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美国algebra1 Chapter03

美国algebra1 Chapter03
美国algebra1 Chapter03

CHAPTER 3

88C HAPTER

T ABLE OF C ONTENTS 3-1Using Letters to Represent Numbers

3-2T ranslating Verbal Phrases Into Symbols

3-3Algebraic T erms and

Vocabulary

3-4Writing Algebraic

Expressions in Words

3-5Evaluating Algebraic

Expressions

3-6Open Sentences and

Solution Sets

3-7Writing Formulas

Chapter Summary

Vocabulary

Review Exercises

Cumulative Review ALGEBRAIC EXPRESSIONS AND OPEN SENTENCES An express delivery company will deliver a letter or package locally,within two hours.The company has the following schedule of rates.In addition to the basic charge of $25,the cost is $3 per mile or part of a mile for the first 10 miles or less and $4.50 per mile or part of a mile for each additional mile over 10.

Costs such as those described,that vary according to a schedule,are often shown by a formula or set of formulas.Formulas can be used to solve many different problems.

In this c hapter,you will learn to write algebraic expressions and formulas,to use algebraic expressions and formulas to solve problems,and to determine the solution set of an open sentence.

Using Letters to Represent Numbers89 3-1 USING LETTERS TO REPRESENT NUMBERS

Eggs are usually sold by the dozen,that is,12 in a carton.Therefore,we know

that:

In 1 carton,there are 12 ?1 or 12 eggs.

In 2 cartons,there are 12 ?2 or 24 eggs.

In 3 cartons,there are 12 ?3 or 36 eggs.

In n cartons,there are 12 ?n or 12n eggs.

Here,n is called a variable or a placeholder that can represent different numbers from the set of whole numbers,{1,2,3,...}.The set of numbers that can

replace a variable is called the domain or the replacement set of that variable.

Recall that a numerical expression contains only numbers.An algebraic expression,such as 12n,however,is an expression or phrase that contains one or

more variables.

In this section,you will see how verbal phrases are translated into algebraic expressions,using letters as variables and using symbols to represent operations.

Verbal Phrases Involving Addition

The algebraic expression a?b may be used to represent several different ver-

bal phrases,such as:

a plus

b a added to b a is increased by b

the sum of a and b b is added to a b more than a

The word exceeds means “is more than.”Thus,the number that exceeds 5 by

2 can be written as “2 more than 5”or 5 ?2.

Compare the numerical and algebraic expressions shown below.

A numerical expression:The number that exceeds 5 by 2 is 5 ?2,or 7.

An algebraic expression:The number that exceeds a by b is a?b.

Verbal Phrases Involving Subtraction

The algebraic expression a ?b may be used to represent several different ver-

bal phrases,such as:

a minus

b a decreased by b b less than a

b subtracted from a a diminished by b a reduced by b

the difference between a and b

Verbal Phrases Involving Multiplication

The algebraic expressions a?b,a ?b,(a)(b) and ab may be used to represent

several different verbal phrases,such as:

a times

b the product of a and b b multiplied by a

The preferred form to indicate multiplication in algebra is ab .Here,a product is indicated by using no symbol between the variables being multiplied.The multiplication symbol ?is avoided in algebra because it can be con-fused with the letter or variable x .The raised dot,which is sometimes mistaken for a decimal point,is also avoided.Parentheses are used to write numerical expressions:(3)(5)(2) or 3(5)(2).Note that all but the first number must be in parentheses.In algebraic expressions,parentheses may be used but they are not needed:3(b )(h ) ?3bh .Verbal Phrases Involving Division The algebraic expressions a ?b and may be used to represent several differ-ent verbal phrases,such as:a divided by b the quotient of a and b The symbols a ?4 and mean one-fourth of a as well as a divided by 4.Phrases and Commas In some verbal phrases,using a comma can prevent misreading.For example,in “the product of x and y ,decreased by 2,”the comma after y makes it clear that the x and y are to be multiplied before subtracting 2 and can be written as (xy )?2 or xy ?2.Without the comma,the phrase,“the product of x and y decreased by 2,”would be written x (y ?2).

EXAMPLE 1Use mathematical symbols to translate the following verbal phrases into alge-

braic language:

Answers

a.w more than 3 3 ?w

b.w less than 3 3 ?w

c.r decreased by 2r ?2

d.the product of 5r and s 5rs

e.twice x ,decreased by 102x ?10

f.25,diminished by 4 times n 25 ?4n

g.the sum of t and u ,divided by 6h.100 decreased by twice (x ?5)100 ?2(x ?5)

t 1u 6a 4a b 90Algebraic Expressions and Open Sentences

T ranslating Verbal Phrases into Symbols91

EXERCISES

Writing About Mathematics

1.Explain why the sum of a and 4 can be written as a?4 or as 4 ?a.

2.Explain why 3 less than a can be written as a?3 but not as 3 ?a.

Developing Skills

In 3–20,use mathematical symbols to translate the verbal phrases into algebraic language.

3.y plus 8

4.4 minus r

5.7 times x

6.x times 7

7.x divided by 10

8.10 divided by x

9.c decreased by 610.one-tenth of w11.the product of x and y

12.5 less than d13.8 divided by y14.y multiplied by 10

15.t more than w16.one-third of z

17.twice the difference of p and q18.a number that exceeds m by 4

19.5 times x,increased by 220.10 decreased by twice a

In 21–30,using the letter n to represent “number,”write each verbal phrase in algebraic language.

21.a number increased by 222.20 more than a number

23.8 increased by a number24.a number decreased by 6

25.2 less than a number26.3 times a number

27.three-fourths of a number28.4 times a number,increased by 3

29.3 less than twice a number30.10 times a number,decreased by 2

In 31–34,use the given variable(s) to write an algebraic expression for each verbal phrase.

