1-2 Properties of Real Numbers
Name the sets of numbers to which each number belongs.
2. SOLUTION: The number is a real number. Since can be expressed as a ratio where a and b are integers and b is not 0 it
is also a rational number. It is not a part of the set {…–2, –1, 0, 1, 2, …} so it is not an integer. Since it is not a part
of the set {…0, 1, 2, 3, …} it is not a whole number or a natural number.
Q, R
4. –12
SOLUTION: The number -12 is a real number. Since -12 can be expressed as a ratio where a and b are integers and b is not 0
it is also a rational number. It is part of the set {…–2, –1, 0, 1, 2, …} so it is an integer. It is not part of the set {…0,
1, 2, 3, …} so it is not a whole number and since it is not a whole number it is not a natural number either.
Z, Q, R
Name the property illustrated by each equation.
6. SOLUTION: Distributive Property; the Distributive Property states that there is no difference between a term multiplied by each
term in a group and the term multiplied by the group.
8. SOLUTION: Distributive Property; the Distributive Property states that there is no difference between a term multiplied by each
term in a group and the term multiplied by the group.
Find the additive inverse and multiplicative inverse for each number.
10. SOLUTION: Since
Since
, the additive inverse of
is , the multiplicative inverse of
.
is .
12. SOLUTION: Since
, the additive inverse of
Since
, the multiplicative inverse of
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Simplify
expression.
14. is is .
.
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Since
, the additive inverse of
1-2 Properties
of Real, Numbers
Since
the multiplicative inverse of
is is .
.
Simplify each expression.
14. SOLUTION: 16. SOLUTION: Name the sets of numbers to which each number belongs.
18. SOLUTION: The number
is a real number. Since
can be expressed as a ratio where a and b are integers and b is not 0
it is also a rational number. It is not a part of the set {…-2, -1, 0, 1, 2, …} so it is not an integer. Since it is not a part
of the set {…0, 1, 2, 3, …} it is not a whole number or a natural number.
Q, R
20. SOLUTION: Since
= 5, this is a real number. Since 5 can be expressed as a ratio where a and b are integers and b is not
0 it is also a rational number. It is part of the set {…-2, -1, 0, 1, 2, …} so it is an integer. It is part of the set {…0, 1,
2, 3, …} so it is a whole number and since it is not 0 it is also a natural number.
N, W, Z, Q, R
22. SOLUTION: The number
= 3 and is a real number. Since 3 can be expressed as a ratio
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where a and b are integers and b is
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not 0 it is also a rational number. It is part of the set {…-2, -1, 0, 1, 2, …} so it is an integer. It is part of the set {…0,
1, 2, 3, …} so it is a whole number and since it is not 0 it is also a natural number.
N, W, Z, Q, R
Since
= 5, this is a real number. Since 5 can be expressed as a ratio
where a and b are integers and b is not
0 it is also a rational number. It is part of the set {…-2, -1, 0, 1, 2, …} so it is an integer. It is part of the set {…0, 1,
2, 3, …} so of
it isReal
a whole
number and since it is not 0 it is also a natural number.
1-2 Properties
Numbers
N, W, Z, Q, R
22. SOLUTION: The number
= 3 and is a real number. Since 3 can be expressed as a ratio
where a and b are integers and b is
not 0 it is also a rational number. It is part of the set {…-2, -1, 0, 1, 2, …} so it is an integer. It is part of the set {…0,
1, 2, 3, …} so it is a whole number and since it is not 0 it is also a natural number.
N, W, Z, Q, R
24. SOLUTION: The number = 3 and is a real number. Since 3 can be expressed as a ratio where a and b are integers and b is
not 0 it is also a rational number. It is part of the set {…–2, –1, 0, 1, 2, …} so it is an integer. It is part of the set {…
0, 1, 2, 3, …} so it is a whole number and since it is not 0 it is also a natural number.
N, W, Z, Q, R
Name the property illustrated by each equation.
26. SOLUTION: Additive Inverse Property; the Additive Inverse Property states that a number added to its opposite is zero.
28. SOLUTION: Associative Property of Addition; the Associative Property of Addition states that the way the factors are grouped
does not affect the sum.
Find the additive inverse and multiplicative inverse for each number.
30. –8
SOLUTION: Since −8 + 8 = 0, the additive inverse of −8 is 8.
Since
, the multiplicative inverse of −8 is
.
32. –0.25
SOLUTION: Since −0.25 + 0.25 = 0, the additive inverse of −0.25 is 0.25.
, the multiplicative inverse of −0.25 is −4.
Since
34. SOLUTION: Since
, the
additive
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Since
inverse of
is .
, the multiplicative inverse of
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is .
SOLUTION: Since −0.25 + 0.25 = 0, the additive inverse of −0.25 is 0.25.
