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2.1 Introduction to Variables 37

CHAPTER 2 UNDERSTANDINGVARIABLES ANDSOLVINGEQUATIONS

2.1 Introduction to Variables2.1 Section Exercises

1. is a variable. is a constant.

- % -

%


. ' .

'


$ 7 7$


% 8 8

%

5. is a variable. is a coefficient.

&2 2

&


$= =

$


is a constant.

#- "! -

#

"!

8. is a variable. is a coefficient. is a constant.

', " ,

'

"

9. Both and are variables.B C B C

10. Both and are variables.BC B C


is a constant.

'1 * 1

'

*


is a constant.

"!5 "& 5

"!

"&

13. Expression (rule) for ordering robes: 1 "!

(a) Evaluate the expression when there are '&%graduates.

1 "! 1 '&%

'&% "!

''%

Replace with .Follow the rule and add.ðóñóò

robes must be ordered.

Evaluate the expression when there are (b) #!)graduates.

1 "! 1 #!)

#!) "!

#")

Replace with .Follow the rule and add.

robes must be ordered.ðóñóò

Evaluate the expression when there are (c) *&graduates.

1 "! 1 *&

*& "!

"!&

Replace with .Follow the rule and add.ðñò

robes must be ordered.

14. Expression (rule) for degrees: - $(

(a) is degrees.%& $( )#

is degrees.(b) $$ $( (!

is degrees.(c) &) $( *&

15. Expression (rule) for finding perimeter ofequilateral triangle of side length : = $=

(a) Evaluate the expression when , the side=length, is inches.""

$= = ""

$

$$

Replace with .Follow the rule and multiply.

inches is the perimeter.î• ""

(b) Evaluate the expression when , the side=length, is feet.$

$= = $

$

*


feet is the perimeter.í• $

16. Expression (rule) for perimeter: &=

(a) & • #& "#& meters is meters.

inches is inches.(b) & • ) %!

17. Expression (rule) for ordering brushes: $- &

(a) Evaluate the expression when , the class size,-is ."#

$- & - "#

$ &

$' &

$"

Replace with .Multiply before subtracting.

brushes must be ordered.

îï• "#

(b) Evaluate the expression when , the class size,-is ."'

$- & - "'

$ &

%) &

%$

Replace with .Multiply before subtracting.

brushes must be ordered.

îï• "'

18. Expression (rule) for doughnuts: #8 %

(a) # • "$ % ## is doughnuts.

is doughnuts.(b) # • ") % $#

38 Chapter 2 Understanding Variables and Solving Equations

19. Expression (rule) for average test score, where is:the total points and is the number of tests: > :Î>

(a) Evaluate the expression when , the total:points, is and , the number of tests, is .$$# > %

:

>: $$# > %

$$#

%)$

Replace with and with .

Follow the rule and divide.

points is the average test score.

Evaluate the expression when , the total(b) :points, is and , the number of tests, is .'$( > (

:

>: '$( > (

'$(

(*"


Follow the rule and divide.

points is the average test score.

20. Expression (rule) for buses: :

,

(a) is buses."('%% %

is buses.(b) (#$' #

21. Expression Expression

is

Valueof B

"#

!

B B B B %B

"# "# "# "#

%)% •

•

•

"# %)

! ! ! !

!% ! !

& & & &

#!% & #!

is

is is

is is &

22. Expression Expression

is is

is

is is

is is

Valueof C

"!

!

$C C #C

$Ð"!Ñ $!"! #Ð"!Ñ

"! #! $!

$Ð $Ñ *$ #Ð $Ñ

$ ' *

$Ð!Ñ œ !! #Ð!Ñ

!

$

! ! is

23. ExpressionValue Valueof of B C

&

!

#

%

' #

)

B C

#Ð

%Ñ & ) & "$#Ð 'Ñ #

"# # "!#Ð!Ñ ) ! ) )

is is

is is is is

24. ExpressionValue Valueof of B C

&

!

#

%' #

)

BC

#

• •• •

• •

% & %!# ' # #%# ! ) !

is is is

25. A variable is a letter that represents the part of arule that varies or changes depending on thesituation. An expression expresses, or tells, therule for doing something. For example, is an- &expression, and is the variable.-

26. The number part in a multiplication expression isthe coefficient. For example, is the coefficient in%%=. A constant is a number that is added orsubtracted in an expression. It does not vary. Forexample, is the constant in .& - &

27. Let represent "a number." Multiplying a number,by leaves the number unchanged."

or , "• •" œ , , œ ,

28. Adding to any number leaves the number!unchanged.

or , ! œ , ! , œ ,

29. Let represent "any number." Any number,divided by is undefined.!

, is undefined or is undefined.

!, ƒ !

30. Multiplication distributes over addition.

+Ð, -Ñ œ + • •, + -

31. written without exponents is-'

- • • • • •- - - - -

32. is . .( • • • • • •. . . . . .

33. written without exponents isB C% $

B • • • • • •B B B C C C

34. is - . -# & • • • • • •- . . . . .

35. $$+ , can be written as $ • • • •+ + + ,.The exponent applies only to the base .$ +

36. #)7 8 is ) • • •7 7 8

37. *BC *# can be written as • • •B C C.The exponent applies only to the base .# C

38. is &+, &% • • • • •+ , , , ,

39. &#- . can be written as # • • • • • •- - - - - ..The exponent applies only to the base .& -

40. $%B C is % • • • •B B B C

41. + ,-$ # can be written as + + + , - -• • • • • .The exponent applies only to the base .$ +The exponent applies only to the base .# -

42. is B CD B# ' • • • • • • • •B C D D D D D D

2.1 Introduction to Variables 39

43. Evaluate when is > ># %.

.>

> >

"'

# meansReplace with Multiply.

••>

%

% %ï44. < œ <# • <

Replace with < $. $ $ œ *•

45. Evaluate when is <= <$ $ = # and is .

and with .<=

< <

$ meansReplace with Multiply left to right.

• • •• • •

• •

•

= = =

# # #

# #

#

$ = #

$

'

"#

#%

îîï

46. = > œ =% • • • •= = = >Replace with and with = # > %.# '%• • • •# # # % œ

47. Evaluate when is $<= < $ = # and is .

and with .$<=

$ <

$

meansReplace with Multiply left to right.

• •• •

•

< =

#

#

$ = #

$

*

")

îî

48. '=>Replace with and with = # > %.' % œ• •# %)

49. Evaluate # # #= > = # > % when is and is .

means

.

# #

#= >

#%

# % %

% % %

) % %

%

"#)

• • • •

• • • •

• • •

• •

•

= = > >

# #

#

Replace with and with Multiply left to right.

= #

>

$#

îîï

ï50. % %<= œ % • • • • •< = = = =

Replace with < $ = # and with . % $ #• • • • •# # # œ "*#

51. Evaluate when is < = > <# & $ $ = # > %, is , and is ,using a calculator.

, with ,.< = >

<

>

Ð C

Ð*ÑÐ$#ÑÐ

# & $

B

Replace with and with Use the key.

Multiply left to

# & $

$ = #

%

$Ñ Ð#Ñ Ð %Ñðóóóóóñóóóóóò'%Ñðóóóñóóóò right.

ðóóñóóòÐ#))ÑÐ

'%Ñ

") %$#,

52. Use a calculator.< = >$ % #

Replace with , with , and with .< = # > $ %Ð '*"# $ % #$Ñ Ð#Ñ Ð %Ñ œ Ð#(ÑÐ"'ÑÐ"'Ñ œ

53. Evaluate & ( "!< = < $ = # when is and is , using acalculator. & (

&

"!< =

"! $Ñ

"!Ð #%$Ñ

Replace with < $ = # and with .

î íðóóñóóò

ðóóñóóò

Ð Ð#Ñ C

Ð"#)Ñ

#%$!Ð"#)Ñ

( BUse the key.

Multiply left to right.

$"" !%!,

54. Use a calculator. ' &&= >Replace with and with = # > %. ' & &Ð#Ñ Ð %Ñ œ &Ð'%ÑÐ "!#%Ñ œ $#( ')!,

55. Evaluate k k k kBC BCD B % C when is , is # D, and is .'

, and with .

k k k k

k k ¹ ¹k k

BC BCD

%

Replace with , with

Multiply left to rightwithin the absolutevalue signs.

B %

C

%

#

D '

# # '

)

ï î• • •

¹ ¹k k k k

ï

ï

) '

)

•

%)Evaluate the absolutevalues.

Add.) %)

&'

56. B k k k k k k k kC BD œ B C B# • •C DReplace with , with B % C # D ', and with .%

œ % % #%

œ $#

k k k k k k k k # # ' #%• • % œ % %


57. Evaluate when is D

D#

$C D' C # and is .

D D

C

Ð

$'

!Ð

#

#

#

$C D

'

#

'Ñ

$Ð #Ñ '

'Ñ œ ' ' œ $'

Replace with and with

Follow the order of operations.

Numerator: Denominator:

.

•$Ð #Ñ ' œ ' '

Undefined Division by is undefined.!

58. Replace with ,and with

, which is undefined.

C C

B #C B #œ

B %

C

œ% #

œ%

%

œ%

!

# •

•

•

•

C

C

#

# #

#

%

.

59. when is seconds.(a) Evaluate =

&= "&

Replace with .

Divide.

miles

=

&= "&

"&

&$

Evaluate (b)=

&= "!when is seconds.

Replace with .

Divide.

miles

=

&= "!

"!

&#

Evaluate (c)=

&= &when is seconds.

=

&= &

&

&"

Replace with .

Divide.

mile

60. (a) Using part (c) of Exercise , the distance&*covered in seconds is half of the distance#"

#

covered in seconds, or mile.& "#

Using part (a) of Exercise , the time to(b) &*cover miles is half the time to cover miles, or" $"

#

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

number halfway between seconds and & "!seconds.

Using parts (a) and (b) of Exercise , find the(c) &*number halfway between seconds and "! "&seconds; that is seconds."#"

#

2.2 Simplifying Expressions2.2 Section Exercises

1. and are the only like terms in the#, ,# #

expression. The variable parts match; both are .,#

The coefficients are and .# "

2. and are like terms. The coefficients are B #B "$ $

and .#

3. are the like terms in the expression.BC #BC and The variable parts match; both are . TheBCcoefficients are and ." #

4. he coefficients # #+ , $+ , and are like terms. Tare and . " $

5. and (ß $ß % are like terms. There are no variableparts; constants are considered like terms.

6. like terms.&ß "ß %and are There are no variableparts; constants are considered like terms.

7. These are like terms.Add the coefficients.

The variable part, ,stays the same.

'< '<

Ð' 'Ñ<

"#<<

8. %> "!> œ Ð% "!Ñ>

œ "%>

9. These are like terms.Rewrite as .Add the coefficients.


B &B

B "B

"B &B

Ð" &ÑB

'BB

# #

# #

# #

#

##

10. *C C œ *C "C

œ Ð* "ÑC

œ "!C

$ $ $ $

$

$

11. These are like terms.Rewrite as .Change subtraction toadding the Add the coefficients.

: &:

: ":

": &:

":

Ð"

opposite.

&:

&Ñ:

%:The variable part, ,stays the same.

:

12. 8 $8 œ "8

œ Ð"

œ

$8

$Ñ8

#8

2.2 Simplifying Expressions 41

13. These are like terms.Rewrite as .Change subtraction toadding the Add the coefficien

$ $

$ $

$ $

#+ +

#+ "+

#+ "+

+ "+$ $

opposite.ts.


