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2.1 Introduction to Variables 43 43 Chapter 2 Understanding Variables and Solving Equations

International Management Managing Across Borders and Cultures Text and Cases

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CHAPTER 2 UNDERSTANDING VARIABLES AND SOLVING EQUATIONS

2.1 Introduction to Variables

2.1 Margin Exercises

1.

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Value of

Value of

Expression

is

is

is

is

(b)

6. (a) Multiplying any number by gives a product of .

Any

The expression is .

number times zero

The variable is . It represents the class limit.

The constant is .

•

2. (a) Evaluate the expression when is . (b) Changing the grouping of addends , , does not change the sum.

Order books.

Replace c with 25. 7. (a)

can be written as • • • •

is used as a factor times.

(b) Evaluate the expression when is . (b) can be written as • • • •

Replace c with 60. (c)

(d)

8. (a)

can be written as • • • •

can be written as • • • • • •

means

Order books.

3. (a) Evaluate the expression when is feet.

• •

• •

Replace y with 5.

Multiply left to right.

• feet

feet

Replace s with 3 feet. •

The perimeter of the square table is feet. (b) means

(b) Evaluate the expression when is miles. • • • Replace r with 6 and s with 3.

Replace s with 7 miles. • • • Multiply left to right.

• miles

miles

The perimeter of the square park is miles.

• •

•

4. Evaluate the expression

when is .

(c)

means Replace x with 4

Replace a with 40.

Divide.

• • •

• • •

• •

and y with 3.



Add.

•

The approximate systolic blood pressure is .

5. (a) Evaluate the expression when is and

(d)

means

is .

Replace with and with .

Divide.

• • • •

• • • •

• • •

• •

•

Replace c with 2.


Your average score is .


2.1 Section Exercises 14. Expression (rule) for degrees:

1. is the variable; (a) is degrees.

2.

is the constant.

is the variable;

is the constant.

(b) is degrees.

(c) is degrees.

15. Expression (rule) for finding perimeter of an

3. is the variable; equilateral triangle of side length :

is the constant.

4. is the variable;

(a) Evaluate the expression when , the side

length, is inches.

is the constant.

5. is the variable;

•

Replace s with 11.

Follow the rule and multiply.

is the coefficient.

6. is the variable;

is the coefficient.

inches is the perimeter.

(b) Evaluate the expression when , the side

length, is feet.

7. is the variable;

is the coefficient.

•

Replace s with 3.


is the constant.

8. is the variable;

is the coefficient.

is the constant.

feet is the perimeter.

16. Expression (rule) for perimeter:

(a) • meters is meters.

(b) • inches is inches.

9. Both and are variables.

17. Expression (rule) for ordering brushes:

10. Both and are variables.

(a) Evaluate the expression when , the class size,

11.

12.

is the variable;

is the coefficient;

is the constant.

is the variable;

is .

•

Replace c with 12.

Multiply before subtracting.

is the coefficient;

is the constant.

13. Expression (rule) for ordering robes:

brushes must be ordered.

(b) Evaluate the expression when , the class size,

is .

(a) Evaluate the expression when there are

graduates.

•

Replace c with 16.

Multiply before subtracting.

Replace g with 654.

Follow the rule and add.

brushes must be ordered.

(b)

robes must be ordered.

Evaluate the expression when there are

18. Expression (rule) for ordering doughnuts:

(a) • is doughnuts must be ordered.


graduates.

Replace g with 208.


(b) • is doughnuts must be ordered.

19. Expression (rule) for average test score, where is

the total points and is the number of tests:

(c)

robes must be ordered.

Evaluate the expression when there are

(a) Evaluate the expression when , the total

points, is and , the number of tests, is .

graduates.

Replace g with 95.


Replace p with 332 and t with 4.

Follow the rule and divide.

robes must be ordered. points is the average test score.


Value

of

Expression

Expression

is

• is

is

• is

is

• is

Value

of

Expression

Expression

is is

, or

is is

, or

is is

, or

Value

of

Value

of

Expression

is , or

is , or

is

, or

Value

of

Value

of

Expression

• • is

• • is

• • is

(b) Evaluate the expression when , the total 27. Multiplying a number by leaves the number

points, is and , the number of tests, is . unchanged. Let represent "a number."

Replace p with 637 and t with 7.

•

or •

Follow the rule and divide. 28. Adding to any number leaves the number

unchanged. Let represent "any number."

points is the average test score. or

20. Expression (rule) for buses:

29. Any number divided by is undefined. Let

represent "any number."(a) is buses.

(b) is buses. is undefined or

is undefined.

21. 30. Multiplication distributes over addition. Let , ,

and represent variables.

• •

22.

31.

32.

33.

34.

written without exponents is

• • • • •


• • • • • •


• • • • • •


• • • • • •

23.

24.

25. A variable is a letter that represents the part of a

rule that varies or changes depending on the

situation. An expression expresses, or tells, the

rule for doing something. For example, is an

expression, and is the variable.

26. The number part in a multiplication expression is


35. can be written as • • • • .

The exponent applies only to the base .

36. can be written as • • • .


37. can be written as • • • .


38. can be written as • • • • • .


39. can be written as • • • • • • .


40. can be written as • • • • .


41. can be written as • • • • • .



42. can be written as • • • • • • • • .



43. Evaluate when is .

means

the coefficient. For example, is the coefficient in

. A constant is a number that is added or

subtracted in an expression. It does not vary. For

example, is the constant in .

•

•

Replace t with 4.

Multiply.


44. • Replace with . 52. Use a calculator. Replace with ,

•

45. Evaluate when is and is .

means

with , and with .

53. Evaluate when is and is , using a

calculator.

• • •

• • •

• •

Replace r with 3 and s with 2.


Replace r with 3

and s with 2.

Use the yx key.

•

46. • • • •

with .

Replace with and

,


• • • •

54.

Use a calculator. Replace with


means

and with .

,

• •

• •

•

Replace r with 3 and s with 2.


55. Evaluate when is , is , and

is .

Replace x with 4,

48.