1-2 Properties
of Real Numbers
Since
, the multiplicative inverse of −0.25 is −4.
34. SOLUTION: Since
Since
, the additive inverse of
is .
, the multiplicative inverse of
is .
36. CONSTRUCTION Jorge needs two different kinds of concrete: quick drying and slow drying. The quick-drying
concrete mix calls for
pounds of dry cement, and the slow-drying concrete mix calls for
pounds of dry cement. He needs 5 times more quick-drying concrete and 3 times more slow-drying concrete than the mixes make.
a. How many pounds of dry cement mix will he need?
b. Use the properties of real numbers to show how Jorge could compute this amount mentally. Justify each step.
SOLUTION: a. Write an expression. Jorge needs 5 times the amount of dry cement,
amount of dry cement,
He will need
, for the quick-drying mix plus 3 times the
, for the slow-drying mix.
pounds of dry cement.
b.
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Since
, the additive inverse of
is .
1-2 Properties
of Real, Numbers
Since
the multiplicative inverse of
is .
36. CONSTRUCTION Jorge needs two different kinds of concrete: quick drying and slow drying. The quick-drying
concrete mix calls for
pounds of dry cement, and the slow-drying concrete mix calls for
pounds of dry cement. He needs 5 times more quick-drying concrete and 3 times more slow-drying concrete than the mixes make.
a. How many pounds of dry cement mix will he need?
b. Use the properties of real numbers to show how Jorge could compute this amount mentally. Justify each step.
SOLUTION: a. Write an expression. Jorge needs 5 times the amount of dry cement,
amount of dry cement,
He will need
, for the quick-drying mix plus 3 times the
, for the slow-drying mix.
pounds of dry cement.
b.
Simplify each expression.
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38. SOLUTION: Page 5
1-2 Properties of Real Numbers
Simplify each expression.
38. SOLUTION: 40. SOLUTION: 42. SOLUTION: 44. PETS The chart shows the percent of dogs registered with the American Kennel Club that are of the eight most
popular breeds.
a. Illustrate the Distributive Property by writing two expressions to represent the number of registered dogs of the
top four breeds.
b. Evaluate the expressions you wrote to find the number of registered dogs of the top four breeds.
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SOLUTION: Page 6
1-2 Properties of Real Numbers
44. PETS The chart shows the percent of dogs registered with the American Kennel Club that are of the eight most
popular breeds.
a. Illustrate the Distributive Property by writing two expressions to represent the number of registered dogs of the
top four breeds.
b. Evaluate the expressions you wrote to find the number of registered dogs of the top four breeds.
SOLUTION: a. Expression representing the number of registered dogs of the top four breeds is 870,192(0.142 + 0.056 + 0.05 +
0.049).
Use the Distributive Property to rewrite the expression.
870,192(0.142 + 0.056 + 0.05 + 0.049) = 870,192(0.142) + 870,192(0.056) + 870,192(0.05) + 870,192(0.049)
b. So, the number of registered dogs of the top four breeds is about 258,447.
Simplify each expression.
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1-2 Properties of Real Numbers
So, the number of registered dogs of the top four breeds is about 258,447.
Simplify each expression.
46. SOLUTION: 48. SOLUTION: 50. MODELING Mary is making curtains out of the same fabric for 5 windows. The two larger windows are the same
size, and the three smaller windows are the same size. One larger window requires
smaller window needs
yards of fabric, and one
yards of fabric.
a. How many yards of material will Mary need?
b. Use the properties of real numbers to show how Mary could compute this amount mentally.
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a. Since one larger window requires
yards of fabric and a smaller window requires yards of fabric, the Page 8
1-2 Properties of Real Numbers
50. MODELING Mary is making curtains out of the same fabric for 5 windows. The two larger windows are the same
yards of fabric, and one
size, and the three smaller windows are the same size. One larger window requires
smaller window needs
yards of fabric.
a. How many yards of material will Mary need?
b. Use the properties of real numbers to show how Mary could compute this amount mentally.
SOLUTION: a. Since one larger window requires
yards of fabric and a smaller window requires expression that represents the requirement of total yards of fabric is
So, Mary requires
yards of fabric, the .
yards of fabric.
b.
52. CLOTHING A department store sells shirts for $12.50 each. Dalila buys 2, Latisha buys 3, and Pilar buys 1.
a. Illustrate the Distributive Property by writing two expressions to represent the cost of these shirts.
b. Use the Distributive Property to find how much money the store received from selling these shirts.
SOLUTION: a. 12.50(2 + 3 + 1)
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Use the Distributive Property to rewrite the expression.
12.50(2 + 3 + 1) = 12.50 ∙ 2 + 12.50 ∙ 3 + 12.50 ∙ 1
b.