Ð

+

$

$

# "Ñ+

$+$

14. # # # #

# #

#

#

"!B B œ "!B "B

"!B "B

"! "ÑB

""B

œ

œ Ð

œ

15. Any number minus itself is .

î- -

! !

16. Any number minus itself is .

ï, ,

! !

# #

17. These are like terms.Rewrite as .Change subtraction toadding the Add the c

*BC BC *BC

BC "BC

*BC "BC *BC

*BC "BC

opposite.*BC oefficients.

or The variable part, ,stays the same.

Ð* "

"BC BCBC

*ÑBC

18.

or

< = (< = (< = œ "< =

œ Ð"

œ "< = < =

# # # #

# #

# #

#

(< = (< =

( (Ñ< =

19. These are like terms.Change subtraction toadding the Add the coefficients.

The

&> (> '>

&> (>

Ð& (

'>

% % %

% %

%

opposite. %

%

'>

'Ñ>

variable part, ,stays the same.

>%

20. "!78 *78 $78 œ "!78

œ Ð"!

œ %78

*78 $78

* $Ñ78

21. These are like terms.Write in the under-stood coefficientof .Add the coefficients.

C C C C

"C "C "C "C

"

Ð" " " "ÑC

%C

# # # #

# # # #

#

##The variable part, ,

stays the same.C

22. + + + œ "+ "+ "+

œ Ð" " "Ñ+

œ $+

23.

B 'B B

B "B B "B

"B 'B "B

"B 'B "B

These are like terms. Rewrite

Change subtraction toadding the Add

as and as .

opposite. the coefficients.


Ð

B

" ' "ÑB

)B

24.

C C $C œ "C "C $C

"C "C $C

" " $ÑC

&C

œ

œ Ð

œ

25. )+ %, %+

)+ %+ %,

Use the commutative propertyto rewrite the expression so thatlike terms are next to each other.Add the coefficients of like terms.


Ð) %Ñ+ %,

"#+ %,+

26. 'B &C %C œ 'B Ð& %ÑC

œ 'B *C

27. Use the commutative propertyto put the constants at the end.

The onl

' ) (<=

(<= ' )

(<= "%

Add the coefficients of like terms.y like terms are constants.

28. "! #- "& œ #- "! "&

œ #- #&

# #

#

29. Write in the understoodcoefficient of .

+ +, +,

"

"+ "+, "+,

"+ Ð" "Ñ+,

"+#+,

# #

# #

#

#

Add the coefficients oflike terms.


+,#

or+ #+,#

30.

or

8 78 8 œ "8 "78 "8

œ "78 "8 "8

œ "78 #8 78 #8


31. 'B C )B C

"

'B "C

Write in the understoodcoefficients of .Change subtraction toadding the opposite.Rewrite using theco

)B "C mmutative property.Add the coefficients oflike terms.'B

Ð' B Ð" "Ñ C

)B "C "C

)Ñ

#B #C

#B #C

ðñò ðñò

32. . $- (- $. œ ". $-

œ $-

œ Ð$

œ

(- $.

(- ". $.

(Ñ- Ð" $Ñ.

%- %.

33. ), + , +

"

), "+ ", "+

),

# # # #

# # # #

#

Write in the understoodcoefficient of .Change subtraction toadding the opposite.

# #"+ ", "+

", "+ "+

" "Ñ

#

# # #

Rewrite using thecommutative property.Add the coefficients oflike terms.),

Ð) "Ñ, Ð +

(, !

(, !

#

# #

#

#

ðñò ðñò

• +#

(,#

34. &+, +, $+ , %+, œ &+,

œ &+,

œ Ð&

œ !+, $+ ,

œ $+ ,

#

#

#

#

#

#

"+, $+ , %+,

"+, %+, $+ ,

" %Ñ+, $+ ,

35. $ #B $B $B #There are no like terms. The expression cannot besimplified.

36. + , #+, +, $+ ,# $ $

There are no like terms. The expression cannot besimplified.

37.

*< '> = &< = > '> &= <

*< '> "= &< "

Write in the understood coefficient of .Change subtraction to adding the opposite.

"

= "> '> &= "<

*< &< "< "= "= &= '> "> '>

Rewrite using the commutative property.

Add the coefficients of like terms.ðóóóóóñóóóóóò ðóóóñóóóò ðóóóñóóóòÐ < Ð = Ð' "

* & "Ñ " " &Ñ 'Ñ>

"&< &= ">

"&< &= >

38.

B $C %D B D &C )B C

"B $C %D "B "D &C )B "C

"B "B )B $C &C "C %D "D

" " )ÑB Ð $ & "ÑC Ð% "ÑD

)B "C $D

œ

œ

œ Ð

œ or )B C $D

39. By using the associative property, we can write$Ð"!+Ñ as

Ð$ • • •"!Ñ + œ $! + œ $!+.

So, simplifies to .$Ð"!+Ñ $!+

40. )Ð%,Ñ œ Ð)

œ $#,

• %Ñ,

41. By using the associative property, we can write #%Ð#B Ñ as

Ð % ) )B #• • •#Ñ B œ B œ# # .

So, # #%Ð#B Ñ )B simplifies to .

42. $

$

(Ð$, Ñ œ Ð (

#",

• $Ñ,$

œ

43. By using the associative property, we can write&Ð $%C Ñ as

Ð& %Ñ #! #!C• • • $C œ C œ$ $ .

So, simplifies to .&Ð %C Ñ #!C $ $

44. #Ð

œ

'BÑ œ Ð# 'ÑB

"#B

•

45. By using the associative property, we can write *Ð #-.Ñ as

Ð * #Ñ • • • • •- . œ ") - . œ ")-..

So, . *Ð #-.Ñ simplifies to ")-.

46. 'Ð %<=Ñ œ Ð ' %Ñ<=•œ #%<=

47. By using the associative property, we can write(Ð$+ ,-Ñ# as

Ð( • • • • • • •$Ñ + , - œ #" + , - œ #"+ ,-# # # .

So, .(Ð$+ ,-Ñ #"+ ,-# # simplifies to

48. %Ð#BC D Ñ œ Ð%

œ )BC D

# #

# #

• #ÑBC D# #

49.

"#Ð AÑ

"#Ð "AÑ

"# "ÑA

Write in the understoodcoefficient of .Rewrite using theassociative property.

"

Ð

"#

"#A

••A

2.2 Simplifying Expressions 43

50.

"!Ð 5Ñ œ "!Ð "5Ñ

"! "Ñ5œ Ð

œ "!5

•

51. Distributive property'Ð, 'Ñ

'

', $'

• •, ' '

52. &Ð+ $Ñ œ &

œ &+ "&

• •+ & $

53. Distributive property(ÐB "Ñ

(

(B (

• •B ( "

54. %ÐC %Ñ œ %

œ %C "'

• •C % %

55. Distributive property$Ð(> "Ñ

$

#"> $

• •(> $ "

56. )Ð#- &Ñ œ )

œ "'- %!

• •#- ) &

57. Distributive property

Change addition to subtractionof the opposite.

#Ð&< $Ñ

# #

"!< '

"!< '

• •&< $

58.

or

&Ð'D #Ñ œ & &

$!D "!

$!D "!

• •'D #

œ


Change addition to subtractionof the opposite.

*Ð5 %Ñ

* *

*5 $'

*5 $'

• •5 %

60.

or

$Ð: (Ñ œ $ $

$: #"

$: #"

• •: (

œ

61. Distributive property&!Ð7 'Ñ

&!

&!7 $!!

• •7 &! '

62. #&Ð8 "Ñ œ #&

œ #&8 #&

• •8 #& "


Combine like terms.

"! #Ð%C $Ñ

"! #

"! )C '

)C "! '

)C "'

• •%C # $

Rewrite using thecommutative property.

64. % (ÐB $Ñ œ % (

œ % (B #"

œ (B #&

#

#

#

• •B ( $#


Combine like terms.

'Ð+ #Ñ "&

'

'+ "# "&

'+ $

#

#

#

• •+ ' # "&#

66. &Ð, %Ñ #& œ &

œ &, #! #&

œ &,

œ &, &

• •, & % #&

#! #&


Change subtraction toadding the opposite.

# *Ð7 %Ñ

# *

# *7 $'

# *7

• •7 * %

$'Rewrite using thecommutative property.

*7 #

*7

*7 $%

$'

$%

Add the coefficientsof like terms.Change addition to subtractionof the opposite.

68.

or

' $Ð8 )Ñ œ ' $

œ ' $8 #%

œ ' $8

œ $8 $8 ")

• •8 $ )

#%

")


Add the coefficientso

&Ð5 &Ñ &5

& &

&5 #& &5

&5 &5 #&

• •5 & &5

Rewrite using thecommutative property.

f like terms.

Zero times anynumber is .Zero added to any numberis the number

ðñò

ðñò

Ð 5

!5 !

!

& &Ñ #&

#&

#&

#&

70.

(Ð: #Ñ (: œ ( (

(: "% (:

"%

"%

"%

• •: # (:

œ

œ !:

œ !

œ



Change subtraction toadding the opposite.Combine like terms.

%Ð'B $Ñ "#

%

#%B "# "#

#%B

• •'B % $ "#

"# "# Anynumber plus its oppositeis .!

#%B !

#%B

72. 'Ð$C $Ñ ") œ '

œ ")C ") ")

œ ")C

œ ")C

• •$C ' $ ")

") ")

73. Distributive propertyRewrite as .Change subtraction toadding the opposite.

& #Ð$8 %Ñ 8

& # 8 "8

& '8 ) "8

& '8 )

• •$8 # % 8

"8Rewrite using thecommutative property.

& ) '8

Ð& )Ñ Ð'

"$ &8 &8 "$

"8

"Ñ8

Add the coefficientsof like terms.

or

74. ) )Ð%D &Ñ D œ ) )

œ ) $#D %!

œ $"D %)

• •%D ) & "D"D


Rewrite Change subtraction toadding the opposite.

: 'Ð#: "Ñ &

: '

: "#: ' &

: ":

": "#

• •#: ' " &

as .

: ' &

" "#Ñ: Ð ' &Ñ

"

Add the coefficientsof like terms.

Change addition tosubtraction of theopposite.

Ð

"":

"": "

76.

or

5 $Ð%5 "Ñ # œ "5 $

"5 "#5 $ #

"5 "#5 $ #

"

• •%5 $ " #

œ

œ

œ ""5 ""5 "

77. A simplified expression still has variables, but iswritten in a simpler way. When evaluating anexpression, the variables are all replaced byspecific numbers and the final result is a numericalanswer.

78. &Ð$B #Ñ &Ð# $BÑ

œ & œ &

œ "&B "! œ "! "&B

• • • •$B & # # & $B

The answers are equivalent because of thecommutative property of addition.

79. Like terms have matching variable parts, that is,matching letters and exponents. The coefficientsdo not have to match. Examples will vary.Possible examples: In 'B * B, the terms #'B B %5 $ )5 "! and are like terms. In ,the terms and are like terms.$ "!

80. Add the coefficients of like terms. If no coefficientis shown, it is assumed to be . Keep the variable"part the same. Examples will vary.#B (B )81. ðóñóò

&B )

Keep the variable part unchanged when combininglike terms.

82. Do not change the sign of the first term. Thecorrect answer is %+ &.


Change subtractionto adding the opposite.G

%Ð$CÑ & #Ð&C (Ñ

%

"#C & "!C "%

"#C & "!C "%

• • •$C & # &C # (

roup like termsand add thecoefficients.

