Take the absolute value of each number and subtract the smaller absolute value from the larger.. Take the absolute value of each number and subtract the smaller absolute value from the l
Trang 1CHAPTER R REVIEW OF PREALGEBRA
Section R.1 Operations with Integers
1 The number 0 belongs to the set of whole numbers and all whole numbers also belong
to the set of integers
2 The number 6 belongs to the set of natural numbers All natural numbers also belong to the set of whole numbers and to the set of integers
3 The number −20 belongs to the set of integers
4 The number −7 belongs to the set of integers
5 The number 11 belongs to the set of natural numbers All natural numbers also belong to the set of whole numbers and to the set of integers
6 The number 0 belongs to the set of whole numbers and all whole numbers also belong
to the set of integers
7 The number −9 belongs to the set of integers
8 The number −3 belongs to the set of integers
9 The number 1 belongs to the set of natural numbers All natural numbers also belong to the set of whole numbers and to the set of integers
10 The number −1 belongs to the set of integers
11 The value for the population of Los Angeles is
a natural number The population could be 10,000, 100,000, 1,000,000 Assuming the population would not be 0, the value 0 is not included as a possibility These values are also whole numbers and integers
12 The value for the number of people in a crowd
at the beach is a natural number The number could be 5, 10, 25 Assuming the number would not be 0, because there would not be a crowd if there were 0 people These values are also whole numbers and integers
13 The value for the population of Norfolk, VA is
a natural number The population could be 10,000, 100,000, 1,000,000 Assuming the population would not be 0, the value 0 is not included as a possibility These values are also whole numbers and integers
14 The value for the number of children attending Sullivan Middle School graduation is a natural number The number could be 25, 60, 110
Assuming the number would not be 0 because
animal shelter is a whole number The number
of kittens could be 0, 1, 2, Since the animal shelter could be empty of kittens on any given day (0 kittens), the value 0 is included as a possibility These values are also integers
16 The value for the number of puppies in the animal shelter is a whole number The number
of puppies could be 0, 1, 2, Since the animal shelter could be empty of puppies on any given day (0 puppies), the value 0 is included as a possibility These values are also integers
17 The value for the number of hours worked weekly by a Home Depot employee is a whole number The number of hours worked could be
0, 5, 12, Since the employee could have a week off (0 hours worked), the value 0 is included as a possibility These values are also integers
18 The value for the number of hours worked yearly by an auto worker is a whole number The number of hours worked could be 0, 100,
220, Since the employee could be laid off
or otherwise have no work (0 hours worked), the value 0 is included as a possibility These values are also integers
19 This is an example from the set of integers The average daily high temperature in Missoula could range from negative numbers
to positive numbers
20 This is an example from the set of natural numbers The average daily high temperature
in Honolulu has never been recorded below
12 FD so this belongs to the set of natural numbers These values are also whole numbers and integers
21. The set of all integers between –5 and 1,
including –5 and 1 is as follows: {-5, -4, -3, -2,
-1, 0, 1}
22. The set of all integers between −2 and 4,
including −2 and 4 is as follows: {-2, -1, 0, 1,
2, 3, 4}
23 The set of all natural numbers between –5 and
5, not including –5 and 5 is as follows: {1, 2,
3, 4} Notice that negative values and 0 are not included because this is a set of only natural numbers
24 The set of all natural numbers between –6 and
4, not including –6 and 4 is as follows: {1, 2,
3} Notice that negative values and 0 are not included because this is a set of only natural
NOT FOR SALE
Trang 225 The set of all whole numbers between –4 and
0, including –4 and 0 is as follows: {0}
Notice that negative values are not included because this is a set of only whole numbers
26 The set of all whole numbers between –3 and
3, including –3 and 3 is as follows: {0, 1, 2, 3}
Notice that negative values are not included
because this is a set of only whole numbers
27 The tick marks on the number line are 4 units