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1-2 Properties of Real Numbers
52. CLOTHING A department store sells shirts for $12.50 each. Dalila buys 2, Latisha buys 3, and Pilar buys 1.
a. Illustrate the Distributive Property by writing two expressions to represent the cost of these shirts.
b. Use the Distributive Property to find how much money the store received from selling these shirts.
SOLUTION: a. 12.50(2 + 3 + 1)
Use the Distributive Property to rewrite the expression.
12.50(2 + 3 + 1) = 12.50 ∙ 2 + 12.50 ∙ 3 + 12.50 ∙ 1
b.
So, the store received $75.
54. CHALLENGE If 12(5r + 6t) = w, then in terms of w, what is 48(30r + 36t)?
SOLUTION: 56. REASONING Determine whether the following statement is sometimes, always, or never true. Explain your
reasoning.
An irrational number is a real number underneath a radical sign.
SOLUTION: Sometimes; π and e are two examples of irrational numbers that do not involve the radical symbol but 117 is a real
number while
is irrational.
OPEN ENDED The set of all real numbers is dense, meaning between any two distinct members of the
set there lies infinitely many other members of the set. Find an example of (a) a rational number, and (b)
an irrational number between the given numbers.
58. 2.45 and 2.5
SOLUTION: Sample answer:
a. 2.46
b. 2.48448444844448…
60. and
SOLUTION: Sample answer:
a. 2.001
b. 2.001000100001…
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EXTENDED
RESPONSE
62. Page 10
Lenora bought several pounds of cashews and several pounds of almonds for a party.
The cashews cost $8 per pound, and the almonds cost $6 per pound. Lenora bought a total of 7 pounds and paid a
total of $48. Write and solve equations to determine the pounds of cashews and the pounds of almonds that Lenora
SOLUTION: Sample answer:
a. 2.001 of Real Numbers
1-2 Properties
b. 2.001000100001…
62. EXTENDED RESPONSE Lenora bought several pounds of cashews and several pounds of almonds for a party.
The cashews cost $8 per pound, and the almonds cost $6 per pound. Lenora bought a total of 7 pounds and paid a
total of $48. Write and solve equations to determine the pounds of cashews and the pounds of almonds that Lenora
purchased.
SOLUTION: Let c be the number of pounds of cashews and a be the number of pounds of almonds.
Eight pounds of cashews and 6 pounds of almonds costs $48. So, 8c + 6a = 48.
Lenora bought 7 pounds of cashews and almonds. So, c + a = 7.
Substitute c = 7 – a in the equation 8c + 6a = 48.
Substitute a = 4 in the equation c = 7 – a.
Therefore, Lenora bought 3 pounds of cashews and 4 pounds of almonds.
64. GEOMETRY What are the coordinates of point A in the parallelogram?
F (b – a, c) H (b, c)
G (a – b, c) J (c, c)
SOLUTION: Since the points A and B are in the horizontal line segment, the y-coordinate of the point A is equal to the ycoordinate of B. From the figure, the x-coordinate of A is negative. Since OABC is a parallelogram, the distance
between O and C is same as the distance between A and B. Since b > a, the x-coordinate of A is a − b. So, the
coordinate of A is
A(a − b, c). So, the correct choice is G.
3
66. Evaluate 8(4 – 2) .
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Since the points A and B are in the horizontal line segment, the y-coordinate of the point A is equal to the ycoordinate of B. From the figure, the x-coordinate of A is negative. Since OABC is a parallelogram, the distance
between O and C is same as the distance between A and B. Since b > a, the x-coordinate of A is a − b. So, the
coordinate of
is Numbers
1-2 Properties
ofAReal
A(a − b, c). So, the correct choice is G.
3
66. Evaluate 8(4 – 2) .
SOLUTION: 68. GEOMETRY The formula for the area A of a circle with diameter d is
. Write an expression to represent the area of the circle.
SOLUTION: Substitute d = (x + 3) in the formula
The area of the circle is
.
.
Factor each polynomial.
70. SOLUTION: 2
The GCF of the terms 14x , 10x and 8 is 2.
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2
The GCF of the terms 8x , 16x and 12 is 2 ∙ 2 or 4.
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1-2 Properties of Real Numbers
72. SOLUTION: 2
The GCF of the terms 8x , 16x and 12 is 2 ∙ 2 or 4.
74. SOLUTION: 2
The GCF of the terms 7x , 14x and 21 is 7.
Find each product.
76. SOLUTION: Use the FOIL method to find the product.
78. SOLUTION: Use the FOIL method to find the product.
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Use the FOIL method to find the product.
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Use the FOIL method to find the product.
1-2 Properties of Real Numbers
80. SOLUTION: Use the FOIL method to find the product.
Evaluate each expression if a = 3,
, and c = –1.7.
82. 6b – 5
SOLUTION: 84. 2.3c – 7
SOLUTION: 86. a + b + c
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SOLUTION: 1-2 Properties of Real Numbers
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