ðóóñóóò ðñò

"#C "!C & "%

#C *

#C *

84. 'Ð

œ

œ

$BÑ * $Ð #B 'Ñ œ ")B * $ #B $

")B * 'B ")

#%B *

• • '


Change subtraction to adding the opposite.

"! %Ð $, $Ñ #Ð', "Ñ

"! % $, %

"! "#, "# "#, #

• • • •$ # ', # "

"! "#, "# "#, #

"#, "#, "! "# #

Group like terms and add the coefficients.ðóóñóóò ðóóóóñóóóóò

!, !

!

86. "# #Ð%+ %Ñ %Ð

œ "# #

œ "# )+ )

œ "# )+

œ !

#+ "Ñ

#+ %

)+ %

) )+ %

• • • •%+ # % % "

Summary Exercises on Variables and Expressions 45


Change subtraction to adding

&Ð B #Ñ )Ð BÑ $Ð #B #Ñ "'

& B & B $ #B $

"! )B 'B ' "'

• • • • •# ) # "'

&B

the opposite.

Group like terms and add the coefficients.&B

&B

"! )B 'B ' "'

)B 'B "! ' "'

*B !

*B

ðóóóóóñóóóóóò ðóóóóñóóóóò

88.

(Ð CÑ 'ÐC "Ñ $Ð #CÑ ' C

'C ' C

' 'C ' "C

œ (C 'C '

œ (C 'C

œ 'C

Summary Exercises on Variables andExpressions

1. Expression (rule) for finding the perimeter of anoctagon of side length : = )=

(a) Evaluate the expression when , the side=length, is yards.%

)= = %

)

$#


yards is the perimeter.í• %

(b) Evaluate the expression when , the side=length, is inches."&

)= = "&

)

"#!


inches is the perimeter.î• "&

2. Expression (rule) for finding the total cost of a carwith down payment , monthly payment , and. 7number of payments : > . 7>

(a) Evaluate the expression when the downpayment is $ , the monthly payment is $ ,$!!! #)!and the number of payments is .$'

Replace with $ , with $ , and with .Multiply before adding.

. 7>. $!!! 7

#)! > $'ðóóóóóóñóóóóóóòðóóóóóóñóóóóóóò$ $

$ $

$!!! #)!

$!!! "

• $'

! !)!,

$ , is the total cost of the car."$ !)!

Evaluate the expression when the down(b)payment is $ , the monthly payment is $ ,"(&! %#*and the number of payments is .%)

Replace with $ , with $ , and with .Multiply before adding.

. 7>. "(&! 7

%#* > %)ðóóóóóóñóóóóóóòðóóóóóóñóóóóóóò$ $

$ $

"(&! %#*

"(&! #

• %)

! &*#,

$ , is the total cost of the car.## $%#

3. ABC

Replace with , with A & B # C ', and with .

Multiply left to right.îðñò& • •

•

# '

"! '

'!

4. B œ B$ • •B B

Replace with B #.

Multiply left to right.ïî

# # #

#

)

• •

•%

5. %AC

Replace with and with .A & C '

Multiply left to right.îðñò

% '

#! '

• •

•

&

"#!

6. $BC œ $# • • •B C C

Replace with B # C ' and with .

Multiply left to right.îï

ï$

$'

• • •

• •

•

# ' '

' ' '

'

#"'

7. A B œ A# & • • • • • •A B B B B B

Replace with and with A & B #.

Multiply left to right.íïðñòðñòðóñóòðñò

&

#&

"!!

%!!

• • • • • •

• • • • •

• • • •

• • •

• •

•

&

# # # # #

# # # # #

&! # # # #

# # #

#!! # #

#

)!!


8. % $ (AB C œ ( • • • • • • • •A B B B B C C C

Replace with , with A & B # C ', and with .

îMultiply left to right.ðñò

ïðóñóò

ðñò

( # # # # ' ' '

$& # # # # ' ' '

# # # ' ' '

"%! # # ' ' '

• • • • • • • •

• • • • • • •

• • • • • •

• • • • •

&

(!

#)! • • • •

• • •

• •

•

# ' ' '

&'! ' ' '

' '

#! "'! '

"#! *'!

ðóñóòðóñóòðóóóñóóóò$$'!

,

,

9. "!, %, "!, œ Ð"! % "!Ñ,

œ #%,

10.

$B & "# "!B œ $B "!B & "#

$ "!ÑB Ð & "#Ñœ Ð

œ (B (

11.

or

)Ð- %Ñ œ ) )

)- $#

)- $#

• •- %

œ

12. *BC *BC œ Ð * *ÑBC

œ !BC

œ !

13. # %Ð $- .Ñ œ Ð % $Ñ• • - .#

œ "#- .#

14. $0 &0 %0 œ $0

œ Ð$

œ

&0 %0

& %Ñ0

'0

15. #Ð$A %Ñ œ #

œ Ð#

œ 'A )

• •• •$A # %

$ÑA # %

16.

or

+ ', + œ + ', +

+ + ',

"+ "+ ',

" "Ñ ',

#+ ',

#+ ',

œ

œ

œ Ð

œ

• +

17. $ # "! &B C œ Ð "! &Ñˆ ‰ • •B C$ #

œ &!B C$ #

18. &< #< #< &<

œ &< &<

œ &< Ð&

œ &< $< #<

# # $

$ #

$

$ #

#

#

#< #<

#Ñ< #<

19. #" ( 2 $ œ #" (

œ #" (2 #"

œ (2

ˆ ‰#

#

#

• •2 ( $#

20.

$Ð7 $Ñ $7 œ $ $

$7 $7 *

$ $Ñ 7 *

*

*

*

• •

•

7 $ $7

œ

œ Ð

œ !7

œ !

œ

21.

%Ð)C &Ñ & œ % %

% #! &

$#C #! &

$#C #&

• •• •)C & &

)Ñ C œ Ð

œ

œ

22.

or

# "#Ð$B "Ñ œ # "#

œ # Ð"#

œ #

œ

$'B "!

• •• •$B "# "

$Ñ B "#

"# $'B

"! $'B

23.

8 &Ð%8 #Ñ "" œ 8 &

8 Ð&

8 #!8 "! ""

" #!Ñ

• •• •

•

%8 & # ""

%Ñ 8 "! ""

8 "

œ

œ

œ Ð

œ "*8 "

24.

or

#

#

' $Ð#2 %Ñ "!2 & 2 #

' $ "!2 &

' '2 "# "!2 &2 "!

"!Ñ ' "# "!Ñ

%2 %

ˆ ‰œ

œ

œ &2 Ð'

œ &2

&2 %2 %

• • • •

•

#2 $ % 2 & #

2 Ð

#

#

#

#

2.3 Solving Equations Using Addition2.3 Section Exercises

1. 8 &! œ )

&) &! 8 &)

) œ )

&)

Given equation

Replace with .

Yes, is the

œ )

%# '!

?

solution. (No need to check and .)

2. Given equation< #! œ &Replace with and .< "&ß $!ß #&

"& #! & $! #! & #& #! &

"& "! Á & & œ &

œ œ œ

œ

? ? ?

?

#! &

& Á &

is the solution.#&

2.3 Solving Equations Using Addition 47

3. Given equationReplace with .

No, is not the solution.

Yes, is

'

' % "! %'

%

"' "! "' '

"

œ C "!

CÁ '

'œ

'

œ

œ C '

?

? Replace with .

the solution.

4. Given equation% œ B "$Replace with and .B "(ß ß "( *

% "( "$ % "( "$ % * "$% Á $! % œ % % Á %

œ œ œ? ? ?

"( is the solution.

5. Given equationReplace with .

No, is not the solution.

Yes, is the

> "# œ !

! "# > !"# Á !

!

"#! œ !

œ !

œ ! >

?

? Replace with .

sol

"# "#

"# ution.

6. Given equation

Yes, is the

, ) œ !) ) œ

! œ !)

! , )Replace with .

solution.

7. Add the opposite of , to both sides.

: & œ *

& &

: ! œ %: œ % %

The solution is .

& &,

Check: : & œ * : %% & œ *

* œ *

Replace with .

Balances

8. + $ œ "#$ $

+ ! œ *+ œ * *

Add

The solution is .

$ to both sides.

Replace with .

Balances

Check: + $ œ "# + ** $ œ "#

"# œ "#

9. Add the opposite.Add the opposite of , to both sides.

) œ < #

) œ <

# #

"! œ < !

"! œ < "!

# # #,

The solution is .

Replace with .

Balances

Check: ) œ < # < "!

) œ "! #

) œ )

10. Add the opposite.$ œ , &$ œ , && &

) œ , !) œ , )

& Add

The solution is .

to both sides.

Replace with .

Balances

Check: $ œ , & , )$ œ ) &$ œ $


& $

$ $

)) )

œ 8 $

œ 8 !œ 8

$ ,

.The solution is

Check:

& œ 8 $ )& œ ) $& œ &

Replace with

Balances

8 .

12.

" )) )

**

œ + )

œ + !œ + *

Add

The solution is .

to both sides.

Check:

" œ + ) *" œ * )" œ "

Replace with

Balances

+ .


% 5 %

)

œ "%

% %

! 5 œ ")5 œ ") "

%,

.The solution is

Check:

% 5 œ "%% ") œ "%

Replace with .

Balances

5 ")

"% œ "%

14. * C œ ( ** *

! C œ "'C œ "' "'

Add

The solution is .

to both sides.

Check:

* C œ (

* "' œ (Replace with .

Balances

C "'

( œ (

15. Add the opposite.Add the opposite of , to both sides.

C ' œ !C œ !

' '

C ! œ 'C œ ' '

' ' ',

The solution is .

Check: C ' œ ! C '' ' œ !

' ! œ !

Replace with .Add the opposite.

Balances' œ !


16. Add the opposite.5 "& œ !

5 œ ! "&

"& "&

5 ! œ "&

5 œ "& "&

"& Add

The solution is .

to both sides.

Check: 5 "& œ ! 5 "&

"& "& œ !

! œ !

Replace with .

Balances


( œ < "$

œ < !

œ <

"$

"$

"$ "$

'

' '

,

.The solution is

Check: ( œ < "$ <

( œ

( œ (

Replace with

Balances

'

' "$

.

18. "# œ D "* "*

œ D !

œ D (

Add

The solution is .

to both sides."* "*

(

(

Check: "# œ D "* D

"# œ

"# œ "#

Replace with

Balances

(

( "*

.

19. Add the opposite.Add the opposite of ,

to both sides.

B "# œ

B œ

"# "#

B ! œ ""

B œ "" ""

"

"# " "#

"#,

The solution is .

Check: B "# œ B ""

"" "# œ

""

"

"

"# œ "

" œ "


Balances

20. Add the opposite.7 $ œ

7 œ $

7 ! œ

7 œ '

*

$ *

$ $

'

'

Add

The solution is .

to both sides.

Check: 7 $ œ 7

* '

' $ œ *

' $ œ *

* œ *

Replace with .

Balances

21. Add the opposite of ,, to both sides.

& # > #

$

$ $

œ

# #

œ ! >

œ >

#

The solution is .

Check:

& œ # > $

& œ # $

& œ &

Replace with

Balances

> .

22. " "! Aœ "!

"! "!

* œ ! A

* œ A *

Add

The solution is .

to both sides.

Check:

" œ "! A

" œ "! *

" œ "

Replace with

Balances

A *.

23. The given solution is #.

Replace with Add the opposite.

Does not balance

Check: D & œ $ D

#

# & œ $

# & œ $

( Á $

.

Correct solution:

Add the opposite.Add the opposite of ,

to both sides.

D & œ $

D œ $

& &

D ! œ )

D œ ) )

& &

&,

The solution is .