31.the number of baseball cards,if b cards are added to a collection of 100 cards

32.Hector’s height,if he was h inches tall before he grew 2 inches

33.the total cost of n envelopes that cost $0.39 each

34.the cost of one pen,if 12 pens cost d dollars

3-2 TRANSLA TING VERBAL PHRASES INTO SYMBOLS

A knowledge of arithmetic is important in algebra.Since the variables represent

numbers that are familiar to you,it will be helpful to solve each problem by first

using a simpler related problem;that is,relate similar arithmetic problems to the

given algebraic one.

EXAMPLE 1

Represent each phrase by an algebraic expression.

a .a distance that is 20 meters shorter than x meters

b.a bill for n baseball caps,each costing d dollars

c.a weight that is 40 pounds heavier than p pounds

d.an amount of money that is twice d dollars

Solution a.How to Proceed

(1)Think of a similar problem

Think of a distance that is in arithmetic:

20 meters shorter than 50 meters.(2)Write an expression for this

50 ?20arithmetic problem:

(3)Write a similar expression using

x ?20Answer

the letter x in place of 50.b.How to Proceed

(1)Think of a similar problem

Think of a bill for 6 caps,each in arithmetic:

costing 5 dollars.(2)Write an expression for this

6(5)arithmetic problem.Multiply

the number of caps by the cost

of one cap:

(3)Write a similar expression

nd Answer

using n and d :Note:After some practice,you will be able to do steps (1) and (2) mentally.

c.(p ?40) pounds or (40 ?p ) pounds

d.2d dollars

Answers a.(x ?20) meters

b.nd dollars

c.(p ?40) or (40 ?p)pounds

d.2d dollars

92Algebraic Expressions and Open Sentences

T ranslating Verbal Phrases into Symbols93 EXAMPLE 2

Brianna paid 17 dollars for batteries and film for her camera.If the batteries

cost x dollars,express the cost of the film in terms of x.

Solution If Brianna had spent 5 dollars for the batteries,the amount that was left is found by subtracting the 5 dollars from the 17 dollars,(17 ?5) dollars.This

would have been the cost of the film.If Brianna spent x dollars for the batter-

ies,then the difference,(17 ?x) dollars would have been the cost of the film.

Answer(17 ?x) dollars

Note:In general,if we know the sum of two quantities,then we can let x rep-

resent one of these quantities and (the sum ?x) represent the other.

EXERCISES

Writing About Mathematics

1.a.Represent the number of pounds of grapes you can buy with d dollars if each pound costs

b dollars.

b.Does the algebraic expression in part a always represent a whole number? Explain

your answer by showing examples using numbers.

2.a.If x apples cost c cents,represent the cost of one apple.

b.If x apples cost c cents,represent the cost of n apples.

c.Do the algebraic expressions in parts a and b always represent whole numbers?

Explain your answer.

Developing Skills

In 3–18,represent each answer in algebraic language,using the variable mentioned in the problem.

3.The number of kilometers traveled by a bus is represented by x.If a train traveled 200 kilo-

meters farther than the bus,represent the number of kilometers traveled by the train.

4.Mr.Gold invested $1,000 in stocks.If he lost d dollars when he sold the stocks,represent the

amount he received for them.

5.The cost of a mountain bike is 5 times the cost of a skateboard.If the skateboard costs x

dollars,represent the cost of the mountain bike.

6.The length of a rectangle is represented by l.If the width of the rectangle is one-half of its

length,represent its width.

7.After 12 centimeters had been cut from a piece of lumber,c centimeters were left.

Represent the length of the original piece of lumber.

94Algebraic Expressions and Open Sentences

8.Paul and Martha saved 100 dollars.If the amount saved by Paul is represented by x,repre-

sent the amount saved by Martha.

9.A ballpoint pen sells for 39 cents.Represent the cost of x pens.

10.Represent the cost of t feet of lumber that sells for g cents a foot.

11.If Hilda weighed 45 kilograms,represent her weight after she had lost x kilograms.

12.Ronald,who weighs c pounds,is d pounds overweight.Represent the number of pounds

Ronald should weigh.

13.A woman spent $250 for jeans and a ski jacket.If she spent y dollars 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 $25.Represent the amount

for which he sold it.

15.The width of a rectangle is represented by w meters.Represent the length of the rectangle if

it exceeds the width by 8 meters.

16.The width of a rectangle is x centimeters.Represent the length of the rectangle if it exceeds

twice the width by 3 centimeters.

17.If a plane travels 550 kilometers per hour,represent the distance it will travel in h hours.

18.If a car traveled for 5 hours at an average rate of r kilometers per hour,represent the dis-

tance it traveled.

19.a.Represent the total number of days in w weeks and 5 days.

b.Represent the total number of days in w weeks and d days.

Applying Skills

20.An auditorium with m rows can seat a total of c people.If each row in the auditorium has

the same number of seats,represent the number of seats in one row.

21.Represent the total number of calories in x peanuts and y potato chips if each peanut con-

tains 6 calories and each potato chip contains 14 calories.

22.The 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 minute.Represent 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 flyer.Additional

copies cost 6 cents each.Represent the cost of c copies if c is greater than 8.

24.A utility company measures gas consumption by the hundred cubic feet,CCF.The company

has a three-step rate schedule for gas customers.First,there is a minimum charge of $5.00 per month for up to 3 CCF of gas used.Then,for the next 6 CCF,the charge is $0.75 per

CCF.Finally,after 9 CCF,the charge is $0.55 per CCF.Represent the cost of g CCF of gas if

g is greater than 9.

Terms

A term is a number,a variable,or any product or quotient of numbers and vari-

ables.For example:5,x ,4y ,8ab ,,and are terms.An algebraic expression that is written as a sum or a difference has more

than one term.For example,4a ?2b ?5c has three terms.These terms,4a ,2b ,

and 5c ,are separated by ?and ?signs.