apart, so the scale of this number line is 4
28 The tick marks on the number line are 7 units
apart, so the scale of this number line is 7
29. The tick marks on the number line are 0.1
units apart, so the scale of this number line is 0.1
30. The tick marks on the number line are 0.25
units apart, so the scale of this number line is 0.25
37. The number −5 is to the right of −15 on the
number line, therefore −5 is greater than −15
− > −
38. The number −6 is to the left of 0 on the
number line, therefore −6 is less than 0
− <
39 The number −3 is at the same location as −3
on the number line, and is therefore neither less than nor greater than −3 Write an expression using inequality symbols as follows:
40. The number 8 is at the same location as 8 on the number line, and is therefore neither less than nor greater than 8 Write an expression using inequality symbols as follows:
48 0 =0 since there is no distance between 0
and itself on the number line
49 16 =16 since the distance between 16 and 0
14.5, 2, 1.75, , 3.5, 5
4
−
NOT FOR SALE
INSTRUCTOR USE ONLY
Trang 355 First, find any absolute values
in its original notation as follows:
in its original notation as follows:
in its original notation as follows:
23.5, , 4 , 4.2, 53
in its original notation as follows:
3 ( 5) 8
− + − = −
66 The two integers have the same sign, so we add the two numbers using the rule for adding integers and keep the negative sign
4 ( 9) 13
− + − = −
67 The two integers have different signs Adding
a negative number is the same as subtracting the number, so we will subtract 7 from 16
16 ( 7)+ − =16 7− =9
68 These two integers have different signs Take the absolute value of each number and subtract the smaller absolute value from the larger
NOT FOR SALE
Trang 4Attach the sign of the number that is larger in absolute value Since the number that is larger
is 15+ , the answer is 7+ or simply 7
69 These two integers have different signs Take
the absolute value of each number and subtract the smaller absolute value from the larger
Subtract 6 from 13
13 6− = 7Attach the sign of the number that is larger in absolute value The number that is larger in absolute value is 13− , so the answer will be negative Attach a negative sign to 7
− + = −
The final answer is 7−
70 These two integers have different signs Take
the absolute value of each number and subtract the smaller absolute value from the larger
Subtract 12 from 20
20 12− = 8Attach the sign of the number that is larger in absolute value The number that is larger in absolute value is 20− , so the answer will be negative Attach a negative sign to 8
The final answer is 8−
71 The two integers are being subtracted Change
the sign of the second number and add Add the two numbers using the rule for adding integers, keeping the negative sign because both numbers are negative
− + − = −
72 The two integers are being subtracted Change
the sign of the second number and add Add the two numbers using the rule for adding integers, keeping the negative sign because both numbers are negative
6 ( 13) 19
73 Using the rule for subtracting integers, change
the sign of the second term and add the terms
− − − = − + Subtracting a negative is the same as adding so
we now have two integers with different signs
Take the absolute value of each number and subtract the smaller absolute value from the larger Subtract 4 from 8
we now have two integers with different signs
Take the absolute value of each number and subtract the smaller absolute value from the larger Subtract 1 from 7
7 1− = 6Attach the sign of the number that is larger in absolute value The number that is larger in absolute value is 7− , so the answer will be negative Attach a negative sign to 6
− + = −
The final answer is 6−
75 Using the rule for subtracting integers, change the sign of the second term and add the terms
12 ( 11) 12 11
Subtracting a negative is the same as adding so
we now have two integers with different signs
Take the absolute value of each number and subtract the smaller absolute value from the larger Subtract 11 from 12
12 11 1− =Attach the sign of the number that is larger in absolute value The number that is larger in absolute value is 12− , so the answer will be negative Attach a negative sign to 1
The final answer is 1−
76 Using the rule for subtracting integers, change the sign of the second term and add the terms
17 ( 18) 17 18
Subtracting a negative is the same as adding so
we now have two integers with different signs
Take the absolute value of each number and subtract the smaller absolute value from the larger Subtract 17 from 18