Check: D & œ $ D )

)

)

$

Replace with Add the opposite.

Balances

. & œ $

& œ $

œ $

24. The given solution is "$.

Replace with

Balances

Check: B * œ % B

"$ *

%

"$

œ %

œ %

.

"$ is the correct solution.

25. The given solution is ").

Replace with

Balances

Check: ( B œ B

(

""

")

") œ ""

"" œ ""

.

") is the correct solution.

26. The given solution is &.

.Check: # 5 œ 5

#

( Replace with

Does not balance

&

& œ (

$ Á (

Correct solution:

Add the opposite of , to both sides.

# 5 œ

! 5 œ

5 œ

(

#

# #

*

* *

#

,

.The correct solution is


Check: # 5 œ 5

#

( Replace with

Balances

*

* œ (

( œ (

.

27. The given solution is ."!

Replace with .

Does not balance

Check: "! œ "! , , "!

"! œ "! "!

"! Á !

Correct solution:


"! "! , "!

"! "!

œ

! œ ! ,

! œ , !

"!,

.The solution is

Check:

"! "! ,

"! "! !

"! "!

œ , !

œ

œ

Replace with

Balances

.

28. The given solution is .!

Replace with .

Does not balance

Check: ! œ + !

!

!

"% +

œ "% !

Á "%

Correct solution:


! "% + "%

"% "%

œ

"% œ ! +

"% œ + "%

"%,

.The correct solution is

Check: ! "% +

! "% "%

! !

œ + "%

œ

œ

Replace with

Balances

.

29. Simplify the right side.Add the opposite.Add to both sides.

The solution is .

- % œ

- % œ #

- œ # %

% %

- ! œ '

- œ ' '

) "!

%

Replace with .

Balances

Check: - % œ - '

' % œ

# œ #

) "!

) "!

30. , ) œ

, ) œ %

) )

, ! œ "#

, œ "#

"! '

The solution is ."#

Balances

Check: , ) œ

"# ) œ

% œ %

"! '

"! '

31. Simplify the left side.Add the opposite.Add to both sides.

The solution is .

" %

#

œ C #

$ œ C #

$ œ C #

# #

& œ C !

& œ C &


Balances

Check:

"

" %

% œ C # C &

œ & #

$ œ $

" % œ & #

32. # $ œ 5 %

& œ 5 %

% %

* œ 5 !

* œ 5 The solution is .*

Balances

Check: # $ œ 5 %

# $ œ * %

& œ &

33. Add the opposite.Add.Add to both sides.

The solution is .

"! , œ

"! , œ

"! , œ #!

! , œ

, œ

"% '

"% '

"!

"! "!

$!

$! $!

Check:"! , œ ,

"! œ

#! œ #!

"% ' $!

$! "% '

Replace with .

Balances

34. " A œ

" A œ

" A œ

! A œ

A œ

) )

) )

"'

" "

"(

"( The solution is ."(

Balances

Check: " A œ

" œ

œ

) )

"( ) )

"' "'

35. Add the opposites.Simplify the right side.Add to both sides.

The solution is .

> # œ $ &

> œ $

> œ #

# #

> ! œ !

> œ ! !

# &

# #

Replace with .Add the opposites.

Balances

Check: > # œ $ & > !

! # œ $ &

!

# œ $ &

# œ #


36. : ) œ

: ) œ

: œ

) )

: ! œ

: œ

"! #

)

) )

!

! The solution is .!

Balances

Check: : ) œ

! ) œ

œ

"! #

"! #

) )

37. Add the opposite.Combine like terms.

is the same as .The solution is

"!D *D œ

"!D œ

"D œ "D D

D œ

"& )

*D "& )

(

( (.

Check:"!D *D œ D

"!

"& ) (

( * ( œ "& )

(! '$ œ (

(! '$ œ (

( œ (

Replace with

Add the opposite.

Balances

.• •

38. #< < œ & "!

#<

"< œ

< œ

"< œ & "!

&

& The solution is .&

Balances

Check: #< < œ & "!

# •

& & œ & "!

"! & œ & "!

& œ &

39. Rearrange andcombine like terms.

Add .

The solution is .

&A # 'A % *

&A 'A % *

#

œ

# œ

"A # œ &

# #

"A ! œ $

A œ $ $

ðóñóò ï to both sides

Check:

&A # 'A % *

& % *

"& # ") œ % *

œ &

œ A $

œ

&

Replace with .

Balances

• •$ # ' $

40.

#> % $>

(

"

% %

&

&

œ ' (

"> % œ '

> % œ

> ! œ

> œ The solution is .&

Check:

#> % $> œ ' (

#Ð &Ñ % $Ð &Ñ œ ' (

"& œ ' (

"& œ "

" œ "

"! %

"%

Balances

41.

$ $ % $B %B

$ $ $B %B

' %

% %

"!

œ

œ %

œ % "B

œ ! "B

œ "

Add the opposites.Combine like terms.

Add .ðñò ðóñóò

to both sides

"! B "B B

œ B

is the same as .The solution is . "! "!

42.

& & # ', (,

& & # ', (,

"! # ",

"! # ,

# #

)

)

œ

œ

œ

œ

œ ! ,

œ , The solution is .)

43. Add the opposite.

$ ( % œ #+ $+

$ ( % œ #+ $+ Combine like terms.The solution is .! œ + !

44. ' "" & œ

'

! œ "-

! œ -

)- *-

"" & œ )- *-

The solution is .!

45. C (& œ

C œ (&

(& (&

C ! œ

C œ

"!!

(& "!!

#&

#&

Add the opposite.Add .

The solution is .

to both sides

#&

46. + #!! œ

+ œ

#!! #!!

+ ! œ "!!

+ œ "!!

"!!

#!! "!!

The solution is 1 .!!

47. Rearrange andcombine like terms.

B $ #B ")

") $

$ $

&

&

œ

B $ œ

B ! œ "

B œ " "

Add .

The solution is .

to both sides

&

48.

= #= % "$

"= #= %

%

œ

œ "$

"= "$

% %

"= ! œ "(

= œ "( The solution is ."(


49. )# œ

""$ œ ! 5

""$ œ 5 ""$

$" 5 $"

$" $"

Add .

The solution is .

to both sides

50.

& œ (# A

(# (#

œ A

(( The solution is .((

51. Add the opposite.Rearrange andcombine like terms.Add to both sides.

The solut

# ""

# "" * ,

*

œ #, * ,

œ #,

* œ , *

* *

") œ , !

") œ , ion is .")

52.

' (

' ( " "2

"

œ #2 " 2

œ #2

" œ "2

" "

# œ "2 !

# œ 2 The solution is .#

53. Add the opposites.Combine like terms.Add to both sides.

The solution is .

< ' œ ( "! )

< œ (

< œ '

' '

< ! œ

< œ

' "! )

' ""

&

& &

54. 7 œ # * "

7 œ #

7 œ

& &

7 ! œ

7 œ

&

& * "

& '

"

"

The solution is ."

55. Add to both sides.

The solution is .

"%

*" *"

"!&

"!& "!&

œ 8 *" *"

œ 8 !

œ 8

56. '' œ B #)

'' œ B

*% œ B !

*% œ B

#)

#) #)

The solution is .*%

57. Combine like terms.Add to both sides.

The solution is .

* *

&

& &

&

& &

œ & 2

! œ & 2

œ ! 2

œ 2

58. ") ") œ ' :

! œ ' :

œ ! :

œ :

' '

'

' The solution is .'

59. No, the solution is "%, the number used toreplace in the original equation.B

60.

Does not balance

Check:

$ ' œ 8 &

$ ' œ # &

* Á (

To correct the errors, change $ ' $ ' to .Then, add to both sides, not . The correct& &

solution is .%

61. Add the opposite of,

1 œ $!&

"!

1 ! œ #*&

1 œ #*&

"!

"!

, to both sides.10 10

There were graduates this year.#*&

62. 1 œ #()

1 ! œ #')

1 œ #')

"! 10 10

There were graduates last year.#')

63. Add .*# œ - $(

&& œ - !

&& œ -

$(

$( $(

to both sides

When the temperature is degrees, a field cricket*#chirps times (in seconds).&& "&

64. (( œ - $(

%! œ - !

%! œ -

$( $(

When the temperature is degrees, a field cricket((chirps times (in seconds).%! "&

65. Add the opposite.Add .

: '& œ %&

: œ %& '&

'& '&

: ! œ ""!

: œ ""!

'& to both sides

Ernesto's parking fees average $ per month in""!winter.


66. Add .

: &' œ *)

&' &' &'

: ! œ "&%

: œ "&%

to both sides

Aimee's parking fees average $ per month in"&%winter.

67. "( " #' $) œ $ 7 ) #7Change all subtractions to adding the opposite.Write the understood coefficient of ." "( " #' $) œ $ "7 ) #7Use the commutative property to group like termson the right side. "( " #' $) œ $ ) "7 #7Combine like terms on each side. $! œ "" "7To get by itself, add the opposite of , , to7 "" ""

both sides.

$! "" "7

"*

"* "*

œ

"" ""

œ "7

œ 7 The solution is .

68.

The solution is .

"* $) * "" œ ' #> '

"* œ

œ ">

"# "#

œ "> !

œ >

>

$) * "" "> ' #> '

"( "#

&

& &

69. 'B #B ' &B œ ' &k k k k! * Change subtraction within absolute value toadding the opposite and rearrange the terms. 'B #B &B ' œ * ' &k k k k! Simplify inside absolute value signs. Collect liketerms.B ' œ k k k k * "Evaluate absolute values.B ' œ * "Change subtraction to adding the opposite.B ' œ *

B œ ) '

' '

B ! œ #

B œ # #

"

' Add .

The solution is

to both sides

.

70.

2 * * "# & !

2 * * '2 "# &

"2 ") '2 "# &

"2 ") )2 '2 "# &

"(

") "(

k k k kk k k kk k )2 '2

)2

)2

œ

œ

œ

œ

"2 ") œ

"2 œ

") ")

2 ! œ "

2 œ "

The solution is ."

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

The solution is .

8 " œ

8 œ "

" "

8 ! œ #

8 œ #

$

" $

Add the opposite.Add . to both sides

#

Add the opposite of) œ B "!

"!ß

"! "!

œ B !

œ B

"!

#

#

, to both sides.

The solution is .#

(b) Equations will vary. Some possibilities are:

The solution is .

C ' œ '

'ß

C ! œ !

C œ !

Add the opposite of

'

' '

, to both sides.

!

Add the opposite of,

& &

&

œ ,

ß &

& &

! œ ! ,

! œ ,

to both sides.

The solution is .!

72. Add the opposite of(a)

The solution is .

B " œ "

"ß

B ! œ

B œ

"#

"#"#

"

" "

, to both sides.

"#

The solution is .

(b) "%

"%

"%" &% %

œ C "

œ C "ß

" "

" œ C !

" œ C C œ

Add the opposite.Add the opposite of

or

"

", to both sides.

&%

The solution is .

(c) $ $ Add the opposite of$ $ ,

$ $$ $

$ $

#Þ&! 8 œ $Þ$&

#Þ&!ß #Þ&!

#Þ&! #Þ&!

! 8 œ !Þ)&

8 œ !Þ)&

to both sides.

!Þ)&

(d) Equations will vary. Some possibilities are:

The solution is .B # œ " $$ $& &

+ $ $ The solution is $ .(Þ$# œ *Þ"' "'Þ%)&- ""Þ#! œ %- #Þ!! *Þ#!$ $ The solution is $ .