Factors of a Term

If a term contains two or more numbers or variables,then each number,each

variable,and each product of numbers and variables is called a factor of the

term,or factor of the product.For example,the factors of 3xy are 1,3,x ,y ,3x ,

3y ,xy ,and 3xy .When we factor whole numbers,we write only factors that are

integers.

Any factor of an algebraic term is called the coefficient of the remaining

factor,or product of factors,of that term.For example,consider the algebraic

term 3xy :

3 is the coefficient of xy

3x is the coefficient of y 3y is the coefficient of x xy is the coefficient of 3

When an algebraic term consists of a number and one or more variables,the

number is called the numerical coefficient of the term.For example:

In 8y ,the numerical coefficient is 8.

In 4abc ,the numerical coefficient is 4.

When the word coefficient is used alone,it usually means a numerical coef-

ficient.Also,since x names the same term as 1x ,the coefficient of x is under-

stood to be 1.This is true of all terms that contain only variables.For example:

7 is the coefficient of b in the

1 is the coefficient of b in the term b .term 7b .

2.25 is the coefficient of gt in

1 is the coefficient of gt in the term gt .

the term 2.25gt .Bases,Exponents,and Powers

You learned in Chapter 2 that a power is the product of equal factors.A power

has a base and an exponent.

The base is one of the equal factors of the power.5c k 253-3 ALGEBRAIC TERMS AND VOCABULARY

Algebraic T erms and Vocabulary 95

The exponent is the number of times the base is used as a factor.(If a term

is written without an exponent,the exponent is understood to be 1.)

42?4(4):base ?4exponent ?2power ?42?16x 3= x (x )(x ):

base ?x exponent ?3power ?x 335m :base ?m exponent ?1power ?m 1or m 5d 2= 5(d )(d ):base ?d exponent ?2power ?d 2

An exponent refers only to the number or variable that is directly to its left,as

seen in the last example,where 2 refers only to the base d .To show the product

5d as a base (or to show any sum,difference,product,or quotient as a base),we

must enclose the base in parentheses.

(5d )2?(5d )(5d ):base ?5d exponent ?2power ?(5d )2

(a + 4)2?(a + 4)( a + 4):base ?(a + 4)exponent ?2power ?(a ?4)2

Note that –d 4is not the same as (?d )4.

–d 4?–1(d )(d )(d )(d ) is always a negative number.(?d )4?1(?d )(?d )(?d )(?d ) is always a positive number since the exponent is even.

EXAMPLE 1

For each term,name the coefficient,base,and exponent.

Answers

a.4x 5

coefficient ?4base ?x exponent ?5b.?w 8

coefficient ?–1base ?w exponent ?8c.2p r coefficient ?2p base ?r exponent ?1

Note:Remember that coefficient means numerical coefficient ,and that 2p

is a real number.

Writing About Mathematics

1.Does squaring distribute over multiplication,that is,does (ab )2= (a 2)(b 2)? Write (ab )2as (ab )(ab ) and use the associative and commutative properties of multiplication to justify your answer.

2.Does squaring distribute over addition,that is,does (a ?b )2?a 2?b 2? Substitute values for a and b to justify your answer.EXERCISES

96Algebraic Expressions and Open Sentences

Developing Skills

In 3–6,name the factors (other than 1) of each product.

3.xy

4.3a

5.7mn

6.1st

In 7–14,name,in each case,the numerical coefficient of x .

7.8x

8.(5 ?2)x 9.10.x 11.?1.4x 12.2 ?7x 13.3.4x 14.?x

In 15–22,name,in each case,the base and exponent of the power.

15.m 2

16.?s 317.t 18.(?a )419.10620.(5y )421.(x ?y )522.12c 3

In 23–29,write each expression,using exponents.

23.b ?b ?b ?b ?b

24.p ?r ?r 25.a ?a ?a ?a ?b ?b 26.7 ?r ?r ?r ?s ?s 27.(6a )(6a )(6a )28.(a ?b )(a ?b )(a ?b )

29.the fourth power of (m +2n )

In 30–33,write each term as a product without using exponents.

30.r 631.5x 432.4a 4b 233.(3y )5

In 34–41,name,for each given term,the coefficient,base,and exponent.

34.?3k

35.?k 3

36.p r 237.(ax )538.39.0.0004t 1240.41.(?b )3

Applying Skills

42.If x represents the cost of a can of soda,what could 5x represent?

43.If r represents the speed of a car in miles per hour,what could 3r represent?

44.If n represents the number of CDs that Alice has,what could n ?5 represent?

45.If d represents the number of days until the end of the year,what could represent?

46.If s represents the length of a side of a square,what could 4s represent?

47.If r represents the measure of the radius of a circle,what could 2r represent?

48.If w represents the number of weeks in a school year,what could represent?

49.If d represents the cost of one dozen bottles of water,what could represent?

50.If q represents the point value of one field goal,what could 7q represent?d 12w 4d 732a 4"2y 12x Algebraic T erms and Vocabulary 97

In Section 1 of this chapter,we listed the words that can be represented by each

of the four basic operations.We can use these same lists to write algebraic

expressions in words and to write problems that can be represented by a given

algebraic expression.

For an algebraic expression such as 2n ?3,n could be any real number.That

is,associated with any real number n ,there is exactly one real number that is the

value of 2n ?3.However,if n and 2n ?3 represent the number of cans of tuna

that two customers buy,then n must be a whole number greater than or equal

to 2 in order for both n and 2n ?3 to be whole numbers.

For this situation,the domain or replacement set would be the set of whole

numbers.

EXAMPLE 1

If n represents the number of points that Hradish scored in a basketball game

and 2n ?3 represents the number of points that his friend Brad scored,describe

in words the number of points that Brad scored.What is a possible domain for

the variable n ?

Solution The number of points scored is always a whole number.In order that 2n ?3

be a whole number,n must be at least 2.

