraic expression3-3Term • Factor • Coefficient • Numerical coefficient3-5Evaluating an algebraic expression3-6Open sentence • Solution set3-7FormulaREVIEW n the difference between an algebraic expression and an open

n the difference between 2a2and (2a) 3–6,use mathematical symbols to translate the verbal phrases into ed by ased by d4.4 less than r6.3 more than twice gIn 7–14,find the value of each expression.7.6acϪdwhen aϭ10,cϭ8,and dϭ58.4b2when bϭ2.5

112Algebraic Expressions and Open Sentences9.3bϩcwhen bϭ7 and cϭn aϭ5,bϭ3,and cϭϪ9 when kϭ15 and mϭ0.612.2a2Ϫ2awhen

a51413.(2a)2Ϫ2awhen

a51414.a(bϩc) when aϭ2.5,bϭ1.1,and cϭ an algebraic expression for the total number of cents in n nickelsand the term 2xy3what is the coefficient? the term 2xy3what is the exponent of y? the term 2xy3,what is the base that is used 3 times as a factor? distance is the product of rate and time,write a formula for distance,d,in terms of rate,r,and time, is the smallest member of the solution set of 19.4 ՅyϪ29 if thedomain is {46.25,47.9,48,48.5,49,50,51.3}? is the smallest member of the solution set of 19.4 ՅyϪ29 if thedomain is the set of whole numbers? is the smallest member of the solution set of 19.4 ՅyϪ29 if thedomain is the set of real numbers? a baseball game,the winning team scored nruns and the losing teamscored 2nϪ5 be in words the number of runs that the losing team could have been the score of the game? Is there more than oneanswer? are the possible values for n?24.A mail order book club offers books for $8.98 each plus $3.50 for shippingand handling on each cost of Bethany’s order,which totaledless than $20,can be expressed as 8.98bϩ3.50 Ͻ20 where b representsthe number of books Bethany could be the domain for this problem? is the solution set for this open sentence? two algebraic expressions to represent the cost of sending anexpress delivery based on the rates given in the chapter opener on page88,the first if the delivery distance is 10 miles or less,and the second if thedelivery distance is more than 10 miles.26.A list of numbers that follows a pattern begins with the numbers

2,5,8,11,....

Chapter the next number in the a rule or explain how the next number is is the 25th number in the list? of the numbers given below is different from the others,that is,itbelongs to a set of numbers to which the others do not nwhy each is oranges cost as much as five orange costs the same as abanana and an many apples cost the same as three bananas? a three-digit multiple of 11 by multiplying any whole numberSTEPSTEPSTEPSTEPSTEPSTEPfrom 10 to 90 by the digits in the hundreds and the ones the sum is greater than or equal to 11,subtract e thisresult to the digit in the tens the procedure for otherthree-digit multiples of a three-digit number that is not a multiple of 11 by adding anycounting number less than 11 to a multiple of 11 used in step thedigits in the hundreds and the ones the sum is greater than orequal to 11,subtract e this result to the digit in the the procedure with another on steps 1 and 2,can you suggest a way of determining whetheror not a three-digit number is divisible by 11? a four-digit multiple of 11 by multiplying any whole number from91 to 909 by the digits in the hundreds and ones thedigits in the thousands and tens one sum is greater than orequal to 11,subtract e these the procedurefor another four-digit multiple of a four-digit number that is not a multiple of 11 by adding anycounting number less than 11 to a multiple of 11 used in step thedigits in the hundreds place and ones the digits in the thou-sands place and tens one sum is greater than or equal to 11,subtract e these the procedure starting withanother on steps 4 and 5,can you suggest a way of determining whetheror not a four-digit number is divisible by 11? a rule for determining whether or not any whole number is divis-ible by 11.

114Algebraic Expressions and Open SentencesPart IAnswer all questions in this correct answer will receive 2 ial credit will be of the following is the set of negative integers greater than –3?(1){Ϫ4,Ϫ5,Ϫ6,Ϫ7,...}(3){Ϫ3,Ϫ2,Ϫ1}(2){Ϫ3,Ϫ4,Ϫ5,Ϫ6,...}(4){Ϫ2,Ϫ1} exact value of the rational number

53can be written as(1)1.6(2)1.6(3)1.666666667(4)d to the nearest hundredth,"5is approximately equal to(1)2.23(2)2.236(3)2.24(4) of the following numbers is rational?(1)p(2)"2(3) of the following inequalities is a true statement?(1)0.026 Ͻ0.25 Ͻ0.2(2)0.2 Ͻ0.026 Ͻ0.25(3)0.2 Ͻ0.25 Ͻ0.026(4)0.026 Ͻ0.2 Ͻ0.25(4)" length of a rectangle is given as 30.02 measure has howmany significant digits?(1)1(2)2(3)3(4) of the following is not a prime?(1)7(2)23(3) arithmetic expression 8 Ϫ5(Ϫ0.2)2Ϭ10 is equal to(1)0.9(2)0.78(3)7.98(4)51(4) of the following is a correct application of the distributive prop-erty?(1)4(8 ϩ0.2) ϭ4(8) ϩ4(0.2)(3)8(5 ϩ4) ϭ8(5) ϩ4(2)6(3 Ϫ1) ϭ6(Ϫ1 ϩ3)(4)8(5)(4) ϭ8(5) ϩ8(4) additive inverse of 7 is(1)–7(2)0(3)17(4)͉7͉Part IIAnswer all questions in this correct answer will receive 2 y indicate the necessary steps,including appropriate formula substitu-tions,diagrams,graphs,charts, all questions in this part,a correct numer-ical answer with no work shown will receive only 1 credit.

Cumulative the 80 students questioned about what they had read in the pastmonth,35 had read nonfiction,55 had read fiction,and 22 had read nei-ther fiction nor many students had read both fiction andnonfiction? is the largest number that is the product of three different two-digitprimes?Part IIIAnswer all questions in this correct answer will receive 3 y indicate the necessary steps,including appropriate formula substitu-tions,diagrams,graphs,charts, all questions in this part,a correct numer-ical answer with no work shown will receive only 1 formula for the area of an equilateral triangle (a triangle with three32sides of equal measure) is

A5" the area of an equilateral

triangle if the measure of one side is 12.6 s youranswer to the number of significant digits determined by the given data.14.A teacher wrote the sequence 1,2,4,...and asked what the next numbercould students each gave a different answer and the teacher saidthat all three answers were said n what rule Adam said n what rule Bette said n what rule Carlos IVAnswer all questions in this correct answer will receive 4 y indicate the necessary steps,including appropriate formula substitu-tions,diagrams,graphs,charts, all questions in this part,a correct numer-ical answer with no work shown will receive only 1 aϭ7,bϭ–5,and

c513,evaluate the expression

c

1

3(b 2 2).Donot use a each step in your le bought material to make a vest and used half of thematerial to make the skirt and two-thirds of what remained to make had

114yards of material many yards of material did she buy? many yards of material did she use for the vest? many yards of material did she use for the skirt?