2.4 Solving Equations Using Division 53

2.4 Solving Equations Using Division2.4 Section Exercises

1. 'D œ "#'D "#

' 'œ

D œ #

Divide

The

both sides by .

solution is .

'

#

Balances

Check: Replace with .'D œ "#

'

"# œ "#

D #

• # œ "#

2.

Balances

)5 œ #% )5 œ #%

)5 #%

) )œ

5 œ $

)

#% œ #%

Check:• $ œ #%

The solution is .$

3. %) œ "#<%) "#<

"# "#œ

% œ <

Divide sides by .

The solution is .

both "#

%

Balances

Check: Replace with .%) œ "#<

%) œ "#

%) œ %)

< %

• %

4.

Balances

** œ ""7 ** œ ""7** ""7

"" ""œ

* œ 7

** œ ""

** œ **

Check:• *

The solution is .*

5. $C œ !$C !

$ $œ

C œ !

Divide sides by .

The solution is .

both $

!

Replace with .

Balances

Check: $C œ ! C !

$

! œ !

• ! œ !

6.

Balances

&+ œ ! &+ œ !&+ !

& &œ

+ œ !

&

! œ !

Check:• ! œ !

The solution is .!

7.

(5 œ (! (

(5

( ("! "!

Divide sides by .

The solution is .

both

œ(!

5 œ

Replace with .

Balances

Check:

(5 œ (! "!

( "! œ (!

5

(! œ (!

•

8.

Balances

'C œ $' 'C œ $''C

' ''

' ' œ $'œ

$'

C œ$' œ $'

Check:•

The solution is .'

9.

&% œ *< *&% *<

* *

Divide sides by .

The solution is .

both

œ

' œ < '

Replace with .

Balances

Check:

&% œ *<

&% œ *

&% œ &%

< '

• '

10.

Balances

$' œ %: $' œ %:$' %:

% %

$' œ %

$' œ $'œ

* œ :

Check:• *

The solution is .*

11.

#& œ &,

#&

& œ , &

Divide sides by .

The solution is .

both &

& &œ

&,

Replace with .

Balances

Check:

#& œ &,

#& œ & &

#& œ #&

, &

•

12.

Balances

(! œ "!B (! œ "!B(!

( œ B

(! œ "! (

(! œ (!"! "!œ

"!BCheck:

•

The solution is .(

13. Combine like terms.#< œ

#< œ '#< '

# #œ

< œ $

( "$

Divide sides by .

The solution is .

both #

$

Replace with .

Balances

Check: #< œ < $

#

' œ '

( "$

• $ œ ( "$

14.

Balances

'C œ #) % 'C œ #) %

'C œ #% ''C #%

' 'œ

C œ %

#% œ #%

Check:• % œ #) %

The solution is .%


15. Add the opposite.Rewrite as Combine like terms.

"# œ &: :

"# œ &: : ":

"# œ &: ":

"# œ %:"#

$ œ :

:

% %œ

%:

.

Divide sides by .both %

The solution is .$

Replace with .

Add the opposite.

Balances

Check:

"# œ &: : $

"# œ & $ $

"# œ "& $

"# œ "& $

"# œ "#

:

•

16.

Balances

#! œ D ""D #! œ D ""D

#! œ "D #! œ

#! œ #! œ#!

œ#! œ

#! œ #!

""D # "" #

"!D # ##

"! "!

"!D

# œ D

# ##

Check:•

The solution is .#

17. Add the opposite.Combine like terms.

$ #) œ &+

$

& &œ

&+

#) œ &+#&

& œ + &

Divide sides by .

The solution is .

both &

18.

&& ( œ )8

%) œ )8%)

' œ 8 '

Divide sides by .

The solution is .

both )

) )œ

)8

19. Add the opposite.Rewrite as .Combine like terms.

B *B œ )!

B B "B

"B

œ)!

B œ

*B œ )!

*B œ )!

)B œ )! ))B

) )"!

Divide sides by .

T

both

he solution is ."!

20. %- - œ

%- "- œ

$- œ$-

$ $œ

- œ

#(

#(

#(#(

*

Divide sides by .

The solution is .

both $

*

21. Add the opposite.Rewrite as .Combine like terms.

is the same as .

"$ "$ œ #A A

"$ A "A

"$

! œ "A "A A

! œ A

"$ œ #A A

"$ œ #A "A

The solution is .!

22.

"" "" œ )> (>

œ )> (>!

! œ ">

! œ > The solution is .!

23. $> *> œ #! "! #'

$> *> œ #!

"#> œ $'"#> $'

"# "#œ

> œ $

Add the opposite.Combine like terms."! #'

Divide sides by .

The solution i

both "#

s .$

24. '7 '7 œ %! #! "#

"#7 œ '!

"#7 œ %)

"#7 %)

"# "#œ

7 œ %

"#

Divide sides by .

The solution is .

both"#

%

25. ! œ!

œ

! œ >

*> *

* *

*>Divide sides by .

The solution is .

both

!

26.

"! œ "!,

"!

" œ , "

Divide sides by .

The solution is .

both "!

"! "!œ

"!,

27.

"%7 )7 œ ' '!

"%7 )7 œ ' '!

'7 œ &% ''7 &%

' '

Add the opposite.Combine like terms.Divide sides by .

The solution is

both

œ

7 œ * *.

28. (A "%A œ " &!

(A

œ

A œ (

"%A œ " &!

(A œ %*

((A %*

( (

Divide sides by .

The solution is .

both

(


The solution i

"!! *' œ $"C $&C

"!!

% œ

%œ

*' œ $"C $&C

%C%

% %

%C

" œ C

Combine like terms.Divide sides by .

both

s ".

2.4 Solving Equations Using Division 55

30. "&! "$* œ #!B *B

"&!

"" œ ""B

"" ""B

"" ""œ

" œ B

"$* œ #!B *B

Divide sides by .

The solution is .

both""

"

31. To multiply on the left, usethe associative property.

$Ð#DÑ œ

Ð$

'D œ'D

' 'œ

D œ

$!

$!

$!$!

&

• •#Ñ D œ

Divide sides by .

The solu

both '

tion is .&

32. To multiply on the left, usethe associative property.

#Ð%5Ñ œ "'

Ð#

)5 œ "'

)5 "'

) )œ

5 œ #

• •%Ñ 5 œ "'

Divide sides by .

The solution

both )

is .#

33. To multiply on the right, usethe associative property.

&! œ

&! œ Ð

&! œ&!

œ

&Ð&:Ñ

&

#&: #&

#& #&

#&:

# œ

• •&Ñ :

Divide sides by .both

: #The solution is .

34. To multiply on the right, usethe associative property.

'! œ %Ð

'! œ Ð%

'! œ'!

œ

$+Ñ

$Ñ

"#+ "#

"# "#

"#+

& œ

• • +Divide sides by .both

+ &The solution is .

35. Associative property

#Ð %5Ñ œ &'

# %ÑÐ œ &'

)5 œ &'

)5 &'

) )œ

5 œ (

• • 5Divide sides by .

The solution is .

both )

(

36.

&Ð%<Ñ œ )!

#!< œ )! #!#!< )!

#! #!

Divide sides by .

The solution is .

both

œ

< œ % %

37. Associative property

*! œ "!Ð $,Ñ

*! œ Ð "! $Ñ

*! œ $!,

*!

$ œ , $

• • ,Divide sides by .

The solution is .

both $!

$! $!œ

$!,

38.

*! œ &Ð #CÑ

*! œ "!C*!

* œ C *

Divide sides by .

The solution is .

both "!

"! "!œ

"!C

39. Write in the understood

B œ $# "

"B œ $# ""B

" "$# $#

.Divide sides by .

The solution is .

both

œ$#

B œ


- œ #$ "

"- œ #$ ""-

" "#$ #$

.Divide sides by .

The solution is .

both

œ#$

- œ


# œ A "

# œ "A "# "A

" "

.Divide sides by .

The solution is .

both

œ

# œ A #


(& œ > "

(& œ "> "(& ">

" "

.Divide sides by .

The solution is .

both

œ

(& œ > (&


8 œ &! "

"8 œ &! ""8 &!

" "

.Divide sides by .

The solution is .

both

œ

8 œ &! &!


B œ " "

"B œ " ""B "

" "

.Divide sides by .

The solution is .

both

œ

B œ " "

45. Write in the understood "! œ

"! œ"!

œ

: "

": "

" "

":

"! œ : "!

.Divide sides by .

The solution is .

both

46. Write in the understood "!! œ

"!! œ

"!!œ

5 "

"5 "

" "

"5

"!! œ 5 "!!

.Divide sides by .

The solution is .

both


47. Each solution is the opposite of the number in theequation. So the rule is: When you change thevariable from negative to positive, then change thenumber in the equation to its opposite.In B œ & & & B œ &, the opposite of is , so .

48. You can divide both sides of an equation by thesame nonzero number and keep the equationbalanced.

49. Divide by the coefficient of , which is , byB $ notthe opposite of .$

The correct solution is .

$B œ "' "

$B œ "&$B "&

$ $œ

B œ & &

50. Equations will vary. Some possibilities are&B œ #! "# #! œ #B and .

51. $= œ %&$= %&

$ $œ

= œ "&

Divide sides by .both $

The length of one side is feet."&

52. $= œ '$$= '$

$ $œ

= œ #"

Divide sides by .both $

The length of one side is inches.#"

53. "#! œ &="#! &=

& &œ

#% œ =

Divide sides by .both &

The length of one side is meters.#%

54. $$& œ &=$$& &=

& &œ

'( œ =

Divide sides by .both &

The length of one side is yards.'(

55. )* ""' œ %Ð %CÑ *Ð#CÑ C

Use the associative property to simplify theproducts.

)* ""' œ Ð % %Ñ • • • •C Ð* #Ñ C C)* ""' œ "'C ")C C

Change subtraction to adding the opposite.Combine like terms.

)*

""' œ "'C ")C C

#( œ "C

Divide both sides by the coefficient, " C, to get by itself.

#( "C

" "œ

#( œ C The solution is .#(

56. &) #!) œ

&)

œ

$ œ ,

, )Ð $,Ñ &Ð &,Ñ

#!) œ ", #%, #&,

"&! œ &!,

"&! &!,

&! &!

The solution is .$

57. $(Ð"%BÑ #)Ð#"BÑ œ "'' *'k k k k(# (#

Simplify within the absolute values. $(Ð"%BÑ #)Ð#"BÑ œ (!k k k k!

Simplify the absolute values.$(Ð"%BÑ #)Ð#"BÑ œ ! (!

Use the associative property to multiply on theleft.

Ð $(

&")B &))B œ (!

• • • •"%Ñ B Ð#) #"Ñ B œ ! (!

Combine like terms.

(!B œ (!

Divide both sides by the coefficient, , to get (! Bby itself.

The solution is .

(!B (!

(! (!œ

B œ " "

58. '+ "!+ $Ð#+Ñ œ

'+

'+

œ"!

+ œ

k kk kk k

#& #&

"!+ '+ œ #& #&

"!+ '+ œ &!

"!+ œ &! %!

"!+ œ "!"!+

"! "!"

&Ð)Ñ

%!

%!

The solution is ."

2.5 Solving Equations with SeveralSteps

2.5 Section Exercises1. (: & œ "# (:

& & &

(: ! œ (

(: œ ( ((: (

( (œ

: œ " "

To get by itself, add to both sides.

Divide sides by .

The solution is .

both

2.5 Solving Equations with Several Steps 57

Balances

Check: (: & œ "#

(Ð"Ñ & œ "#

( & œ "#

"# œ "#

2.