Answer The number of points that Brad scored is 3 less than twice the number that

Hradish scored.A possible domain for n is the set of whole numbers

greater than or equal to 2.

EXAMPLE 2

Molly earned d dollars in July and dollars in August.Describe in words

the number of dollars that Molly earned in August.

Answer In August,Molly earned 10 more than half the number of dollars that she

earned in July.

EXAMPLE 3

Describe a situation in which x and 12 ?x can be used to represent variable

quantities.List the domain or replacement set for the answer.

Solution If x eggs are used from a full dozen of eggs,there will be 12 ?x eggs left.

Answer The domain or replacement set is the set of whole numbers less than or equal to 12.12d 1 103-4 WRITING ALGEBRAIC EXPRESSIONS IN WORDS

98Algebraic Expressions and Open Sentences

Another Solution The distance from my home to school is 12 miles.On my way to school,after I

have traveled x miles,I have 12 ?x miles left to travel.

Answer The domain or replacement set is the set of non-negative real numbers that

are less than or equal to 12.

Many other answers are possible.

Writing About Mathematics

1.a.If 4 ?n represents the number of books Ken read in September and 4 ?n represents

the number of books he read in October,how many books did he read in these two

months?

b.What is the domain of the variable n ?

2.Pedro said that the replacement set for the amount that we pay for any item is the set of rational numbers of the form 0.01x where x is a whole number.Do you agree with Pedro?Explain why or why not.

Developing Skills

In 3–14:a.Write in words each of the given algebraic expressions.b.Describe a possible domain for each variable.

3.By one route,the distance that Ian walks to school is d miles.By a different route,the dis-tance is d ?0.2 miles.

4.Juan pays n cents for a can of soda at the grocery store.When he buys soda from a machine,he pays n ?15 cents.

5.Yesterday Alexander spent a minutes on leisurely reading and 3a ?10 minutes doing homework.

6.The width of a rectangle is w meters and the length is 2w ?8 meters.

7.During a school day,Abby spends h hours in class,hours at lunch and hours on sports.

8.Jen spends d hours at work and hours driving to and from work.

9.Alicia’s score for 18 holes of golf was g and her son’s score was 10 ?g .

10.Tom paid d cents for a notebook and 5d ?30 cents for a pen.

11.Seema’s essay for English class had w words and Dominic’s had words.

12.Virginia read r books last month and Anna read 3r ?5 books.

13.Mario and Pete are playing a card game where it is possible to have a negative score.Pete’s

score is s and Mario’s score is s ?220.34w 1 80d 12h 3h 6EXERCISES

Writing Algebraic Expressions in Words 99

100Algebraic Expressions and Open Sentences

14.In the past month,Agatha has increased the time that she walks each day from m minutes

to 3m?10 minutes.

3-5 EVALUA TING ALGEBRAIC EXPRESSIONS

Benjamin has 1 more tape than 3 times the number of tapes that Julia has.If

Julia has n tapes,then Benjamin has 3n?1 tapes.

The algebraic expression 3n?1 represents an unspecified number.Only when the variable n is replaced by a specific number does 3n?1 become a spe-

cific number.For example:

If n ?10,then 3n?1 ?3(10) ?1 ?30 ?1 ?31.

If n?15,then 3n?1 ?3(15) ?1 ?45 ?1 ?46.

Since in this example,n represents the number of tapes that Julia has,only

whole numbers are reasonable replacements for n.Therefore,the replacement

set is the set of whole numbers or some subset of the set of whole numbers.

When we substitute specific values for the variables in an algebraic expres-sion and then determine the value of the resulting expression,we are evaluating

the algebraic expression.

When we determine the number that an algebraic expression represents for specific values of its variables,we are evaluating the algebraic expression.

EXAMPLE 1

Evaluate 50 ?3x when x?7.

Solution How to Proceed

(1)Write the expression:50 ?3x

(2)Replace the variable by its given value:

50 ?3(7)(3)Multiply:

50 ?21(4)Subtract:29

Answer 29

EXAMPLE 2

Evaluate 2x 2?5x ?4 when:a.x = ?7 b.x = 1.2

Solution How to Proceed

a.(1) Write the expression:

2x 2?5x ?4(2)Replace the variable by the value ?7:

2(?7)2?5(?7) ?4(3)Evaluate the power:

2(49) ?5(?7) ?4(4)Multiply:

98 ?35?4(5)Add:

137b.(1) Write the expression:

2x 2?5x ?4(2) Replace the variable by the value 1.2:

2(1.2)2?5(1.2) ?4(3) Evaluate the power:

2(1.44) ?5(1.2) ?4(4) Multiply:

2.88?6?4(5) Add and subtract:0.88

Answers a.137

b.0.88

EXAMPLE 3Evaluate when a ??4,n ?10,and d ?3.Solution How 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:Answer

25252135 1 2655

2135 1 27285 1(9)(3)2(24)5 1 (10 – 1)(3)2a 5 1 (n 2 1)d 2a 5 1 (n 2 1)d

Evaluating Algebraic Expressions 101

The values given for the variables can be stored in the calculator. ENTER:

4

10

3

DISPLAY:

ENTER:2 5 1

DISPLAY:

Answer

EXAMPLE 4

Evaluate (2x)3?2x3when x? ?0.40.

Solution How to Proceed

(1)Write the expression:(2x)3?2x3

(2)Replace the variable by its given value:[2(?0.40)]3?2(?0.40)3

(3)Simplify the expression within brackets:[?0.80]3?2(?0.40)3

(4)Evaluate the powers:?0.512?2(?0.064)

(5)Multiply:?0.512 ?0.128

(6)Subtract:?0.384

Answer?

0.384

25255

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ALPHA

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ALPHA

ENTER

ALPHA

STO?

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Solution

102Algebraic Expressions and Open Sentences

Writing About Mathematics

1.Explain why,in an algebraic expression such as 12ab ,12 is called a constant and a and b are called variables?