Replace with and with .

• •

• • •

y with 2, and

z with 6.

Multiply left to

right within the

abs. value bars.


means

Replace s with 2

•

Evaluate the

• • • •

and t with 4.

absolute values.

Add.

• • • • Multiply left to right.

• • •

• •

•

56. • •

Replace with , with , and with .

• •

50. • • • • •

and with .

Replace with

• • • • • 57. Evaluate when is and is .


51. Evaluate when is , is , and is ,

using a calculator.

Replace r with 3, s with 2,

and t with 4. x

Replace z with 6

and y with 2.

Follow the order of operations.

,

Use the y key.


Undefined

Numerator: •

Denominator:

Division by 0 is undefined.

2.2 Simplifying Expressions 47 47 Chapter 2 Understanding Variables and Solving Equations


(b)

The like terms are the constants, and .

•

•

Replace x with 4

and y with 2.

Follow the order of operations.

Numerator: •

Denominator:

•

There are no variable parts.

(c)

The like terms are and since the variable parts match; both are .

The coefficients are and .

(d)

The like terms are and since the variable

parts match; both are .


Undefined Division by 0 is undefined.

Relating Concepts (Exercises 59–60)

2. (a)

These are like terms.

59. (a) Evaluate when is seconds.

Add the coefficients.

The variable part, b,

Replace s with 15.

Divide.

(b)

stays the same.


Rewrite y3 as 1y3 .

(b)

miles

Evaluate when is seconds.


The variable part, y3 ,

stays the same.

Replace s with 10.

Divide.

(c)


Rewrite n as 1n.

Change to addition.

(c)

miles

Evaluate when is seconds.


The variable part, n,

stays the same.

Replace s with 5.

Divide.

(d)


Change to addition.


mile

60. (a) Using part (c) of Exercise 59, the distance

covered in seconds is half of the distance

covered in seconds, or mile.

(e)

The variable part, c,

stays the same.


Rewrite xy as 1xy

(b) Using part (a) of Exercise 59, the time to Add the coefficients.

cover miles is half the time to cover miles, or

seconds. Or, using parts (b) and (c), find the

number halfway between seconds and

The variable part, xy,

stays the same.

seconds (f)



(a) Change to addition.

The like terms are and since the variable

parts match; both are .


Zero times anything is zero.


(c) Using parts (a) and (b) of Exercise 59, find the Change to addition.

number halfway between seconds and

seconds; that is seconds.


The variable part, p2,

2.2 Simplifying Expressions or

stays the same.


1.

(g)


Rewrite ab as 1ab


3. (a)

(b)

Rewrite using the

commutative property.

Combine 3b2 7b2


(c) can be written as

• •

•

Change to addition.

Rewrite using the


Rewrite b as 1b.

Add the coefficients

of like terms.

(d)

• •

•

Rewrite x as 1x.

can be written as

(c)

5. (a) can be written as

• •

(d)

Rewrite using the



of like terms.

Zero times anything

is zero.

Change to addition.

(b) can be written as

• •

(c) can be written as

• •

• •

•

Rewrite y as 1y.

Rewrite using the

commutative prop .


of like terms.

(d)

• •

Multiply.

Change addition

or to subtraction.

(e)

Change to addition.

Rewrite using the

commutative prop.


of like terms.

(e)

• •

6. (a)

Multiply.

Change addition

to subtraction.

Distributive property

4. (a) means • • . Using the associative • •

property, it can be rewritten as

• •

•

Rewrite using the


Combine constants.


(b)

can be written as

or

• •

(b)

• •


Multiply.

• Combine constants.


(c)

• •

or

(d)

• •


Multiply.

Change to addition.

Rewrite using the



of like terms.


Multiply.

7.

8.

9.



The variable part, r,

stays the same.



The variable part, t,

stays the same.


Change to addition.

Rewrite x2 as 1x2 .

Rewrite using the


Combine constants.


The variable part, x2 ,

stays the same.

(e)

10.

• •


Rewrite y as 1y.

Change to addition.


of like terms.

11.


Rewrite p as 1p.

Change to addition.


or

The variable part, p,

2.2 Section Exercises

stays the same.

1.

and are the

12.

only like terms in the expression. The variable

parts match; both are . The coefficients are

and .

2.

and are like 13. These are like terms.

Rewrite a3 as 1a3 .

terms. The variable parts match; both are . The

coefficients are and .

Change to addition.


3. and

are the like terms in the expression. The variable parts match; both are . The coefficients

are and .

The variable part, a3 ,

stays the same.

4.

and 14.


are like terms. The variable parts match;

both are . The coefficients are and .

5. , , and are

like terms. There are no variable parts; constants

are considered like terms. 15.

Any number minus itself is .

6. , , and

16.

are like terms. There are no variable parts;

constants are considered like terms.

Any number minus itself is .


17.


Rewrite xy as 1xy.

Change to addition.

26.


The variable part, xy,

27.

Use the commutative property

to put the constants at the end.

Add the coefficients of like terms.

or stays the same. The only like terms are constants.

18.

28.

or

19.


Change to addition.


The variable part, t4 ,

stays the same.

29.

,

Write in the understood

coefficients of 1.

Add the coefficients of

like terms.

The variable part, ab 2,

stays the same.

20.

30.

or

21. These are like terms.

Write in the under-

or

stood coefficients

of 1.

Add the coefficients. 2

31.


coefficients of 1.

Change to addition.

The variable part, y ,

stays the same.

22.


Rewrite using the



like terms.

23.

Rewrite x as 1x

and x as 1x.

Change to addition.


The variable part, x,

stays the same.

32.

Write in the under-

24.

]


33.

stood coefficient of 1. Change to

addition. Rewrite using the


25. Use the commutative property


to rewrite the expression so that

like terms are next to each other.


The variable part, a,

stays the same.

•

of like terms.


34.

44. •

45. By using the associative property, we can write

as

• • • • • .

So, simplifies to .

35. There are no like terms.

46.

•

The expression cannot be simplified.