Balances

'5 $ œ "&

$ $

'5 ! œ "#

'5 "#

' 'œ

5 œ #

Check: '5 $ œ "&

'Ð#Ñ $ œ "&

"# $ œ "&

"& œ "&

The solution is .#

3. Add the opposite.# œ )C '

# œ )C )C

' ' '

) œ )C ! )) )C

) )œ

" œ C

To get by itself, add to both sides. Divide sides by .

The

'

both

solution is ."

Balances

Check: # œ )C '

# œ )Ð"Ñ '

# œ ) '

# œ #

4.

Balances

"! œ "": "#

"! œ "":

"# "#

## œ "": !## "":

"" ""œ

# œ :

"#

Check: "! œ "": "#

"! œ ""Ð#Ñ "#

"! œ ## "#

"! œ "!

The solution is .#

5.

$7 $7

$7

$$7

$ $

" œ "

" " "

! œ !

$7 œ !

œ!

7 œ !


Divide sides by .

The solution is

both

.!

Balances

Check:

$7 " œ "

$Ð!Ñ " œ "

! " œ "

" œ "

6.

Balances

%5 %5 & œ &

%5

%5

% %

%Ð!Ñ & œ &

& œ &

& &

! œ !

œ!

5 œ !

Check:

! & œ &

& œ &

The solution is .!

7. #) œ

"! "! "!

") œ

") œ")

œ

œ +

*+ "! *+

*+ !

*+ *

* *

*+

#


Divide sides by .

The so

both

lution is .#

Balances

Check: #) œ

#) œ

#) œ ") "!

#) œ #)

*+ "!

*Ð #Ñ "!

8. (& œ

#& #&

&! œ&! !A

œ

œ A

"!A #&

"!A !

"! "!

"

&

Check: (& œ

(& œ

(& œ &! #&

(& œ (&

"!A #&

"!Ð &Ñ #&

Balances

The solution is .&


&B %

&B % &B

&B !

&&B

&

œ "'

œ "'

% % %

œ #!

&B œ #!

œ#!


Divide sides by . both

&% %B œ The solution is .

Balances

Check:

&B % œ "'

&Ð %Ñ % œ "'

#! % œ "'

"' œ "'

10.

Balances

"#, $ "#, $ œ #"

"#, $ "#Ð #Ñ $ œ #"

"#, !

"#,

"# "##

œ #"

œ #"

$ $

œ #%

œ#%

, œ

Check:

#% $ œ #"

#" œ #"

The solution is .#


11. Solve with the variable on the left side.

Add

Add

Divide sid

': # œ %: '

': œ %:

#: œ ! '

#: œ ' #

# #

#: ! œ )

#: œ )

Add the opposite.

# ' %:

%: %:

#

#

to both sides.

to both sides.

both es by .

The solution is .

##: )

# #œ

: œ % %

side.Solve with the variable on the right

Add

Add

Divide

': # œ %: '

': œ %:

': ':

! œ '

# œ #: '

) œ #: !

œ

Add the opposite.

# ' ':

# #:

'

' '

) #:

to both sides.

to both sides.

sides by .

The solution is .

both

#

# #

#:)œ

% œ : %

Balances

Check: ': # œ %: '

'Ð%Ñ # œ %Ð%Ñ '

#% # œ "' '

## œ ##

12. Left side: Right side:

&C & œ #C "! &C & œ #C "!

$C & œ ! "! ! & œ "!

$C œ "! œ "!

& &

$C ! œ "& œ$C "&

$ $ $ $œ œ

C œ & & œ C

#C #C &C &C

& & $C

$C

"! "!

"& $C !"& $C

Balances

Check: &C & œ #C "!

&Ð&Ñ & œ #Ð&Ñ "!

#& & œ "! "!

#! œ #!

The solution is .&


Add

Add

Divide

#5 '

#5 ''

'5 '

)5 '

)5 ''

)5 !

œ '5 "!

œ '5 "!5

5

œ ! "!

œ "!

' '

œ "'

Add the opposite. to

both sides.

toboth sides.

both sidesby .

The solution is .

)

) )# #

)5 "'œ

5 œ


Add

Add

Divide

#5 '

#5 '

5 #

'

'"!

"! "!

"'

œ '5 "!

œ '5 "!#5

# 5

! œ )5 "!

œ )5 "!

œ )5 !

Add the opposite. to

both sides.

toboth sides.

both sidesby .

The solution is .

)

"'

) )# #

œ)5

œ 5

Balances

Check:

#5 ' œ '5 "!

#Ð #Ñ ' œ 'Ð #Ñ "!

' œ "# "!

# œ #

%

14. Left side: Right side:

&B % œ &B % œ

$B $B B

)B % œ ! % ! % œ %

)B % œ % œ

)B ! œ ) œ)B )

) ) ) )œ œ

B œ œ B

$B % $B %

% )B %

% % % %

) )B !) )B

" "

& &B

)B

Balances

Check: &B % œ

&Ð

$B %

"Ñ % œ $Ð "Ñ %

& % œ $ %

" œ "

The solution is ."



Add

Add

The

") (+#+

#+ #+

") &+

") &+

œ #+ (

œ ! (

œ (")

") ")

! &+ œ #&

&+ œ #&

&+ #&

& &œ

+ œ &

toboth sides.

toboth sides.

Divide bothsides by .&

solution is .&


Add

Add

") (+(+

(+ (+

") ! &+

") &+ ((

( (

#& &+ !

#& &+

#&

œ #+ (

œ (

œ

œ

œ

&œ

toboth sides.

toboth sides.

Divide bothsides by .&

&+

&& œ + & The solution is .

Balances

Check:

") (+ œ #+ (

") (Ð&Ñ œ #Ð&Ñ (

") $& œ "! (

"( œ "(

16. Left side:

The solution is .

* #D

*D *D

* (D

* (D

(D(D

( ($

œ *D "#

œ ! "#

œ "#

* *

! œ #"

œ#"

D œ $

Right side:

The solution is .

* #D

#D #D

* !

*

"# "#

#"#"

( ($

œ *D "#

œ (D "#

œ (D "#

œ (D !

œ(D

œ D $

Balances

Check:

* #D œ *D "#

* #Ð $Ñ œ *Ð $Ñ "#

* ' œ #( "#

"& œ "&

17. Distribute.Add the opposite.Add to both sides.

)ÐA #Ñ œ $#

)A "' œ $#

)A œ $# "'

"' "'

)A ! œ %)

)A œ %))A %)

) )œ

A

"'

Divide sides by .both )

œ ' The solution is .'

18. *Ð, %Ñ œ #(

*, $' œ #(

*, œ #(

$' $'

*, ! œ '$

*, '$

* *œ

, œ (

$'

The solution is .(

19. Distribute.Add to both sides.

"!

"! )

) )

")

")")

* *

œ #ÐC %Ñ

œ #C )

œ #C !

œ #C

# #œ

#C

œ C

Divide sides by .

The solution is .

both #

20.

$

$

") ")

#"#"

(

œ $ÐB 'Ñ

œ $B ")

œ $B !

$ $œ

$B

œ B The solution is .(


21. Distribute.Add to both sides.

%Ð> #Ñ

%> ) )

) )

%> !

%> %%>

% %& &

œ "#

œ "#

œ #!

œ #!

œ#!

> œ

Divide sides by .

The solution is

both

.

22.

&Ð5 $Ñ

&5 "&

"& "&

&5 !

&5

& &)

œ #&

œ #&

œ %!

œ%!

5 œ The solution is .)

23. Distribute.Add the opposite.Add to both sides.

'ÐB &Ñ œ $!

'B $! œ $!

'B œ $!

$! $!

'B ! œ !

'B œ !'B !

' 'œ

B

$! $!

Divide sides by .both '

œ ! The solution is .!

24. (Ð< (Ñ œ

(< %* œ

(< œ

%* %*

(< ! œ !(< !

( (œ

< œ !

%*

%*

%* %*

The solution is .!

25. Distribute. Add the opposite. Add to both sides.

"#

"#

"# #%

œ "#Ð2 #Ñ

œ "#2 #%

œ "#2 #%

#% #%

"# œ "#2 !

"# œ "#2 Divide sides bboth y .

The solution is .

"#

"

"# "#2

"# "#œ

" œ 2

26.

""

""

"" $$

œ ""Ð- $Ñ

œ ""- $$

œ ""-

$$ $$

## œ ""- !## ""-

"" ""œ

# œ - The solution is .#

27. Distribute. Add to both sides.

! œ

! œ %

% %

% œ

œ

œ#C

œ C

#ÐC #Ñ

#C %

#C !

% #C #%

# ## #

Divide sides by .

The solution is .

both

28. ! œ

! œ

* *

* œ

*œ

œ ,

*Ð, "Ñ

*, *

*, !

* *

*,

" The solution is ."

29. To get by itself, add'7 ") œ ! '7

'7 œ

'7 œ'7

' 'œ

7 œ

") ") ")

! ")

")")

$ $

to both sides.

Divide sides by .

The solution is

both '

.

30. ): %! œ !

): œ !

%! %!

): ! œ %): %!

) )œ

: œ &

%!

!

The solution is .&

31. Add the opposite. Add to both sides.

' œ *A "#

' œ *A "#

"# "#

") œ *A !

") œ *A") *A

* *œ

œ A

"#

#

Divide sides by .

The solution is

both *

#.

32. ) œ )2 #%

œ )2 !

) )œ

)2

œ 2

#% #%

"'

"'

# The solution is .#


33. Add to get the variable term on one side.

&B œ $B "!

#B œ ! "!

#B œ "!#B "!

# #œ

B œ &

$B

$B $B

to both sides

Divide sides by .

The

both #

solution is .&

34. (8 œ

(8 œ

#8 #8

*8 œ*8

* *œ

8 œ

#8 $'

#8 $'

! $'$'

% The solution is .%

35. Add the opposite. Add

Add to both sides.

#+ "" œ )+ (

#+ "" œ )+

! "" œ '+

"" œ '+ (

( (

") œ '+ !

") œ '+

( #+

#+ #+

(

(

to both sides.

Divide sides by .

The solution is .

both '

$

") '+

' 'œ

$ œ +

36. < "! œ "!< )

"< œ "!< )

! "! œ *< )

"! œ *< )

") œ *< !") *<

* *œ

œ <

"!

"< "<

) )


37. Add the opposite. Add

Add to both sides.

( &, œ #) #,

( œ #) #, &,

&, &,

( ! œ #) (,

( œ #) (,

œ ! (,

&,

#)

#) #)

#"

#"

to both sides.

œ (,

( (œ

(,

$ œ ,

Divide sides by .

The solution is .

both (

$

#"

38. " )> œ

" œ

)> )>

" ! œ * &>

" œ * &>

* *

"! œ ! &>"! &>

& &œ

œ >

* $>

)> * $>


39.

#! #5

#! #5 %5

#! #5 $5 #5

#5 #5

#! ! &5

#! &5

œ 5 %5

œ 5

œ

œ

œ

Add the opposite.

Add Combine like terms.

Divi

to both sides.

de sides by .

The solution is .

both

&

#! &5

& &œ

% œ 5 %

40. 'C C œ

'C œ

&C œ

%C œ%C

% %œ

C œ

"' C

"C "' "C

"' "C

"C "C

"' !"'

% The solution is .%

41. "!Ð- 'Ñ % œ # - &)

"!- '! % œ # - &)

"!- œ # -

"!- œ #

Distribute. Add the opposites. Group like terms.

'! % &)

'! % &) - Combine like terms.