18 17− = 1Attach the sign of the number that is larger in absolute value Since the number that is larger
is positive, the answer is positive
17 18 1
NOT FOR SALE
INSTRUCTOR USE ONLY
Trang 577 The temperature can be found by subtracting
78 The temperature can be found by adding
6 20
− + These integers have different signs
The number with the larger absolute value is
20, therefore subtract 6 from 20
20 6− =14The number with the larger absolute value is positive 20, therefore the result is positive and the final temperature is 14 FD
79 Beginning with the depth of 30 feet below the surface, or −30 feet, and going another 20 feet deeper, or −20 feet This can be represented
Ray and Karin are at a depth of −50 feet
80 The starting elevation is 282 feet below see level, or −282 feet, and the ending elevation is
8360 feet above sea level, which is positive
8360 feet We need to find the difference between the two elevations which translates to subtraction and we are looking for the total change in elevation, a positive value, so we will use the absolute value and compute
282 8360
282 ( 8360)8642
81 To find the distance between these two points
we will use the formula b a− Let a= and 72
b= − and substitute into the formula
7 ( 2)
7 2 9 9
b− = − −a
= +
=
=
82 To find the distance between these two points
we will use the formula b a− Let a= and 56
b= − and substitute into the formula
5 ( 6)
5 6 11 11
b− = − −a
= +
=
=
83 To find the distance between these two points
we will use the formula b a− Let a = − 1and b= − and substitute into the formula 9
1 ( 9)
1 9 8 8
b− = − − −a
= − +
=
=
84 To find the distance between these two points
we will use the formula b a− Let a= − 8and b= −22 and substitute into the formula
8 ( 22)
8 22 14 14
b− = − − −a
= − +
=
=
85 To find the distance between these two points
we will use the formula b a− Let a= − 16and 35b= − and substitute into the formula
16 ( 35)
16 35 19 19
b− = − − −a
= − +
=
=
86 To find the distance between these two points
we will use the formula b a− Let a= − 26and b= − and substitute into the formula 5
26 ( 5)
26 5 21 21
Trang 6103. − ÷ ⋅ 24 3 7This expression has 1 term
104. 100 ( 2) 5⋅ − ÷ This expression has 1 term
105. 3+ 22 11÷ − ÷ 16 4This expression has 3 terms
106. 42 5÷ − −( 6)7+12This expression has 3 terms
107. 92 ( 8)⋅ − + 26 13÷ − +1 7( 3)−This expression has 4 terms
108. ( 5) ( 6)− ⋅ − − ÷ − +4 ( 2) 10 − −4( 2)This expression has 4 terms
109. This expression has 1 term
16 ( 8) 3 Outline the term
2 3 Work left to right
6
÷ − ⋅
= − ⋅
= −
110. This expression has 1 term
( 25) 5 ( 4) Outline the term
5 ( 4) Work left to right
20
= − ⋅ −
=
111. This expression has 2 terms
9 3 5 Outline the terms
9 15 Do operation inside each term
6
− + ⋅
= − +
=
112. This expression has 2 terms
Do operation inside each term.
6 ( 2) 18 Outline the terms
12 18 6
⋅ − +
= − +
=
113. This expression has 3 terms
Do operation inside each term.
Add from left to right.
9 24 3 3 ( 2) Outline the terms
114. This expression has 3 terms
Do operation inside each term.
Add/subtract from left to right.
56 ( 8) 3 5 ( 7) Outline the terms
115. This expression has 3 terms
Do operation inside each term.
Add/subtract from left to right.
Outline the terms.
14 2 5 4 10 ( 1)
7 20 10 1
7 2 1 4
= − + +
= −
NOT FOR SALE
INSTRUCTOR USE ONLY
Trang 7116. This expression has 3 terms
Do operation inside each term.
Add/subtract from left to right.
Outline the terms.
9 4 ( 6) ( 16) 8 2
6 16 16 26
117. This expression has 4 terms
Do operation inside each term.
Add from left to right.
Outline terms.
100 50 3 2 ( 7) 4 2 ( 2)
2 3 14 8 2
6 14 8 2 30
=
118. This expression has 4 terms
Do operation inside each term.
Add from left to right.
Outline terms.
32 ( 8) 3 7 9 48 12 5
4 21 9 4 5
4 21 9 20 12
Trang 812 12= ⋅ ⋅3 2 2
20= ⋅ ⋅5 2 2 The GCF= ⋅ =2 2 4
13 60= 5 3 2 2⋅ ⋅ ⋅
15= 5 3⋅ The GCF= ⋅ =5 3 15
14. 20= ⋅ ⋅5 2 2
36= ⋅ ⋅ ⋅3 3 2 2 The GCF= ⋅ =2 2 4
15 9= 3 3⋅
27= 3 3 3⋅ ⋅
18= 3 3 2⋅ ⋅ The GCF= ⋅ =3 3 9
16 21= 7 3⋅
49= 7 7⋅
14= 7 2⋅Only one factor in common, so the GCF is 7
17 48= 3⋅ ⋅ ⋅ ⋅2 2 2 2
36= 3 3 2⋅ ⋅ ⋅ 2
60= ⋅ ⋅ ⋅5 3 2 2 The GCF= ⋅ ⋅ =3 2 2 12
18 32= 2⋅ ⋅ ⋅ ⋅2 2 2 2
64= 2⋅ ⋅ ⋅ ⋅ ⋅2 2 2 2 2
16= 2⋅ ⋅ ⋅2 2 2 The GCF= ⋅ ⋅ ⋅ =2 2 2 2 16
19. 51 17 3= ⋅
27= ⋅ ⋅3 3 3Only one factor in common, so the GCF is 3
20. 72= 3 3⋅ ⋅ ⋅ ⋅2 2 2
18= 3 3⋅ ⋅2 The GCF= ⋅ ⋅ =3 3 2 18
=
25= 45
NOT FOR SALE
INSTRUCTOR USE ONLY
Trang 9= −
31 Comparing the denominators of 3 and 15, we have that 3 5⋅ =15 Therefore, we multiply the numerator and denominator by 5
Trang 1051
2 4 Add the numerators.