36. There are no like

terms. The expression cannot be simplified.

37.

Write in the understood coefficients of 1.

Change to addition.

Rewrite using the commutative property.

47. By using the associative property, we can write as

• • • • • • • .

So, simplifies to .

48. •



49.

•

•

coefficient of 1.

Rewrite using the

associative property.

38.

50.

•

51.

• •


52.

• •


or


• • • .

So, simplifies to .

53.

• •


40. •

54. • •


• • • .

55.

• •



So, simplifies to .

42. •

56. • •

57. Distributive property


• • • .

So, simplifies to .

• •

Change addition

to subtraction

of the opposite.


58. • •

69.

• •


or

59.

• •


Change addition

to subtraction

of the opposite.

Rewrite using the



of like terms.

Zero times any

number is 0.

60. • • Zero added to any

or

number is the number


• • 70. • •

62. • •


• • •

Rewrite using the 71. Distributive property


Combine like terms.

• •

Change to addition.

64. • •

Combine like terms.

Any number plus its

opposite is 0.

65.

• •


Combine like terms.

72. • •

66. • •


67.

• •


Change to

addition.

Rewrite using

the

commutative

property. Add

the coefficients

of like terms.

Change addition to subtraction


• •

Rewrite n as 1n. Change

to addition. Rewrite using

the



like terms.

of the opposite.

or

68.

• •

74. • •

or

Summary Exercises Variables and Expressions 53

75.

• •


Rewrite p as 1p.

Change to addition.


of like terms.

Change addition to

subt. of the opposite.

84.

• •

85.


• • • •

Change to addition.

76. Group like terms and add the coefficients.

• •

or

77. A simplified expression usually still has variables,

but it is written in a simpler way. When evaluating

an expression, the variables are all replaced by

specific numbers and the final result is a numerical

answer.

78.

86.

• • • •

• • • •

The answers are equivalent because of the

commutative property of addition.

79. Like terms have matching variable parts, that is,

matching letters and exponents. The coefficients

do not have to match. Examples will vary.

Possible examples: In , the terms

and are like terms. In ,

the terms and are like terms.

80. Add the coefficients of like terms. If no

coefficient is shown, it is assumed to be . Keep

the variable part the same. Examples will vary.

81.

Keep the variable part unchanged when combining

like terms. As shown above, the correct answer is

.

82. In the last step, do not change the sign of the first

term. The correct answer is .

87.


• • • •

•

Change to addition.

Group like terms and add the coefficients.

88.

Summary Exercises

83.

• • •

Distributive prop. Variables and Expressions

Change subtrac-

tion to adding

the opposite.

1.

or 1

is the variable;

1 is the coefficient;

is the constant.

Group like terms

and add the

coefficients.

2.

and are the variables;

is the coefficient.

3. is the variable;

is the coefficient;

is the constant.

54 Chapter 2 Understanding Variables and Solving Equations 2.3 Solving Equations Using Addition 54

4. Expression (rule) for finding the perimeter of an 11. Replace with and with .

octagon of side length :

(a) Evaluate the expression when , the side

length, is yards.

If is multiplied by any number, the result is

Thus, there is no need to make any calculations

since the result is

•

Replace s with 4.


12.

.

Replace with , with , and with

yards is the perimeter.

(b) Evaluate the expression when , the side

length, is inches.

• •

•


•

Replace s with 15.


13. • •

Replace with .

inches is the perimeter.

5. Expression (rule) for finding the total cost of a car

with down payment , monthly payment and

• •

•


number of payments :

(a) Evaluate the expression when the down 14. Replace with and with .

payment is $ , the monthly payment is $ ,

and the number of payments is .

Replace d with $3000, m

• •

•


$ $ •

$ $ ,

with $280, and t with 36.

Multiply before adding.

15. • • •

with .

Replace with and

$ , is the total cost of the car. • • • Multiply left to right.

(b)

Evaluate the expression when the down • •

payment is $ , the monthly payment is $ ,

and the number of payments is .

Replace d with $1750, m

•

with $429, and t with 48. 16. • • • • • • Replace with

$ $ •

$ $ ,

Multiply before adding. and with . Multiply left to right.

• • • • • •

$ , is the total cost of the car.

6. written without exponents is

• • • •


• • • •


• • • • • • • •


• • • • •

• • • •

• • •

• •

•

9. • • •

Replace with . 17.

Use a calculator. Replace with

• • •

• •

•

Multiply left to right. , with , and with .

,

18.

19. 10. Replace with and with .

If is multiplied by any number, the result is

Thus, there is no need to make any calculations

since the result is


20. • •

•

or

21.

22. • •

33. (a) Simplifying the expression correctly:

• •

The student forgot to multiply • .

23.

24. • •

(b)

Simplifying the expression correctly:

• •

• •

Two negative factors give a positive product.

25.

(c) Simplifying the expression correctly:

•

Keep the variable part unchanged; that is, adding

's to 's gives an answer with 's, not 's.

or

26. • •

27.

34. In the last step, do not change the sign of the first

term; keep as . The correct answer is

2.3 Solving Equations Using Addition


28. • •

1. (a)

?

Given equation

Replace c with 95.

110 is more than 80.

No, is not the solution.

29. • •

?

Replace c with 65.

Balances

•

Yes, is the solution.

(No need to check and .)

30. • •

• •

(b)

?

Given equation

Replace c with 28.



31.

• •

?

Replace c with 20.

• • No, is not the solution.

?

Replace c with 24.

or No, is not the solution. ?

32.

• •

Replace c with 32.

Balances

• • Yes, is the solution.


2. (a) Solve for . Combine like terms.

To get by itself, add the opposite of , which is To get r by itself,

. To keep the balance, add to both sides.

add 1 to both sides.

The solution is .

or

Check

The solution is .

Check

Original equation

Replace y with 7.

• •

Replace r with 2.

Balances, so solution is .

(b) Solve for .

Change to addition.


Balances

1. Replace with , , , and .

To get by itself add the opposite of , which is

, to both sides.

?

?

Given equation

Replace n with 58.