Divide sides by

"!- œ

*- œ

*- œ &'

&' &'

*- ! œ !

&' &' - -

- -

&' &' !

&' &'

Add

Add to both sides.

to both sides.

both *

!

.

The solution is .

*- !

* *œ

- œ !

42. )ÐD (Ñ ' œ D '! "!

)D &' ' œ D '! "!

)D &' œ D '!

)D &! œ "D &!

(D &! œ

' "!

"D "D

! &!

continued


(D &! œ

&! &!

(D ! œ !(D !

( (œ

D œ !

&!

The solution is .!

43. Add theopposites.

Comb

") "$C $

") "$C $

") "$C $ $ #

") $ $ #

œ $Ð&C "Ñ #

œ "&C $ #

œ "&C

"$C œ "&C

Distribute.

Group liketerms.

inelike terms.Add to both sides.

Add to both sides.

Divide bothside

"$C œ "&C

! œ #C

œ #C &

& &

œ #C !

œ #C

"& & "$C

"$C "$C

"& &

"& &

"!

"! s by .

The solutionis .

#

# #œ

#C

œ C

"!

&

&

44. $ &2 * œ %Ð$2 %Ñ "

$ &2 œ "#2 "'

&2 œ "#2 "&

! œ (2 "&

œ (2 "&

œ (2 !

( (œ

(2

œ 2

* "

'

&2 &2

'

'

"& "&

#"

#"

$ The solutionis .$

45. Add the opposites.

Add both sides.Divide both sid

' %8 $8 œ #! $&

' œ #!

' œ

! œ

%8 $8 $&

"8 "& '

' '

"8 #"

Combine like terms. to

esby

""8 #"

" "#"

.

œ

8 œ #" The solution is .

46.

The solution

"* )

"* ) (: &

"" ": &

' ": !' ":

" "

œ ': (: &

œ ':

œ

& &

œ

œ

' œ : is .'

47. Add the opposites.Add both sides.

Add to

'Ð- #Ñ œ (Ð- 'Ñ

'- "# œ (- %#

'- œ (-

! œ -

œ - %#

Distribute.

"# %# '-

'- '-

"# %#

"# %#

to

both sides.%# %#

$! œ - !

$! œ - The solution is .$!

48.

$Ð& BÑ

"& $B

"& !

"& )

((

"

œ %ÐB #Ñ

œ %B )

$B $B

œ (B )

œ (B

) )

œ (B !

( (œ

(B

œ B The solution is ."

49.

&Ð#: #Ñ (

"!: "! (

"!: "! (

"!: "( ':

œ $Ð#: &Ñ

œ ': "&

œ ': "&

œ ': "&

Distribute.Add the opposite.Combine like terms.Add to both

.

': ':

"': "(

"': "(

"': !"'

"':

"' "'#

sides.

Add to both sides.Divide both sidesby

œ ! "&

œ "& "(

"( "(

œ $#

œ$#

: œ The solution is .#

50. %Ð$7 'Ñ œ (# $Ð7 )Ñ

"#7 #% œ (# $7 #%

"#7 œ (# $7

"#7 œ $7 %)

*7 œ ! %)

#% #%

#%

$7 $7

#%

Chapter 2 Review Exercises 63

*7 œ %)

#% #%

*7 œ (#*7 (#

* *œ

7 œ )

#%

!

The solution is .)

51.

', %, (, œ "! , $,

œ "!

œ "! #,

œ "! !

œ "

Add the opposites.Combine like terms.Add both sides.

', %, (, , $,

$, #,

#, #,

&,

&,

to

!

œ"!

, œ

Divide both sidesby .

&

&,

& &# #The solution is .

52. A ) &A A "&A ""A

&A "A "&A ""A

%A ) &A

"A

" "

"A

)

œ

"A ) œ

œ

%A %A

! ) œ)

œ

œ A

The solution is .)

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

#> "!

#> "!

"! &

œ $> &

œ $> & #>

#> #>

! œ &> &

Change subtraction toadding the opposite.Add to both sides(addition property).Add to both sides(addition property).Divide both sides by (division property).

& &

"&

$

œ

& &œ

&> &

œ >

The solution is .$

54. Multiplication distributes over both addition andsubtraction. Examples will vary. Somepossibilities are is and$Ð#C 'Ñ 'C ")&ÐB $Ñ &B "& is .

55. Check:

) %+ œ #+ #

) %Ð$Ñ œ #Ð$Ñ #

) "# œ ' #

% Á )

The check does not balance, so is not the correct$solution. The student added #+ ) to on the leftside, instead of adding to . The correct#+ %+solution, obtained using ) #+ œ #ß #+ œ "!ßis .+ œ &

56. Check: #ÐB %Ñ œ

#Ð

#Ð

"'

"! %Ñ œ "'

'Ñ œ "'

"# Á "'

The check does not balance, so "! is not thecorrect solution.

#ÐB %Ñ œ

#B ) œ #

%

#B ! œ#B

# #œ

B œ

"'

"'

) )

#%#%

"#

Student did notdistribute the over the .

The correct solution is ."#

57. (a) It must be negative, because the sum of twopositive numbers is always positive.

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

58. (a) It must be positive, because the sum of twonegative numbers is always negative.

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

59. (a) It must be positive. When the signs are thesame, the product is positive, and when the signsare different, the product is negative.

(b) The product of and a negative number is8negative, so must be positive.8

60. (a) It must be negative also. When the signs aredifferent, the product is negative, and when thesigns match, the product is positive.

(b) The product of and a negative number isCpositive, so must be negative.C

Chapter 2 Review Exercises1. (a) In the expression ariable, $ %5 5, is the v %

is the coefficient, and constant term$ is the .

(b) The term that has as the constant term and#! * *C #! as the coefficient is .

2. (a) Evaluate when is .%- "! - "&

Order test tubes.

%- "! - "&

% "!

'! "!

(!

Replace with .îðñò• "&

(!


(b) Evaluate when is .%- "! - #%

Order test tubes.

%- "! - #%

% "!

*' "!

"!'

Replace with .îðñò• #%

"!'

3. (a) means B C B# % • • • • •B C C C C

(b) means &+, &$ • • • •+ , , ,

4. (a) means8#

8 8

*

••8 Replace with

$

$ $

.ï(b) means8$

8 8

*

• •• •

•

8 8 Replace with

$

$ $ $

$

#(

.ïî

(c) #%7: means

%

%

)

$#

"#)

• • •

• • •

• •

•

7 : :

# % %

% %

%


7 #

: %îïï

(d) means&7 8% #

& 7 #

8

&

"!

#!

%!

)!

(#!

• • • • • •

• • • • • •

• • • • •

• • • •

• • •

• •

•

7 7 7 7 8 8

# # # #

# # #

# #

#


$

$ $

$ $

$ $

$ $

$ $

#%! $

íîîîïðóñóò

5.

or

+, +, #+,

"+, +, #+,

$+, +, +, $+,

#

#

# #

Combine like terms.

6. Rewrite as .Add the opposite.

or

$B #C B (

$B #C "B (

$B #C "B (

%B #C ( %B #C (

B "B

Combine like terms.

7. Associative property $

)Ð #1 Ñ

) #ÑÐ

"'

"'1

• •

•

1

1

$

$

$

8. Associative property%Ð$< >Ñ

Ð%

"#

"#< >

#

#

• •

•

$ < >

< >

Ñ #

#

9. Distribute.&Ð5 #Ñ

&

&5 "!

• •5 & #

10. Distribute.

or

#Ð$, %Ñ

# #

', ) ', )

• •$, %

11. Distribute.$Ð#C %Ñ "#

$ $ "#

'C "# "#

'C

'C !

'C

î í• •#C %

"# "#

12. Distribute.

or

% 'Ð%B "Ñ %B

% #%B ' %B

% #%B ' %B

# #!B #!B #

13. Expressions will vary. One possibility is'+ + $+ '$ # .

14. Add "' 8 œ &

! 8 œ

8 œ

"'

"' "'

""

""

to both sides.

The solution is .""

Check: "' 8 œ &

"'

& œ &

"" œ &

Balances

15.

% #

% # ' "+

# '

œ #+ ' +

œ #+

œ "+

' '

% œ "+ !

% œ +

Balances

Check:

% # œ #+ ' +

% # œ #Ð%Ñ ' %

# œ ) ' %

# œ #

# œ # %

The solution is .%

16. %) œ%)

œ

'7 '

' '

'7

) œ 7

Divide sides by .

The solution is .

both

)

Chapter 2 Review Exercises 65

17. 5 &5 œ

"5 &5 œ

"5

œ

5 œ "!

%!

%!

&5 œ %!

%5 œ %! %

%5 %!

% %

Combine like terms.Divide sides by .

The solution is .

both

"!

"( "" ' œ (>18. ðóóóñóóóò! œ (>

! (>

( (œ

! œ >

Divide sides by .

The solution is .

both (

!

19.

#: &: œ $ #"

#: &: œ $ #"

")")

'

$: œ$:

$ $œ

: œ

Divide sides by .

The solution is .

both $

'

20.

$! œ $Ð &<Ñ

$! œ "&< "&$! "&<

"& "&

Divide sides by .

The solution is .

both

œ

# œ < #

21. "# œ

"# œ

"#œ

2

"2

" "

"2

"# œ 2 The solution is ."#

22.

Add toboth sides.

"#A % œ )A "#

"#A œ )A "#

%A œ ! "#

%A œ "#%

% %

%A ! œ "'

%A œ "'

%A "'

% %œ

A œ %

%

)A )A

%

%

Divide sides by .

The sol

both%

ution is .%

23. Distribute.

Add toboth sides.

! œ

! œ

! œ

))

) œ

%Ð- #Ñ

% %

%- )

)

%- !

• •- #

) œ

)œ

œ -

%-

% %

%-

#

Divide sides by .

The solution is .

both%

#

24. Add toboth sides.$% œ #8 %

%

$! œ #8 !

$! œ #8

$! #8

# #œ

"& œ 8

% %

Divide sides by .

both#

The number of employees is ."&

25. [2.5] "# (+ œ %+ $

"# $+ œ ! $

"# $+ œ

! $+ œ

$+

$ $œ

+ œ

Add toboth sides.

%+

%+ %+

$

"# "#

"&

"&

&

Divide sides by .

The solution

both$

is .&

26. [2.5] Distribute.Add toboth sides.

#Ð: $Ñ "%

#: ' "%'

' '

#: ! #!#: #!

# #

œ

œ

œ

œ

: œ "! The solution is ."!

27. [2.5] Add toboth sides.

"!C œ 'C #!

%C œ ! #!%C #!

% %œ

C œ &

'C

'C 'C

The solution is .&

28. [2.5] Add theopposites.

#7 (7 œ & #!

#7

œ

7 œ $

(7 œ & #!

&7 œ "&&

&7 "&

& &

Combinelike terms.Divide sides by .

The solution i

both

s .$


29. [2.5] #! œ $B (

#! œ $B

( (

#( œ $B !

$ $œ

$B

* œ B

(

#(

The solution is .*

30. [2.5] , ' œ $, )

œ ! )

œ

œ

œ

, œ (

$, $,

#, '

#, ' )

' '

#, ! "%

#, "%

# #The solution is .(

31. [2.3]

D $ œ !

D ! œ

D œ

$ $

$

$ The solution is .$

32. [2.5] Distribute.

Add toboth sides.

$Ð#8 "Ñ œ $Ð8 $Ñ

'8 $ œ $8 *

$8 $ œ ! *

$8 $ œ *$

$ $

$8 ! œ "#$8 "#

$ $œ

8 œ %

$8 $8

The solution is .%

33. [2.5] Distribute.