2.Explain why,in step 2 of Example 1,parentheses were needed when x was replaced by its value.Developing Skills

To understand this topic,you should first evaluate the expressions in Exercises 3 to 27 without a cal-culator.Then,store the values of the variables in the calculator and enter the given algebraic expres-sions to check your work.

In 3–27,find the numerical value of each https://www.doczj.com/doc/533043887.html,e a ?8,b ??6,d ?3,x ??4,and y ?0.5.

3.5a

4. 5.0.3y 6.a ?37.b ?2

8.ax 29.10.5x ?2y 11.7xy 3

12.ab ?dx 13.14.0.2d ?0.3b 15.16.(3y )217.18.a 2?3d 2

19.(ay )3

20.x (y ?2)21.4(2x ?3y )22.23.3y ?(x ?d )

24.2(x ?y ) ?525.(x ?d )5

26.(2a ?5d )227.(2a )2?(5d )2Applying Skills

28.At one car rental agency,the cost of a car for one day can be determined by using the alge-

braic expression $32.00 ?$0.10m where m represents the number of miles driven.

Determine the cost of rental for each of the following:

a.Mike Baier drove the car he rented for 35 miles.

b.Dana Morse drove the car he rented for 435 miles.

c.Jim Szalach drove the car he rented for 102 miles.

29.The local pottery co-op charges $40.00 a year for membership and $0.75 per pound for fir-

ing pottery pieces made by the members.The algebraic expression 40 ?0.75p represents the yearly cost to a member who brings p pounds of pottery to be fired.Determine the

yearly cost for each of the following:

a.Tiffany is an amateur potter who fired 35 pounds of work this year.

b.Nia sells her pottery in a local craft shop and fired 485 pounds of work this year.

30.If a stone is thrown down into a deep gully with an initial velocity of 30 feet per second,the

distance it has fallen,in feet,after t seconds can be found by using the algebraic expression 16t 2?30t .Find the distance the stone has fallen:

a.after 1 second.

b.after 2 seconds.

c.after 3 seconds.12x (y 1 0.1)214x 2y 34x 325a 1 15b 3bd 912x EXERCISES

Evaluating Algebraic Expressions 103

104Algebraic Expressions and Open Sentences

31.The Parkside Bread Company sells cookies and scones as well as bread.Bread (b) costs

$4.50 a loaf,cookies (c) cost $1.10 each,and scones (s) cost $1.50 each.The cost of a bakery order can be represented by 4.50b?1.10c?1.50s.Determine the cost of each of the fol-lowing orders:

a.six cookies and two scones

b.three loaves of bread and one cookie

c.one loaf of bread,a dozen cookies,and a half-dozen scones

32.A Green Thumb volunteer can plant shrubbery at a rate of 6 shrubs per hour and a

Friendly Garden volunteer can plant shrubbery at a rate of 8 shrubs per hour.The total

number of shrubs that g Green Thumb volunteers and f Friendly Garden volunteers can

plant in h hours is given by the algebraic expression 6gh?8fh.Determine the number of shrubs planted:

a.in 3 hours by 2 Green Thumb and 1 Friendly Garden volunteers.

b.in 2 hours by 4 Green Thumb and 4 Friendly Garden volunteers.

3-6 OPEN SENTENCES AND SOLUTION SETS

In this chapter,you learned how to translate words into algebraic expressions.

The value of an algebraic expression depends on the value of the variables.

When the values of the variables change,the value of the algebraic expression

changes.For example,x?6 is an algebraic expression.The value of x?6

depends on the value of x.

If one value is assigned to an algebraic expression,an algebraic sentence is formed.These sentences may be formulas,equations,or inequalities.For exam-

ple,when the value 9 is assigned to the algebraic expression x?6,we can write

the sentence “Six more than x is 9.”This sentence can be written in symbols as

x?6 ?9.

Every sentence that contains a variable is called an open sentence.

x?6 ?93y?122n?0x?5 ?8

An open sentence is neither true nor false.

The sentence will be true or false only when the variables are replaced by numbers 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 elements of the solut ion set of the open sentence.A solution set,as seen below,can

contain one or more numbers or,at times,no numbers at all,from the replace-

ment set.

EXAMPLE 1

Using the domain {0,1,2,3},find the solution set of each open sentence:

a.x?6 ?9

b.2n?0

Solution a.Procedure:Replace x in the open

sentence with numbers from the

domain {0,1,2,3}.

x + 6 ?9

Let x ?0.

Then 0 ?6 ?9 is false.

Let x ?1.

Then l ?6 ?9 is false.

Let x ?2.

Then 2 ?6 ?9 is false.

Let x ?3.

Then 3 ?6 ?9 is true.

Here,only when x ?3 does the

open sentence become a true

sentence.Answer:a.Solution set ?{3}.EXAMPLE 2

Find the solution set for the open sentence 3y ?12 using:

a.the domain ?{3,5,7}

b.the domain ?{whole numbers}

Solution a.Procedure:Replace y with 3,5,

and 7.

If y ?3,then 3(3) ?12 is false.

If y ?5,then 3(5) ?12 is false.

If y ?7,then 3(7) ?12 is false.

When y is replaced by each of

the numbers from the domain,no

true statement is found.The

solution set for this domain is the

empty set or the null set,written

in symbols as {} or as .Answer:a.The solution set is {} or .

ллOpen Sentences and Solution Sets 105

b.Procedure:Replace n in the open sentence with numbers from the domain {0,1,2,3}.2n ?0Let n ?0.Then 2(0) ?0 or 0 ?0 is false.Let n ?1.Then 2(l) ?0 or 2 ?0 is true.Let n ?2.Then 2(2) ?0 or 4 ?0 is true.Let n ?3.Then 2(3) ?0 or 6 ?0 is true.Here,three elements of the domain make the open sentence true.