The solution is .


(No need to check , , and .)

Check

Original equation 2. Replace with , , , and .

Replace b with 4.

Balances

?

?

?

?

3. (a)

Rewrite both sides by

?

?

changing subtraction to addition. Combine like ? ?

terms.

The check for balances, so is the solution.

To get by itself add the opposite of , which is

, to both sides.

)

3.

and .

?

Replace with , , ,

Given equation

Replace y with 4.



The solution is . ? Replace y with 16.

Check

Original equation

Replace k with 4.


(No need to check and .)

( ) ( ) 4. Replace with , , ,

Balances and .

(b) Change to ?

?

addition.

Rewrite the left side by using the commutative

?

property. is the solution. (No need to check .)


5. (a) Add to both sides because 12.

gives on the left side. Add 8 to both sides.

(b) Add to both sides

because gives on the right side. The solution is .

6. (a) Add to both sides Check

because gives on the left side. Replace a with 9.

(b) Add to both sides because Balances

gives on the right side. 13.

7.

Add the opposite of

5, 5, to both sides.

Add the opposite of


The solution is .

The solution is . Check

Check

Replace p with 4.

Balances

14.

Replace k with 18.

Balances

8.

Add 3 to both sides.


The solution is .

The solution is .

Check

Check

Replace a with 9.

Balances

Replace y with 16.

Balances

15.

9.

Change to addition.

Add the opposite of


Change to addition.

Add the opposite of


The solution is .

The solution is .

Check

Check

10.

Replace r with 10.

Balances

Change to addition. Add 5 to

both sides.


16.

Replace y with 6.

Balances

Change

to

addition.

Add 15

to both

sides.

The solution is .

The solution is .

Check

Check

Replace b with 8.

Balances

17.

Replace k with 15.

Balances

11. Add the opposite of

Add the opposite of


The solution is .


The solution is .

Check

Replace n with 8.

Check

Replace r with 6.

Balances Balances


18. Correct solution:


The solution is .

Change to addition.

Add the opposite of

Check

Replace z with 7.


Balances The solution is .

19.

Change to addition.

Add the opposite of


Check

Replace z with 8.

Change to addition.

Balances

The solution is .

24.

The given solution is .

Check

Replace x with 0.

Change to addition.

Check

Replace x with 13.

Balances

Balances is the correct solution.

20. 25. The given solution is .

Change to addition.


Check

Replace x with 18.

Balances

The solution is .

is the correct solution.

Check

Replace m with 0.

26.

The given solution is .

21.

Balances

Check

Replace k with 5.

Does not balance

Add the opposite of


The solution is .

Correct solution:

Add the opposite of


Check

Replace t with 3.

The correct solution is .

22.


Balances Add 10 to both sides.

Check

Replace k with 9.

Balances

27. The given solution is .

The solution is .

Check

Check

Replace w with 9.

Balances

Correct solution:

Replace b with 10.

Does not balance


Check

Replace z with 2.

Change to addition.

Does not balance

Add the opposite of


The solution is .


Replace b with 12.

Balances

Replace t with 0.

31. Change to addition.

Simplify the left side. Balances

Change to addition. 36.


Check

Replace b with 0.

Balances

32.



The solution is .

Check

Replace a with 0.

Does not balance

Check

Balances

Correct solution: 33.

Change to addition.

Add.

Add the opposite of Add 10.



Check

The solution is .

Check

Replace a with 14.

Replace b with 30.

Balances Balances

29.

Simplify the right side.

Change to addition.

34.

Add 4 to both sides. Add 1 to both sides.

The solution is .

The solution is .

Check

Replace c with 6.

Balances

Check

Replace w with 17.

Balances

30. 35.

Change to addition.

Simplify the right side.


The solution is .

Check

Check

The solution is .


Check

The solution is .


The solution is .

Replace y with 5.

Change to addition.

Balances

Check

Replace p with 0.

Balances


37.

Change to addition.

42.

Check

Combine like terms.

1z is the same as z.

The solution is .

Add 2.

• •

Replace z with 7.

Change to add.

Balances

The solution is .

43.

Change to addition.

38.

The solution is .

44.

Combine like terms.

The solution is .

Check Change to addition.

•

Replace r with 5.

Balances

45.

Combine like terms.

The solution is .

Rearrange and Change to addition.

39.

combine like terms. Add 75 to both sides.

The solution is .

Add 2 to both sides. 46.

Change to addition.

Check

The solution is .


• •

40.

Replace w with 3.

Balances

47.

The solution is 1 .

Rearrange and

combine like terms.


Add 4 to both sides. The solution is .


The solution is .

48.

Check

Let t = 5.

Balances


The solution is .

49.

41. Change to addition. Add 31 to both sides.

Combine like terms.

The solution is .



1x is the same as x.

The solution is .

The solution is .


51.

Change to

addition.

Rearrange and

60. Check

combine like terms.


Does not balance

52.

The solution is .

To correct the errors, change to . Then, add to both sides, not .


61.

Add the opposite of

Add 1.

0 0


The solution is .

53.

There were graduates this year.

62.

( ) ( ) Change to addition.

0 Add the opposite of 0

Combine like terms. 10, 10, to both sides.

54.


The solution is .

There were graduates last year.

63.

Change to addition. Add 37 to both sides.

Combine like terms.


55.

The solution is . When the temperature is degrees, a field cricket chirps times (in seconds).



56.

The solution is .

Change to addition.

When the temperature is degrees, a field cricket

Add 28 to both sides. chirps times (in seconds).

65.


The solution is . Change to addition.

57.

Combine like terms.



The solution is .

Ernesto's parking fees average $ per month in

winter.

58.


66.

Change to addition.


The solution is .

59. No, the solution is , the number used to

replace in the original equation. Aimee's parking fees average $ per month in

winter.


67.

Add the opposite of


Change all subtractions to additions.

The solution is .

Commutative property

(b) Equations will vary. Some possibilities are:

Combine like terms.


Add the opposite of


The solution is .

The solution is .

68.