3 3 6 Reduce to lowest terms.
3 2
8 8 8 Reduce to lowest terms.
8 1
5 5 3 Answer is in lowest terms.
14 14 8 Reduce to lowest terms.
14 4 7
15 15 9 Reduce to lowest terms.
15 3 5
24 24 18 Reduce to lowest terms.
24 3 4
4 4 20 Reduce to lowest terms.
4 5
5 5 58 Reduce to lowest terms.
5 3 11 5
3 3 10 Reduce to lowest terms.
3 1 3
6 6 26 Reduce to lowest terms.
3 2 6
4 5 Add numerators.
6 6 9 Reduce to lowest terms.
4 2 8
2 3 Add numerators.
8 8 5 Answer is in lowest terms.
8 3 3 8
12 8 Add/subtract numerators.
24 24 4 Reduce to lowest terms.
24 1 6
NOT FOR SALE
INSTRUCTOR USE ONLY
Trang 1164. The factorization of 6 is 6= ⋅ and 4 is 3 2
12 12 7 Answer is in lowest terms.
5 2 10
2 7 Subtract numerators.
10 10 9 Answer is in lowest terms.
Rewrite as improper fractions
Rewrite each fraction over LCD.
6 3 2
7 22 Add numerators.
6 6
29 5 Answer is in lowest terms.
Rewrite as improper fractions
Rewrite each fraction over LCD.
Rewrite as improper fractions
Rewrite each fraction over LCD.
4 2 2
9 14 Subtract numerators.
4 4
23 3 Answer is in lowest terms.
Rewrite as improper fractions
Rewrite each fraction over LCD.
71 Add the two amounts of paint
Rewrite as improper fractions.
1 1
2 1
4 4
9 5 Add numerators.
Rewrite as improper fractions
Rewrite each fraction over LCD.
6 6
13 1 Answer is in lowest terms.
Trang 1273. Add the two amounts of salt Looking at the
denominators, the factorization of 4 is 4= ⋅ 2 2and 2 is prime TheLCD = ⋅ =2 2 4
Rewrite each fraction over LCD.
1 1
4 2
1 1 2
4 2 2
1 2 Add numerators.
4 4 3 Answer is in lowest terms.
74. Add the amounts of molding Looking at the
denominators, they are the same so we do not need to find a LCD Be sure to include the
given length for 2 sides of the doorway
Rewrite as improper fractions
75. Subtract the amount he needs from the total
amount he has Looking at the denominators, the factorization of 8 is 8= ⋅ ⋅ and 1 is 2 2 2prime TheLCD = ⋅ ⋅ = 2 2 2 8
Rewrite as improper fractions 3
20 10 8
20 83
1 8
20 8 83 Rewrite each fraction over LCD.
1 8 8
160 83 Subtract numerators.
Ted will have 95
8 feet of lumber left
76. Subtract the amounts she needs from the total
amount she has Looking at the denominators, the factorization of 4 is 4= ⋅2 2 and 2 and 1 are both prime TheLCD = ⋅ = 2 2 4
Rewrite as improper fractions
Hanna will have 3
4 pounds of onions left
77. To find a reciprocal, we invert the fraction To find the reciprocal of 3− , recall that 3 3
78. Find the reciprocal, which is 1
5
−
79. Find the reciprocal of 5
1=
82. Find the reciprocal of 1
12, which is
1212
85. The reciprocal of the number 0 does not exist
If 0 01
= and we invert 0
1, we would have
10which is undefined
86. Find the reciprocal, which is 1 1
4 12
⋅2
⋅ Factor Divide out like factors.
221
3 13
441
= =
NOT FOR SALE
INSTRUCTOR USE ONLY