Answer:b.Solution set ?{1,2,3}

b.Procedure:Of course,you cannot replace y with every whole number,but you can use multiplication facts learned previously.You know that 3(4) ?12.Let y ?4.Then 3(4) ?12 is true.No other whole number would make the open sentence 3y ?12 a true sentence.Answer:b.The solution set is {4}.

Writing About Mathematics

1.For the open sentence x ?7 ?12,write a domain for which the solution set is the empty set.

2.For the open sentence x ?7 ?12,write a domain for which the solution set has only one element.

3.For the open sentence x ?7 ?12,write a domain for which the solution set is an infinite set.

Developing Skills

In 4–11,tell whether each is an open sentence,a true sentence,or an algebraic expression.

4.2 ?3 ?5?0

5.x ?10 ?14

6.y ?4

7.3 ?7 ?2(5)

8.n ?7

9.3 ?2 ?1010.3r ?211.2x ?7 ?15In 12–15,name the variable in each open sentence.

12.x ?5 ?913.4y ?2014.r ?6 ?215.7 ?3 ?a

In 16–23,using the domain {?5,?4,?3,?2,?1,0,1,2,3,4,5},find the solution set for each open sentence.

16.n ?3 ?7

17.x ?x ?018.5 ?n ?219.n ?3 ?920.2n ?1 ?821.22.23.2x ??4

Applying Skills

24.Pencils sell for $0.19 each.Torry wants to buy at least one but not more than 10 pencils and

has $1.50 in his pocket.

https://www.doczj.com/doc/533043887.html,e the number of pencils that Torry wants to buy to write a domain for this problem.

b.The number of pencils that Torry might buy,x ,can be found using the open sentence

0.19x ?1.50.Find the solution set of this open sentence using the domain from part a .

c.How many pencils can Torry buy?

25.The local grocery store has frozen orange juice on sale for $0.99 a can but limits the number

of cans that a customer may buy at the sale price to no more than 5.

a.The domain for this problem is the number of cans of juice that a customer may buy at the sale price.Write the domain.

b.If Mrs.Dajhon does not want to spend more than $10,the number of cans that she

might buy at the sale price,y ,is given by the equation 0.99y ?10.Find the solution set

of this equation using the domain from part a .

c.How many cans can Mrs.Dajhon buy if she does not want to spend more than $10?3x 2,x 2n 1 1354EXERCISES

106Algebraic Expressions and Open Sentences

26.Admission to a recreation park is $17.50.This includes all rides except for a ride called The

Bronco that costs $1.50 for each ride.Ian has $25 to spend.

a.Find the domain for this problem,the number of times a person might ride The

Bronco .

b.The number of times Ian might ride The Bronco ,z ,can be found using the open sen-tence 17.50 ?1.50z ?25.Find the solution set of this open sentence using the domain from part a .

c.How many times can Ian ride The Bronco ?

A formula uses mathematical language to express the relationship between two

or more variables.Some formulas are found by the strategy of looking for pat-

terns.For example,how many square units are shown in the rectangle on the

left? This rectangle,measuring 4 units in length and 3 units in width,contains a

total of 12 square units of area.

Many such examples led to the conclusion that the area of a rectangle is

equal to the product of its length and width.This relationship is expressed by the

formula A ?lw where A ,l ,and w are variables that represent,respectively,the

area,the length,and the width of a rectangle.

A formula is an open sentence that states that two algebraic expressions are

equal.In formulas,the word is is translated into the symbol ?.

EXAMPLE 1

Write a formula for each of the following relationships.

a.The perimeter P of a square is equal to 4 times the length of one side.

b.The total cost C of an article is equal to its price p plus an 8% tax on the

price.

c.The sum S of the measures of the interior angles of an n -sided polygon is

180 times 2 less than the number of sides.

Solution a.Let s represent the length of each side of a square.

P ?4s Answer

b.8% (or 8 percent) means 8 hundredths,written as 0.08 or .

C ?p ?0.08p or Answer

c.“2 less than the number of sides”means (n ?2).

S ?180(n ?2) Answer

C 5p 1 8100p 81003-7 WRITING FORMULAS

Writing Formulas

107

第三章 全球化管理

第三章全球化管理 习题 一、填充题 1.如果一个企业的资源转化活动超越了一国国界,即进行商品、劳务、资本、技术等形式的经济资源的跨国传递和转化,那么这个企业就是在开展__________。P48 1.国际化经营2.跨国界的经营活动主要包括______、______、______、______方和______方式。P48 2.国际贸易,特许,劳务输出,国际间接投资,国际直接投资 3.从历史的发展来看,在国际化经营的初始发展阶段,活动方式主要以______为主。 P48 3.进出口贸易 4.第二次世界大战以后到20世纪80年代,随着______、______的不断深化以及__________的不断扩大,各国企业纷纷以主动的姿态开展国际化经营。4.科技的迅猛发展,国际分工,世界市场 5.在国际化经营的高速发展阶段______,逐渐成为世界经济的核心组织者和最主要的经济活动主体。5.跨国公司 6.在国际化经营的高速发展阶段____,成为企业国际化经营的主导方式。6.对外直接投资7.从企业的发展来看,国际化经营是一个从______到______的过程。7.被动,主动 8.从企业发展的角度来看,其国际化过程可分为______、______、______。P48 8.出口阶段,非股权安排阶段,直接投资阶段 9.______通常是一个企业走向世界的最初方式,也是企业尝试国际化经营的初始阶段。 P48 9.出口 11.在非股权安排阶段,_____、_____、____等非股权安排成为企业国际化经营的新方式。 P49 11.特许经营,合同制造,管理合同 13.在直接投资阶段,企业的国际化经营活动不再仅仅以一般_____为目标,它还希望通过生产的国际化,实现全球范围内的______,并最终实现生产利益和商业利益等综合利益的最大化。13.商业利益,资源优化组合 15.国际经营所面对的环境的____和___往往随着国际化经营的地理范围和目标市场的扩大而愈趋扩大。15.多元性,复杂性 16.跨国公司多元化经营很重要的一个原因在于,多元化的国际经营可以______。16.降低国别风险 17.国际企业管理的一体化的表现之一在于,通过____来保证公司全球战略的实现。 17.分级计划管理 20.____优势是国际化经营的前提。20.所有权 21.跨国公司对外直接投资的动机虽然有着很大的差异,但基本上都是以追求______为基本前提的。21.经济利益 22.一个国家的______、有无______、市场对外来产品的____,都直接关系着海外经营机会的大小。P41 22.市场大小,市场潜力,准入程度 23.文化因素与政治因素不同,政治因素一般带有____,而文化因素则带有______。 P41 23.强制性,习惯性 24.霍夫斯泰德把文化因素分为____、______、_________、____和____五个维度。 P41 24.权力差距,不确定性的避免,个人主义或集体主义,男性化或女性化,长期导向和短期导向 25.______是出口进入方式面临的最大问题,____是出口方式的另外一个缺点。P49 25.非生产性成本支出较大,控制程度低