Add the opposite of


The solution is .

The solution is . 72. (a) Add the opposite of

69.

Change subtraction within absolute value to

addition and rearrange the terms.


The solution is .

Simplify inside absolute value bars. Collect like

terms.

(b)

Change to addition.

Add the opposite of

Evaluate absolute values. 1, 1, to both sides.

Change to addition.

Add 6 to both sides. or The solution is .

The solution is .

(c) $

$

Add the opposite

of $2 50, $2 50,

70. to both sides.

$ $

$

The solution is $ .

(d) Equations will vary. Some possibilities are:

$ $ The solution is $ .


$ $ The solution is $ .

The solution is .


71. (a) Equations will vary. Some possibilities are:

Change to addition.

2.4 Solving Equations Using Division


1. (a) Solve .

Use division to undo multiplication. Divide both

sides by the coefficient of the variable, which is .


The solution is .

The solution is .

2.4 Solving Equations Using Division 63 63 Chapter 2 Understanding Variables and Solving Equations

Check

•

Original equation

Replace s with 11.

Check

Balances Replace p with 1.

(b)

Divide both

Balances

The solution is .

Check

sides by 9.

3. (a)


1 as the coefficient of k.

Divide both

• Replace p with 3.

sides by 1.


(c)

The solution is .

Divide both

sides by 5.

Check

(b)

•

Replace k with 12.

Balances

Check

• Replace x with 8.

Write t as 1t.

Divide both

(d)

Balances

sides by 1.

The solution is .

Divide both

sides by 7.

Check

The solution is .

Check

(c)

•

Replace t with 7.

Balances

•

Replace t with 10.

Balances

Write m as 1m.

Divide both

sides by 1.

2. (a)

Combine like terms.

The solution is .

Divide both

sides by 4.

The solution is .

Check

•

Check

• • Replace n with 7.



1.

Divide both

Balances

sides by 6.

(b)

Change to addition. Rewrite p as 1p.

The solution is .

Check

• Replace z with 2.

Combine like terms.

2.

Balances

Check

•

Divide both sides by 13.

The solution is .

The solution is .

Balances


3. 11.

Divide both Divide both

sides by 12.

The solution is .

sides by 5.

The solution is .

Check

• Replace r with 4.

Check

•

Replace b with 5.

Balances Balances

4.

The solution is .

Check

Balances

•

12.

Check

Balances

•

5.

Divide both

sides by 3.

The solution is .

13.

The solution is .

Combine like terms.

Divide both

Check

sides by 2.

• Replace y with 0. The solution is .

6.

Balances

Check

•

Check

•

Replace r with 3.

Balances

Balances

14.

Check

The solution is . •

7.

Divide both

sides by 7.

Balances

The solution is .

The solution is .

Check

•

Replace k with 10.

Balances

15.

Change to addition.

8. Check

Rewrite p as 1p.

Combine like terms.

•

Divide both


The solution is .

9.

Balances

Check

sides by 4.

The solution is .

Divide both

sides by 9.

• Replace p with 3.

The solution is .

Change to addition.

Check

• Replace r with 6. Balances

Balances 16.

10.

Check

•

Change to addition.

Combine like terms.

Divide both

Balances sides by 10.

The solution is . The solution is .


Check

•

Replace z with 2.

Change to addition.

24.

Original equation

Change to addition.

Combine like terms.

Divide both

sides by 12.


17.

Original equation

Change to addition.

Combine like terms.

Divide both

25.

Original equation

Divide both

sides by 9.

The solution is .

sides by 5.

The solution is . 26. Original equation

18.

Original equation

Combine like terms.

Divide both

sides by 8.

Divide both

sides by 10.

The solution is .

The solution is . 27.

Original equation

Change to addition.

19. Original equation Combine like terms.

Change to addition.

Rewrite x as 1x.

Divide both

sides by 6.

Combine like terms. The solution is .

20.

Divide both

sides by 8.

The solution is .

Original equation

Change to addition.

Rewrite c as 1c.

Combine like terms.

28.

Original eq

Combine.

Divide both

sides by 7.

The solu-

tion is .

Divide both

sides by 3.

The solution is .

29.

Original equation

Change to addition.

Combine like terms.

21.

Original equation

Change to addition.

Divide both

sides by 4.

Rewrite w as 1w.


1w is the same as w.

30.

Original equation

The solution is . Change to addition.


22.

Original equation

Change to addition.

Combine like terms.

Combine like terms.

Divide both

sides by 11.

1t is the same as t. The solution is .

The solution is .

31.

Original equation

23.

Original equation

Change to addition.

Combine like terms.

Divide both

sides by 12.

• •

To multiply on the left, use

the associative property.

Divide both

sides by 6.



32. Original equation 40. Original equation

• • To multiply on the left, use Write in the understood 1.

sides by 12.

The solution is .

Divide both

sides by 1.

The solution is .

35. Original equation

• • Associative property 44. Original equation

Divide both

Write in the understood 1.

Divide both

sides by 8.

sides by 1.


the associative property.

Divide both

sides by 8.

Divide both

sides by 1.

The solution is .

The solution is . 41. Original equation

33.

• •

Original equation

To multiply on the right,

use the associative prop.


Divide both

sides by 1.

The solution is .

Divide both

sides by 25.

42.

Original equation


34.

The solution is .

Original equation

Divide both

sides by 1.

• • To multiply on the right,

use the associative prop. The solution is .

43. Original equation

Divide both Write in the understood 1.

36.

• •

Original equation

Associative property

45.

Original equation


Divide both

sides by 20.

Divide both

sides by 1.

37.

The solution is .

Original equation

46.

The solution is .

Original equation

• •



Divide both


sides by 30. The solution is .

38.

•

•


The solution is . Original

equation Associative property

47. Each solution

is the

opposite of

the number in

the equation.

So the rule is:

When you

change the

sign of the

variable from

negative to

positive, then

change the

number in the

equation to its

opposite.

Divide both

sides by 10. In , the opposite of is , so .

39.

The solution is .