Add toboth sides.

% %' $> 'Ñ

% %' #"> %#

#"> %#%#

%# %#

#"> !

#" #"

#">

œ (Ð

œ

%# œ

! œ!

œ

! œ > The solution is !.

34. [2.5]

' "!. "* œ #Ð$. %Ñ "

' "!. œ '. ) "

œ '. (

œ

Distribute.

"*

"$ "!.

'. '.

"$ %. ! (

Add toboth sides.

"$ %. œ ("$

"$ "$

! %. œ #!

%. #!

% %œ

. œ & The solutionis .&

35. [2.5]

%Ð$, *Ñ

"#, $'

"#, $' #%

"#, $'

$'

$'

œ #% $Ð#, )Ñ

œ #% ', #%

œ #% ',

œ ',"#,

"#, "#,

! œ "),

")

Distribute.

Add toboth sides.

œ"),

œ ,")

# The solutionis .#

Chapter 2 Test1. In the expression (A ' ( A, is the coefficient,

is the variable, and is the constant term.'

2. Evaluate the expression when is and$+ #- + %&- #" is .

$+ #-

$

"$& %#

"((

• •%& # #"

Buy hot dogs."((

3. means B C B& $ • • • • • • •B B B B C C C

4. means %+, %% • • • • •+ , , , ,

5. means ##= >


#&

# & &

&

&!

#!!

• • •

• • •

• •

•

= = >

%

%

%

=

> %

"!

ïïï

6. $A )A A

$A )A "A

$A "A

$ $ $

$ $ $

$ $ðóóóñóóóòðóóóñóóóò

$

$ $

$

)A

&A "A

%A

Chapter 2 Test 67

7. BC BC

"BC "BC

Ð" "ÑBC

!BC

!

8.

or

'- & (- &

'- & (- &

'- (- & &ðóñóò ï

"- !

"- -

#9. $7 $7 $78There are no like terms.The expression cannot be simplified.

10. Associative propertyof multiplication

#

#

"!Ð%, Ñ

"!

%!,

Ð • •%Ñ ,#

11. Associative propertyof multiplication

&Ð $5Ñ

& $ÑÐ

"&5

• • 5

12. Distributive property(Ð$> %Ñ

(Ð$>Ñ (Ð%Ñ

#"> #)


%Ð+ 'Ñ

% %

%+ #%

%+ #%

• •+ '

14.

or

) 'ÐB #Ñ &

) 'B "# &

) 'B "# &

"&

Combine like terms.'B 'B "&

15.

*, - $ * #-

*, "- $ * #-

*, "- $ * #-

*, - '

16.

Balances

% % œ B *

% œ & *

% œ %

œ B *

* *

& œ B !

& œ B

Check:

The solution is .&

17.

Balances

(A œ (( (A œ (((A

( (""

( "" œ ((œ

((

A œ(( œ ((

Check: •

The solution is .""

18.

Balances

: œ "% : œ "%

": œ "% ": œ "%":

" ""%

" "% œ "%œ

"%

: œ"% œ "%

Check:

•

The solution is ."%

19.

Balances

"& $Ð+ #Ñ "& œ $Ð+ #Ñ

"& $+ ' "& œ $Ð$ #Ñ

* $+

$ $

"& œ $Ð&Ñ

"& œ "&

œ

œ

' '

œ

œ

$ œ +

* $+

Check:

The solution is .$

20. '8 ) &8 œ

'8 ) œ !

8 ) œ !

8 œ

% %

&8

) )

)

The solution is .)

21. & #! œ #7 $7

&

œ

"& œ 7

#! œ #7 $7

"& "7

" "

"& œ "7

The solution is ."&

22. Add to both sides.

#B #

* *

(

"

œ &B * #B

#B #B

# œ (B *

( (œ

(B

œ B

The solution is ."

23.

Add to both sides.

$7 & œ (7 "$

! & œ %7 "$

œ %7 "$"$

"$ "$

œ%7

%# œ 7

$7 $7

&

)

%

The solution is .#


24.

Add to both sides.

# (, %% œ

(, %# œ ', "#

", %# œ "#%#

%# %#

", œ &%

, œ &%

$, "# *,

', ',

The solution is .&%

25. Distribute.

Add to both sides.

$- #% œ 'Ð- %Ñ

$- #% œ '- #%

œ $- #%#%

#% #%

! œ $-! $-

$ $œ

! œ -

$- $-

#%

The solution is .!

26. Equations will vary. Two possibilities areB & œ * #% œ 'C and .

Solving:

B & œ * #% œ 'C

& &

B œ %

#%

% œ C

' '

œ'C

Cumulative Review Exercises(Chapters 1–2)

1. $!' !!! !!% #"!, , , in words is three hundred sixbillion, four thousand, two hundred ten.

2. eight hundred million, sixty-six thousand:)!! !'' !!!, ,

3. (a) $ "! lies to the of on the number line,rightso . $ "!

(b) " ! lies to the of on the number line, soleft" !.

4. (a) ' # œ # '

Commutative property of addition:

Changing the order of the addends does notchange the sum.

(b) ! • #& œ !

Multiplication property of zero: Multiplying anynumber by gives a product of .! !

(c) &Ð ' %Ñ œ & ' &• • %

Distributive property:

Multiplication distributes over addition.

5. (a) *!%( ¸ *!!!

Underline the hundreds place: *!%(

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 ." !

6. Change subtraction to addingthe opposite.

! )

œ !

œ

)

)

7. is units from .

k k k k ' ' ' ! % is units from .% % !

œ ' %

œ "!

8. Same sign, positive product$Ð "!Ñ

œ $!

9. Same sign, positive product

Ð

œ

œ #&

#

&Ñ

& &•

10. Same sign, positive quotient

%#

'œ (

11. Addition of a numberand its opposite is zero.

"* "*

œ !

12. ExponentMultiply left to right.

Ð

"'

$

%Ñ

% % %

%

'%

ïï

• •

•

13. is undefined. Division by is undefined."%

!!

14. Different signs, negative product&'!

• "#œ

15. Change subtraction toadding the opposite.

#! #!

#! #!

%!

œ

œ

16. Different signs, negative quotient%&

œ

&*

Cumulative Review Exercises (Chapters 1–2) 69

17. &! #& œ #&

18. Distribute.

"! 'Ð% (Ñ

"!

"! #% %#

"! #% %#

%#

#)

í íðóñóò

ðóñóò

' '

"%

• •% (

19.

# $

#! $Ð &Ñ "'

%Ñ $Ð

Numerator:

Multiply.Add the opposite.Add left to right.

#! $Ð &Ñ "'

#! "& "'

#! "&ðóñóò& "'ðñò

"'

""

Denominator:

Ð

Ð $

"' #(

"'

# $

%Ñ $

%ÑÐ %Ñ

#(

""

ðóñóò ïðóñóò

• •$ $

Last step is division: ""

œ

"""

20. days rounds to .## #!'"' '!! miles rounds to .Average distance "per" day implies division.

Estimate: '!!

#!œ $!

miles days

miles per day

Exact: '"'

##œ #)

miles days

miles per day

21. %) &! degrees rounds to ."Rise" of degrees rounds to .#$ #!

A start temperature of degrees followed by a%)rise of degrees implies addition.#$

Estimate: &! #! œ $! degreesExact: %) #$ œ #& degrees

22. shares rounds to .&# &!$ rounds to $ .#"$# #!!!$ rounds to $ .) "!

Each stock dropped in value by $ and Doug)owned shares. Multiply to find out how much&#money he lost. Then, subtract this amount from theoriginal total value.

Estimate: $ $#!!! Ð&! "&!!• "!Ñ œExact: $ $#"$# Ð&# "("'• )Ñ œ

23. $ rounds to $ .&&# '!!$ rounds to $ .$& %!"# "! months (in one year) rounds to .

Estimate: "!Ð '!! %!Ñ

"!Ð '%!Ñ œ '%!!

$ $$ $

$ $ $ $Exact: "#Ð &&# $&Ñ œ "#Ð &)(Ñ œ (!%%

24. $ # %+, - % means • • • • • •+ , , , - -

25. $BC$ means

.$B

C

$

$!

"#!

• • • •

• • • •

• • •

• •

•

B C C CReplace with and with Multiply left to right.

&

#

& # # #

"& # # #

# #

'! #

îðñò

ïðñò

26. Change subtraction toadding the opposite.Combine like terms.

or

$2 (2 &2

$2 &2

"2 2

ðóñóòðóñóò

(2

%2 &2

27. Write with the understoodcoefficients of .Change subtraction toadding the opposite.

- . - .

"

œ "- . "- .

œ "- .

# #

# #

# #"- . Combine like terms.œ Ð"

œ !

œ !

#"Ñ- .

• - .#

28.

or

%8 %8 ' ) 8

%8

%8 8 '

&8

&8 %8 #

# #

#

# #

#

#

#

%8 ' ) 8

%8 )

%8 #

ðóñóò ï

29. Associative property #

#

"!Ð$, Ñ

"!

$!,

ðñòÐ ,• $Ñ #

30. Distribute.(Ð%: %Ñ

( (

#): #)

î í• •%: %

31.

or

$ &Ð

$

$

# #

# #

# #

# #

#A $Ñ A

"!A "& A

"!A "& A

*A "# *A "#


32.

Balances

$B œ B ) $B œ B )

#B œ ! )

#B œ#B

# #œ

B œ

$Ð

B B %Ñ œ % )

)

%

"# œ % )

"# œ "#)

Check:

The solution is %.

33.

Balances

%% # (C %% œ # (C

%#

%#

'

%% œ # (Ð 'Ñ

%% œ # %#

%% œ %%

œ

# #

œ ! (C

œ (C

( (œ

(C

œ C

%#

Check:

The solution is '.

34.

Balances

#5 &5 œ #5 &5 œ

#5 #Ð(Ñ &Ð(Ñ œ

œ

5 œ (

"% $& œ

"%

#" #"

&5 œ #" #"

$5 œ #" #"

$5 #"

$ $

$& œ #"

#" œ #"

Check:

The solution is (.

35.

The solution is

7 ' œ

#7 #7

$7 ' œ ! '

$7 ' œ '

'

$7 ! œ "#$7 "#

$ $œ

7 œ %

#7 '

'

%.

Balances

Check: 7 ' œ

% ' œ

%

#7 '

#Ð%Ñ '

' œ ) '

# œ #

36. Add toboth sides.

The solution is

% %B œ ") "!B %B

%B %B

% ! œ ") "%B

% œ ") "%B

œ ! "%B

"% "%œ

"%B

œ B

") ")

"%"%

" ".

37.

The solution is

") œ

") œ")

œ

<

"<

" "

"<

") œ < ").


Add toboth sides.

), "" (,

", ""

""

"!

"!

&

œ , "

œ ", "",

", ",

!, "" œ #, "

œ #, ""

" "

œ #, !

# #œ

#,

œ ,

The solution is &.


Add toboth sides.

#Ð> "Ñ

#> #

#> !

#>

œ %Ð" #>Ñ

œ % )>#

# #

œ ' )>

œ ' )>)>

)> )>

'> œ ' !'> '

' 'œ

> œ "

The solution is ".

40.

Add toboth sides.

& 'C #$ œ &Ð#C )Ñ "!

& 'C œ "!C %!

'C œ "!C $!")

") ")

'C ! œ "!C %)

'C œ "!C %)

œ ! %)

œ%)

C œ

#$ "!

")

"!C "!C

%C%C

% %"#

The solution is "#.

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