3--钻井风险识别和评估

第三章钻井作业风险识别和评估 钻井是高风险的行业,在钻井作业的整个活动中,都可能潜在对健康、安全与环境危害的影响因素。识别钻井作业中潜在的HSE风险与危害影响因素,是有效控制和削减钻井过程中给健康、安全与环境带来的危害及影响的重要基础。 第一节钻井作业中HSE危害和影响的确定 一.钻井作业中HSE风险识别的特征 由于钻井作业的特殊性,在识别钻井活动过程中存在的对健康、安全与环境的危害时应掌握有以下主要特征: (1)差异性根据钻井工艺的特点,钻井作业大致分为钻前,钻井和完井施工活动几个阶段。不同施工阶段以及采用不同的钻井工艺对健康、安全与环境的影响不同,存在的危害和风险因素不同。此外,因钻井作业场所的流动性,不同地域(如海上和陆地钻井)的环境、气候条件不同,其危害和风险影响因素也不尽相同。 (2)严重性因人为操作或工艺措施不当以及设备处于不安全运行状态等诸多因素所导致的事故造成的危害极大,如井控失效可能造成井毁人亡的恶性事故,产生的后果甚至是灾难性的。 (3)多样性钻井活动中不仅存在常规的着火、爆炸、电击、运输事故、有害材料化学试剂、工作环境(如滑倒、堕落、噪声、振动)等对健康、安全与环境的危害因素外,还存在设备伤害(如水压和气压、旋转机械)、污水和钻井泥浆以及硫化氢等对健康、安全与环境的影响,其危害是多种多样的。 (4)时间性钻井活动中造成的对健康、安全与环境的危害有的是突发性的,影响时间较短暂,而有的影响时间较长(如噪声危害贯穿整个钻井活动过程中),而有的影响则可能是永久性的(如钻井中井漏造成的对地下水源的危害)。 (5)隐蔽性钻井安全事故的发生受人为因素、设备状况因素、施工作业措施因素以及外界等因素影响,并且存在诸多不确定因素的影响,有较强的隐蔽性。其危害和影响的发生及程度有时难以预料。 (6)变化性钻井作业中的风险具有多变性,往往会因措施或处理不当,可能会由一般事故升级为严重事故甚至恶性事故。如钻井过程中发生井漏,若同时存在高压层,处理井漏措施不当,就可能因井漏液柱压力降低而发生井喷或井喷失控事故,从而由一般事故演变成严重或恶性事故。 二.钻井作业HSE风险因素识别方法 根据钻井作业地区环境调查结果和钻井作业活动中易发生事故环节以及日常管理经验,从人的行为、物理状态、环境因素等方面进行分析,对钻井作业项目的全过程进行风险因素识别。可采用危险点源分级挂牌、危害程度分级挂图、环境监测、关联图等定性方法和定量方法进行风险识别。 钻井作业中HSE风险识别,通常可采用关联图分析法,它是通过一种假设方法用图表示危害如何产生及如何导致一系列后果的危险分析法,如图3-1所示,将顶级事件(指不希望发生的事故,如井喷、高空坠落等)用圆圈表示,并置于关联图中心。

犹太民族是世界上最优秀的民族

犹太民族是世界上最优秀的民族,在许多领域犹太人都取得了令人瞩目的成就,为人类当今世界做出了不可磨灭的贡献。 一、财富积累方面,犹太人中的资本巨头,掌握经济命脉。 犹太人最让世界折服的,是他们惊人的财富和超人的赚钱能力。他们特别会经商,也特别会理财,是名副其实的世界商人。由于犹太人尖钻精明地聚敛财富,成了世界上最富有的团体。有人说得好:全世界的金钱装在美国人的口袋里,而美国人的金钱却装在犹太人的脑袋里。据统计,犹太人占世界总人口约0.3%,却掌握着世界经济命脉。在全世界最富有的企业家中,犹太人占50%以上。美国百万富翁中犹太人占三分之一,福布斯美国富豪榜前40名中有18名是犹太人。无论是过去和现在,在知名的经济巨头中犹太人占有绝对的比例,如第一个亿万巨富、石油大王洛克菲勒,有史以来最大的投资家、“美国股神”巴菲特,“金融资本的颠峰力量代表”、华尔街的缔造者摩根,被称为“美元总统”的、美国联邦储备委员会主席格林斯潘,狙击英镑成功一举成名、曾掀起亚洲金融风暴的超实力金融大鳄索罗斯,“金融超市之父”、花旗集团董事长威尔,“打开个人计算机直销大门”的、戴尔公司创办人戴尔,坐在全球软件头把交椅上、甲骨文公司的创办人艾利森,还有红色资本家哈默,《纽约时报》董事长奥克斯,华纳电影公司创办人华纳,娱乐大鳄迪斯尼,钻石大王彼德森,电影世界的领头羊斯皮尔伯格,他们都是犹太人。其中巴菲特个人财富达430亿美元,但他已拿出85%捐赠与慈善事业;现在最富的要算微软公司的总裁兼首席执行官巴尔默,资产已达250亿美元。在欧洲也是一样,犹太人生财、理财、发财,控制了经济命脉长达200年,如罗斯柴尔德家族至今仍然控制世界黄金市场。这些会赚钱的犹太人为资本主义经济发展和繁荣,作出了巨大的贡献。 二、思想研究方面,犹太人中的学术大师,产生了划时代影响。 在近一百年的人类历史上,犹太人中产生了一大批学术大师,如马克思、爱因斯坦、弗洛伊德、达尔文等,他们对人类的发展作出了划时代的贡献。马克思科学社会主义,爱因斯坦的相对论,弗洛伊德的精神分析学,达尔文的生物进化论,这些理论体系分别表征了人类在自然科学、社会科学、心理学、生物学领域内的思想最高峰。曾经有人写了一本书《影响世界历史的三个犹太人》,书中把马克思、爱因斯坦、弗洛伊德称为上一个千年最伟大的三个天才。马克思曾在英国也被评为“千年思想家”,他和恩格斯一同创立的马克思主义理论是人类历史上划时代的伟大成果,产生了巨大的影响。马克思主义哲学、政治经济学和科学社会主义,代表着19 世纪人类知识的精华;他所提出的价值观在当