Original equation


48. Equations will vary. Some possibilities are (i) and (ii) .

(i)

Divide both

sides by 1.

Divide both

sides by 5.


2.5 Solving Equations with Several Steps 67 67 Chapter 2 Understanding Variables and Solving Equations

(ii)

Change to addition.

Combine like terms.

Divide both

56.

• • • •


sides by 2.

The solution is .

Change to addition.

Combine like terms.

49. Divide by the coefficient of , which is , not by Divide both

the opposite of . sides by 50.


The solution is .

57.

• • • •

50. You can divide both sides of an equation by the

same nonzero number and keep the equation

balanced.

Assoc. prop. Simplify within the absolute values.

Simplify the absolute values.

Combine like terms.51.

Divide both

sides by 3.

Divide both

sides by 70.

The length of one side is feet.

52.

The solution is .

58.

Divide both

sides by 3.

Simplify within the absolute value.

The length of one side is inches. Simplify the absolute value.

53.

Divide both

Combine like terms.

Divide both

sides by 5. sides by 10.

The length of one side is meters.

The solution is .

54.

Divide both

sides by 5.

2.5 Solving Equations with Several Steps


1. (a) To get 2r by itself,


The length of one side is yards.

55.

• • • •

7 to both sides.

To solve for r,


divide both sides by

the coefficient, 2.

Change to addition.


Divide both

sides by 1.

Check

•

Replace r with 3.

The solution is . Balances


(b)

Change to addition.


Rewrite p as 1p.

Add 3p.

Divide both Add 6.

sides by 10.

The solution is .

Check Divide both

•

Replace r with 2.

3. (a)

sides by 2.

The solution is .

Balances • • Distribute on the right.

2. (a) Solve, keeping the variable on left side. Change to addition.

Add 2y to both sides.


Change to addition.


Divide both

sides by 4.

The solution is .

or

The solution is .

Check

Replace y with 2.

Solve, keeping the variable on the right side.

Balances

Add 3y to both sides. (b)

• • Distribute on the left.

Add 7 to both sides. Add 20 to both sides.


sides by 1.

sides by 5.



(b) Solve, keeping the variable on left side.

Rewrite p as 1p.

Check

Replace m with 0.

Balances

Add 1p to both sides.

Change to addition.


Divide both

sides by 2.

The solution is .

(c)

• •

Distribute on the left.

Change to addition.

Add 12 to both sides. Divide both

sides by 6.

Solve, keeping the variable on the right side. The solution is .


Check

Replace t with 5.

Balances

Check

Let p = 1.

Balances

4. (a)

• •

Distribute. 2.

Check

Variables left

Add 2b.

Balances

Add 21.

The solution is .

or

3.

Change to addition.

The solution is . Add 6 to both sides.

Check

•

Divide both

Balances

sides by 8.

(b)

• •

Distribute

Check

The solution is .

Add the

opposite.

Combine

Replace y with 1.

Balances

like terms.

Add 2n.

4.

Check

Balances

Add 6.

The solution is .

Divide both

sides by 6.

The solution is .

5.

To get 9a by itself,


Check

•

Let n = 2.

Divide both

sides by 9.

The solution is .



Balances

Check

Replace a with 2.

Balances

1.

To get 7p by itself,


6.

Check

Divide both

Balances

sides by 7.

The solution is .

The solution is .


7.

To get 3m by itself,


11. Solve, keeping the variable on the left side.

Change to addition.

Divide both Add 4p to both sides.

sides by 3.

The solution is .


Check

Replace m with 0.

Divide both

Balances

sides by 2.

8.

The solution is .


Change to addition.

The solution is .

Check

Add 6p to both sides.


Balances

Divide both

9.

Change to addition.

To get 5x by itself,


sides by 2.

The solution is .

Check

Divide both

sides by 5.

Balances

The solution is .

Check

Replace x with 4.

Balances

12. Left side: Right side:


Check

Balances

Balances

10.

The solution is .

Check

The solution is .


13. Solve, keeping the variable on the left side.

15. simplifies to

.

Change to addition.

Add 6k to

Add 2a to

both sides.

both sides.

Add 6 to

Add 18 to

both sides.

both sides.

Divide both

sides by 8.

The solution is .

Divide both

sides by 5.


The solution is .

Change to addition.

Add 2k to

both sides.

Check

Balances

Add 10 to

16. simplifies to

.

both sides.

Divide both

sides by 8.

Add 10 to

both sides.

Check

The solution is .

Divide both

sides by 5.

Replace k with 2.

Balances

The solution is .

14. Left side: Right side:

Check


Balances

17. Neither side can be simplified, so solve the

equation.

Add 3t to

both sides.

The solution is .

Check

Divide both

sides by 11.

The solution is .

Balances

Check

Replace t with 0.

Balances


23.

Distribute.

Divide both Add 8 to both sides.

sides by 2.

The solution is .

Divide both

sides by 2.

Check The solution is .

24.

Balances Distribute.


simplifies to

Distribute.


18. Neither side can be simplified, so solve the 21.

equation.

Distribute.

Change to addition.

Add 9z to

both sides.

Divide both

sides by 24.

Add 16 to both sides. Divide both

sides by 8.

Check


22.

Replace z with 0. Distribute.

Balances Change to addition.

19. simplifies to .

Add 2 to


Divide both

sides by 9.

both sides. The solution is .

20. .

Divide both

sides by 3.

Add 2z to

both sides.

25.

The solution is .

Add 12 to

both sides.

Divide both

sides by 4.


Divide both

sides by 7.

The solution is .

26.

Distribute.

The solution is . Add 15 to both sides.

Check

Divide both

sides by 5.



27.

Divide both

Distribute. sides by 9.

Change to addition.


The solution is .

33.

Add 18 to b

30.

Distribute.

Change to addition.

Divide both

oth sides.

sides by 6.

The solution is . Divide both

sides by 6.28.

Distribute.

The solution is .

Change to addition. 34.

Add 49 to both sides. Change to addition.

Divide both

sides by 7.


Divide both

The solution is .

sides by 8.

