2.  Evaluate 3y when y = 45.
Solution: 3y means the product of 3 and y.
               If y = 45, then 3y means (3)(45) which means the product of 3 and 45 which is 135.

4.  Evaluate 7.1a when a = 1.5.
Solution: 7.1a means the product of 7.1 and a.
               If a = 1.5, then 7.1a means (7.1)(1.5) which means the product of 7.1 and 1.5 which is 10.65.

8. Evaluate 2a - b when a = 12 and b = 7.
Solution: Substitute 12 for a and 7 for b in the expression 2a - b to obtain (2)(12) - 7 = 17

10. The algebraic expression 1.5x gives the total length of shelf space needed (in inches) for x encylopedias. Find the length of shelf space needed for a set of 30 encyclopedias.
Solution:  Notice that x is a variable which represents the number of encylopedias. To determine the required length of shelf space we replace x with 30, the number of encylopedias. This yields (1.5)(30) = 45 inches.

16. List the elements of the set { x | x is a natural number greater than 6 }
Solution:  Since natural numbers are the common counting numbers this set is { 7, 8, 9, ... }.

18. List the elements of the set { x | x is an odd natural number}
Solution:  Since natural numbers are the common counting numbers and odd numbers are not divisible by 2 the set is { 1, 3, 5, 7, ... }

20. List the elements of the set {x | x is a natural number less than 1}
Solution:  Since natural numbers are the common counting numbers none of them are greater than 1. Therefore the set described has no elements. It is the empty set which is denoted by

22.  List the elements of the set { x | x is an odd whole number less than 9 }
Solution:  Since whole numbers are 0 and the common counting numbers and odd numbers are those not divisible by 2 the set is { 1, 3, 5, 7 }

24. Graph the set { -1, -2, -3 }
Solution:   

26. Graph the set {1, 3, 5, 7 }
Solution:  

28. Graph the set  
Solution:    

31. List the elements of that are whole numbers.
Solution:

32. List the elements of that are integers.
Solution:

33. List the elements of that are natural numbers.
Solution:

34. List the elements of that are rational numbers.
Solution:

35. List the elements of that are irrational numbers.
Solution:

36. List the elements of that are real numbers.
Solution:

38. Determine whether -6 is or is not an element of the set {2, 4, 6, ...}.
Solution:  To determine if a mathematical object is an element of a particular set, it is necessary to determine if the object satisfies the rule for the set. The rule for this particular set {2, 4, 6, ...} is by inference -- by looking at the pattern we are to infer that this set consists of even natural numbers. The mathematical object -6 is an even number but it is not a natural number. Therefore - 6 is not an element of {2, 4, 6, ...} and we write

40. Determine whether 12 is or is not an element of the set {1, 2, 3, …}.
Solution:  To determine if a mathematical object is an element of a particular set, it is necessary to determine if the object satisfies the rule for the set. The rule for this particular set {1, 2, 3, …} is by inference -- by looking at the pattern we are to infer that this set consists of all natural numbers. The mathematical object 12 is a natural number.
Therefore 12 is an element of {1, 2, 3, …} and we write

42. Determine whether is or is not an element of the set {x | x is an irrational number}.
Solution:  To determine if a mathematical object is an element of a particular set, it is necessary to determine if the object satisfies the rule for the set. When the set is specified with set-builder notation (as this set is) the easiest way to determine if the objects satisfies the rule is to replace x in the rule with the object and determine if the statement is true or false.
In this case we obtain the statement;    is an irrational number. This statement is false, so we conclude that is not an element of the set {x | x is an irrational number} and we write

44. Determine whether 0 is or is not an element of the set {x | x is a natural number}.
Solution:  To determine if a mathematical object is an element of a particular set, it is necessary to determine if the object satisfies the rule for the set. When the set is specified with set-builder notation (as this set is) the easiest way to determine if the objects satisfies the rule is to replace x in the rule with the object and determine if the statement is true or false.
In this case we obtain the statement;   0 is a natural number. This statement is false, so we conclude that 0 is not an element of the set {x | x is a natural number} and we write

46.  
Because R has elements which are not in N. For example -7 is in R but is not in N

48.  
Because the elements of Q are all those numbers which can be written as the quotient of two integers. Clearly is a quotient of two integers and is therefore an element of Q.

50.  
Because every element of Z is an element of Q. To see why that is true, recall the rule for Q is all numbers which can be written as the quotient of two integers. Every element of Z is an integer and can be written as fraction with a denominator of 1. Therefore every element of Z can be written as the quotient of two integers and is therefore an element of Q.

52.   is false.
Because π is an irrational number -- the classic example of an irrational number-- and is therefore a real number.

54.   is false.
If there is just a single element of I which is not an element of N, then the definition of subset is violated and I is not a subset of N.   In fact there are many integers (elements of I) which are not natural numbers (elements of N). For example -3 is an element of I but is not an element of N.

56.   is true.
Because every element of N is an element of Q. To see why that is true, recall the rule for Q is all numbers which can be written as the quotient of two integers. Every element of N is an natural number (therefore an integer) and can be written as fraction with a denominator of 1. Therefore every element of N can be written as the quotient of two integers and is therefore an element of Q.

59. -|2| = -2	The absolute value of 2 is 2 because it is 2 units from the origin. Then its opposite is -2	60. |8| = 8	8 is 8 units from the origin
61. |-4| = 4	The absolute value of - 4 is 4 because - 4 is 4 units from the origin.	62. |-6| = 6	The absolute value of - 6 is 6 because - 6 is 6 units from the origin.
63. |0| = 0	Its distance from the origin is 0	64. |-1| = 1	-1 is 1 unit from the origin.
65. -|-3| = - 3	The absolute value of -3 is 3 because -3 is 3 units from the origin. The opposite of 3 is -3	66. -|-11| = - 11	The absolute value of -11 is 11 because -11 is 11 units from the origin. The opposite of 11 is -11
69. The opposite of -6.2 is 6.2. This is written as -(-6.2) = 6.2		70. The opposite of -7.8 is 7.8. This is written as -(-7.8) = 7.8
71.		72.
74.		76. The opposite of 10.3 is -10.3

78. The phrase "Six times a number" may be written as an algebraic expression by using the variable x to represent the number and using juxtaposition as an indication of multiplication to obtain 6x.

80. The phrase "One more than six times a number" may be written as an algebraic expression by using the variable x to represent the number.
Use juxtaposition to indicate multiplication to obtain 6x as an algebraic representation of "six times a number".
The use normal addition to obtain + 1 as an algebraic representation of "one more".
Putting it all together yields 6x + 1 as an algebraic expression for the phrase "One more than six times a number".

82. The phrase "A number minus 7" may be written as an algebraic expression by using the variable x to represent the number and normal subtraction to translate "minus 7".
This yields x - 7 as an algebraic expression for "A number minus 7"

84. The phrase "The difference of twenty-five minus a number" may be translated into an algebraic expression by lettin x represent the number and using normal subtraction to translate "minus". This yields 25 - x.

86. The phrase "The quotient of twice a number divided by 13" may be translated into an algebraic expression by letting; x represent the number, juxtaposition represent multiplication, and the fraction bar represent division. We then obtain

88. The phrase "Four subtracted from a number" may be translated into an algebraic expression by letting x represent the number to obtain x - 4.

90. The phrase "Four subtracted from three times a number" becomes 3x - 4 when translated into an algebraic expression.