美国波士顿 犹太人大屠杀纪念碑 碑文

在美国波士顿犹太人大屠杀纪念碑上, In Germany, they first came for the Communists, and I didn't speak up because I wasn't a Communist. Then they came for the Jews, and I didn't speak up because I wasn't a Jew. Then they came for the trade unionists, and I didn't speak up because I wasn't a trade unionist. Then they came for the Catholics, and I didn't speak up because I was a Protestant. Then they came for me, but by that time,there was no one left to speak up. by Rev. Martin Niemoller, 1945 在德国, 起初他们追杀共产主义者,我没有说话———因为我不是共产主义者; 接着他们追杀犹太人,我没有说话———因为我不是犹太人; 后来他们追杀工会成员,我没有说话———因为我不是工会成员; 此后他们追杀天主教徒,我没有说话———因为我是新教教徒; 最后他们奔我而来,却再也没有人站出来为我说话了。 马丁尼莫拉 这首“短诗”的作者是一位牧师,他的名字叫马丁·尼莫拉。这位牧师是二战前德国的宗教领袖,他曾为战后德国社会道德重建立下了不朽功勋。当年,尼莫拉因反对希特勒的犹太政策和对德国教会的控制,被希特勒亲自下令送进集中营,最后在各方正义力量的协助下才逃过纳粹的迫害。这首短诗是他在晚年忏悔自己当年的道德污点时写下的。后来人们把它当作“碑文”刻在美国波士顿犹太人大屠杀纪念碑上。 这段“碑文”确实寓意深刻,它精辟地阐述了人世间正义与邪恶、自我与社会、助人与被助的辩证统一关系。在当今社会中,一个人、一个群体、一个政党乃至一个国家,如果只顾自己的利益,而不为社会正义而付出哪怕一点点代价,那么最后吃亏和受到惩罚的还将是他自己。

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德国新教神父马丁·尼莫拉的忏悔(美国波士顿犹太人屠杀纪念碑碑文) ——德文版 Als die Nazis die Kommunisten holten, habe ich geschwiegen; ich war ja kein Kommunist. Als sie die Sozialdemokraten einsperrten, habe ich geschwiegen; ich war ja kein Sozialdemokrat. Als sie die Gewerkschafter holten, habe ich nicht protestiert; ich war ja kein Gewerkschafter. Als sie die Juden holten, habe ich nicht protestiert; ich war ja kein Jude. Als sie mich holten, gab es keinen mehr, der protestieren konnte. ——英文版

When the Nazis came for the communists, I remained silent; I was not a communist. When they locked up the social democrats, I remained silent; I was not a social democrat. When they came for the trade unionists, I did not speak out; I was not a trade unionist. When they came for the Jews, I did not speak out; I was not a Jew. When they came for me, there was no one left to speak out. ——中文版 在德国, 起初他们追杀共产主义者,

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各种数据表明,全世界最有钱的企业家中犹太人占近一半,全美200名最有影响的名人中,犹太人占一半;福布斯美国富豪榜前40名中有18名是犹太人,美国名牌大学中五分之一的教授是犹太人(有数据表称三分之一);全美律师中,犹太人占四分之一(正在为前国际货币基金组织总裁卡恩辩护的律师之一本杰明?布拉夫曼也是犹太人);诺贝尔奖犹太人占一半。今日美国其他著名的犹太人还包括:格林斯潘、索罗斯、洛克菲勒、尤伯罗斯、迪斯尼、华纳兄弟、Polo 时装拉夫劳伦、李维斯牛仔李维?施特劳斯、星巴克咖啡霍华德?斯卡兹、谷歌谢尔盖? 布林、戴尔电脑迈克尔?戴尔、ORACLE 软件拉里埃里森、甲骨文公司埃尔森、英特尔公司创始人格鲁夫、赫莲娜、雅诗兰黛、露华浓、麦当劳连锁、肯德基连锁、哈根达斯冰激凌……也难怪美国前总统罗斯福曾感叹“影响美国经济的只有二百多家企业,而操纵这些企业的只有六七个犹太人。”没有了他们,美国还算是美国吗? 不仅是金融领域,就是电影和传媒也都是犹太人的天下。美国著名的电影 巨头:米高梅、派拉蒙、华纳兄弟、环球等等都是犹太人的禁脔。可谓即赚了钱,又遏制了反犹主义。

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