29.

Distribute.

35.

The solution is .

Change to addition. Change to addition.



Divide both

Divide both

sides by 12.

sides by 9.


36.



Divide both


sides by 11. The solution is .

The solution is .

37.

31. Add 3x to both sides.


Divide both

sides by 2.

The solution is .

38.

Change to addition.

Add 2n to both sides

Distribute.

Divide both

sides by 9.

The solution is .

Distribute.

Change to addition.


Divide both

sides by 2.

The solution is .

32.

.

Change to addition.



39.

Change to addition.

Divide both

sides by 5.

Add 2a. The solution is .


44.

Change to addition.

Combine like terms.

Add 1y.


sides by 6. sides by 4.


45. 40. Distribute.

Change to addition. Change to add.

Add 1r. Group terms.

Combine terms.

Add c.

Add 8.

Divide both

sides by 9.

Add 56.

The solution is .

41.

Divide

both sides

by 9.

Change to addition.

Add 5b to both sides.

The solution is .

46.


Divide both

sides by 7.

The solution is .


42.

Change to add.

Add 8t.

The solution is .

Add 9.

47.

Distribute.

Divide both

sides by 5.

The solution is .

Add the

opposites.

Group like

terms.

Combine

43.

like terms.

Change to addition. Add 13y.

Combine like terms.

Add 2k.

Add 5.


52.

Divide

by 2.

Add 3x.

The solution is .

48.

Add 8.

Divide both

sides by 7.

The solution is .

53.

Add 6p.

The solution is .

49.

Add 17.

Change to add. Divide

Combine terms. by 16.

Add 6.

Divide both

sides by 1.

The solution is .

54.

The solution is .

50.

Change to add.

Combine terms.

Add 5.

Divide both


sides by 1.

The solution is .

The solution is .

51.

Distribute.

55.

Change to add. Add 4.

Add 6c.

Add b.

Add 42.

The solution is .

The solution is .


Add 2t to both sides

(addition property). 64.

Add 5 to both sides

(addition property).

56.

(b) The sum of and a positive number is

negative, so must be negative.

62. (a) It must be positive, because the sum of two

Add 1. negative numbers is always negative.

(b) The sum of and a negative number is positive, so must be positive.

The solution is .

57. The series of steps may vary. One possibility is:

63. (a) It must be positive. When the signs are the

same, the product is positive, and when the signs

are different, the product is negative.

Change to addition.

Divide both sides by 5

(b) The product of and a negative number is

negative, so must be positive.

(a) It must be negative also. When the signs are

different, the product is negative, and when the

signs match, the product is positive.

(b) The product of and a negative number is

(division

property).

positive, so must be negative.

The solution is .

58. Multiplication distributes over both addition and

Chapter 2 Review Exercises

1. (a) In the expression , is the variable,

subtraction. Examples will vary. Some is the coefficient, and is the constant term.

possibilities are is and

is .

59. Check

(b) The term that has as the constant term and

as the coefficient is .

2. (a) Evaluate when is .

The check does not balance, so is not the correct solution. The student added to on the left side, instead of adding to . The correct

•

Replace c with 15. Order test tubes.

solution, obtained using , ,

is .

(b) Evaluate when is .

60. Check

•

Replace c with 24. Order test tubes.

The check does not balance, so is not the 3. (a) means • • • • •

correct solution.


Student did not

distribute the

over the .

(

b

)

4.

(a)

means

• • •

•

means

•

•

Repla

ce n

with

3.

(b)

means



61. (a) It must be negative, because the sum of two

positive numbers is always positive.

• •

• •

•

Replace n with 3.

8. Associative property

• •

•

9.

• •

Distribute.

Chapter 2 Review Exercises 77

(c) means

• • •

Replace m with 2

12.

Distribute.

• • • and p with 4.

or

• •

•

13. Expressions will vary. One possibility is .

14. Add 16 to both sides.

(d)

means

• • • • • •

• • • • • •

Replace m with 2

and n with 3.

The solution is .

Check

Replace n with 11.

• • • • • Balances

• • • •

• • •

• •

15.

•

5.

Combine like terms.

The solution is .

or Check

6.

Rewrite x as 1x.

Change to addition.

Combine like terms.

16.

Balances

7.

• •


Divide both

sides by 6.

• The solution is .

17.

Combine like terms.

Divide both

sides by 4.

10.

Distribute.

18.

The

solution is

.

Divide both

• •

sides by 7.

or The solution is .

11.

• •

Distribute.

19.

Divide both

sides by 3.

The solution is .

78 Chapter 2 Understanding Variables and Solving Equations

20.

Divide both

sides by 3.

Divide both The solution is .

sides by 15. 26. [2.5]

The solution is . Distribute.

21.

Add 6 to

both sides.

Divide both

sides by 1.

The solution is .

Divide both

sides by 2.

22. The solution is .

27. [2.5]

Add 8w to

both sides.

Add 6y to

both sides.

Divide both

Add 4 to

both sides.

sides by 4.

The solution is .

The solution is .

Divide both

sides by 4.

28. [2.5]

Add the

opposites.

Combine

like terms.

Divide both

sides by 5.

23.

• • Distribute.

The solution is .

Add 8 to 29. [2.5]

both sides. Divide both

sides by 4.

The solution is .

Add 7 to

both sides.

Divide both

sides by 3.

The solution is .

24. Add 4 to

30. [2.5] Add 3b to

79 Chapter 2 Understanding Variables and Solving Equations

both sides.

both sides.

Divide both

sides by 2.

Add 6 to

both sides.

The number of employees is .

Divide both

25. [2.5]

sides by 2.

Add 4a to

both sides.

The solution is .

31. [2.3]

Add 3 to

both sides.

The solution is .

Chapter 2 Test 79

32. [2.5]

Distribute.

Add 3n to

Chapter 2 Test

1. In the expression , is the coefficient,

both sides. is the variable, and is the constant term.

Add 3 to

both sides.

Divide both

sides by 3.

2. Evaluate the expression when is and

is .

• •

The solution is .

33. [2.5]

Buy hot dogs.

Distribute. 3. means • • • • • • •

Add 42 to

4. means • • • • •

both sides. 5. means

• • •

Divide both

sides by 21.

• • • Replace s with 5

and t with 4.

34. [2.5]

The solution is . • •

•

6.

Add 6d.

The solution is .

Add 13 to

both sides.

Divide

by 4.

7.

35. [2.5]

8.

or

Add 12b to

both sides.

9.

There are no like terms.

The expression cannot be simplified.

Divide 10.

The solution is .

by 18.

• •


of multiplication

80 Chapter 2 Understanding Variables and Solving Equations Cumulative Review Exercises (Chapters 1–2) 80

11.

• •

12.


of multiplication


19.


Divide both

sides by 3.

13.

• • Distributive property

The solution is . Check

Replace a with 3.

14.


20.

Balances

Combine like terms.

or

15.

Combine

The solution is .

Add 8.

16.

like terms.

21.


Divide both

sides by 1.

The solution is .

Check

Replace x with 5.

The solution is .

Balances 22.

17.

Add 2x to both sides.

Divide both

sides by 7.


Divide both

sides by 7.

The solution is .

Check

• )

Replace w with 11.

The solution is .


Balances 23.

18.

Add 3m to

both sides.

Divide both

sides by 1.

The solution is . both sides.

Check

•

Replace p with 14.

Divide both

sides by 4.



24.

Add 6b to

(b) lies to the left of on the number line, so

.

4. (a) Commutative

both sides.

property of addition: Changing the order of the

Add 42 to

both sides.

addends does not change the sum.

(b) • Multiplication property of

zero: Multiplying any number by gives a

product of .

(c) • •

Distributive

The solution is .

25.

Distribute.

property: Multiplication distributes over addition.

5. (a)

Underline the hundreds place:

Add 24 to

both sides.

Divide both

sides by 3.

The next digit is or less, so leave as . Change

and to .

(b) , ,

Underline the thousands place: ,

The next digit is or more, so add to , write the

and add to the ten-thousands place. Change

and to .The solution is .

26. Addition property of equality: Start with a possible

solution, for example, . Now add an

abitrary number, say , to both sides, to give us

the equation .

6.

7.

Change to addition.

Division property of equality: Start with a possible

solution, for example, . Now multiply both

sides by an abitrary number, say , to give us the

equation .

8.

6 is 6 units from 0.

4 is 4 units from 0.

Thus, equations will vary. Two possibilities are Same sign, positive product

Solving:

and

. 9.

•

Same sign, positive product


Divide both

sides by 6.

10.

Same sign, positive quotient

11.

Addition of a number

Cumulative Review Exercises

12.

and its opposite is zero.


(Chapters 1–2)

1. , , , in words is three hundred six

billion, four thousand, two hundred ten.

• •

•

Exponent


2. Eight hundred million, sixty-six thousand:

, ,

13. is undefined. Division by is undefined.

3. (a) lies to the right of on the number line,

so .

14. •

Different signs, negative product


15.

Change to addition.

22. shares rounds to .

$ rounds to $ .

$ stays $ (it's a single digit number).

16.

Different signs, negative quotient

17.

18.

Each stock dropped in value by $ and Doug

owned shares. Multiply to find out how much

money he lost. Then, subtract this amount from

the original total value.

Estimate: $ • $

Change to addition. Exact: $ • $

Multiply.

Add.

His shares are now worth $ .

23. $ rounds to $ .

$ rounds to $ .

19.

Numerator:

Denominator:

Multiply.

Change to addition.

Add left to right.

months (in one year) rounds to .

Estimate: $ $

$ $

Exact: $ $ $ $

She will spend $ for rent and parking in one

year.

24. means • • • • • •

25. means

• • • •

Replace x with 5

20.

• • Exponents

Last step is division:

days rounds to .

26.

• • • •

• • •

• •

•

and y with 2.


miles rounds to . Change to addition.

Average distance "per" day implies division.

Estimate: miles

miles per day days

miles

or

27.

Combine like terms.

Exact:

days miles per day

Write the understood

coefficients of .

The average distance the tiger traveled each day

was miles.

21.

degrees

rounds

to .

"Rise"


of degrees rounds to .

•

Change to addition.

Combine like terms.

A start temperature of degrees followed by a

rise of degrees implies addition.

28.

Estimate: degrees

Exact: degrees

The daytime temperature was degrees. or


33.


37.

29.

• Associative property

35.

Add 2m to both sides.

30. Add 6 to both sides.

• • Distribute.

Divide both

31.

The solution is .

sides by 3.

or

32.

Check

Add x to both sides.

Replace m with 4.

Divide both Balances

sides by 2.

36.

The solution is .

Check

Add 4x to both sides.

Replace x with 4. Add 18 to both sides.

Balances Divide both

sides by 14.

The solution is .

Divide both

sides by 7.

Divide both

sides by 1.


Check

38.

34.


Replace y with 6.

Balances

Add 1b

to both

sides.

Add 1 to

Divide both

sides by 3.

both sides.

Divide both

The solution is .

Check

Replace k with 7.

Balances

The solution is .

sides by 2.


39.

Add 2 to

both sides.

40.

Add 18.

Add 8t to

both sides.

Add 10y

Divide

Divide both

by 4.

The solution is .


sides by 6.

International Management Managing Across Borders and Cultures Text and

Cases 9th Edition Deresky SOLUTIONS MANUAL



borders-and-cultures-text-and-cases-9th-edition-deresky-solutions-manual/

International Management Managing Across Borders and Cultures Text and

Cases 9th Edition Deresky TEST BANK



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international management: managing across borders and cultures, text and cases (9th edition)

pdf

international management managing across borders and cultures 9th edition pdf

international management managing across borders and cultures text and cases (8th edition)

pdf

international management: managing across borders and cultures 8th edition pdf

international management deresky 9th edition pdf

international management: managing across borders and cultures pdf

isbn 9780134376042

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The solution is .

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