1. The amount of all The amount of real rational numbers added numbers to the amount of all irrational numbers
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: The set of real numbers consists of all rational numbers and all irrational numbers. Real, both irrational and rational! The real numbers include all integers (whole number), fractions, and decimals (3 types; end, . The set of real numbers can be represented by a number line called the real number line.
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2. The amount of numbers that can be expressed as integers The amount of or that can be expressed as numbers on the real fractions or that can be number line expressed as decimals
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation The set of real numbers consists of all rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals. Thus, real numbers are expressed as decimals, and irrational numbers are expressed as decimals. The set of real numbers can be represented by a number line called the real number line.
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3. The amount of real The amount of points on numbers the number line
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: Every real number corresponds to a point on the number line, and every point on the number line corresponds to a real number.
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4. Assume the terms “left” and “right” represent directions on the real number line. Assume the unknown number is the same number in both comparison boxes. The amount of all the real numbers to the left of an unknown positive number and greater than the negative form of the unknown number
The amount of all the real numbers to the right of an unknown negative number and lesser than the positive form of the unknown number
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: On the number line, all numbers to the left of 0 are negative and all numbers to the right of 0 are positive. A real number x is less than a real number y if x is to the left of y on the number line, which is written as x < y. Are a lnumber y is greater than x if y is to the right of x on the number line, which is written as y > x.
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5. -!5
-2
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: B Explanation: -!5 < -2
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6. Using the double inequality 2 < x < 3 The sum of the endpoints of the interval
The sum of all the integers contained within the interval
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct:A Explanation: Left: A little more than 2 plus a little less than 3. Right: zero To say that a real number x is between 2 and 3 on the number line means that x > 2 and x < 3, which can also be written as the double inequality 2 < x < 3. The set of all real numbers that are between 2 and 3 is called an interval, and the double inequality 2 < x < 3 is often used to represent that interval. Note that the endpoints of the interval, 2 and 3, are not included in the interval. Sometimes one or both of the endpoints are to be included in an interval. The following inequalities represent four types of intervals, depending on whether the endpoints are included. 2<x<3 2<x<3 2<x<3 2<x<3 There are also four types of intervals with only one endpoint, each of which consists of all real numbers to the right or to the left of the endpoint, perhaps including the endpoint. The following inequalities represent these types of intervals. x<4 x<4 x>4 xâ&#x2030;Ľ4
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7. Using the double inequality 2 < x < 3 to answer question 7. The sum of the endpoints of the interval
The sum of all the integers contained within the interval
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: Left: 2 plus 3. Right: 2 plus 3. To say that a real number x is between 2 and 3 on the number line means that x > 2 and x < 3, which can also be written as the double inequality 2 < x < 3. The set of all real numbers that are between 2 and 3 is called an interval, and the double inequality 2 < x < 3 is often used to represent that interval. Note that the endpoints of the interval, 2 and 3, are not included in the interval. Sometimes one or both of the endpoints are to be included in an interval. The following inequalities represent four types of intervals, depending on whether the endpoints are included. 2<x<3 2<x<3 2<x<3 2<x<3 There are also four types of intervals with only one endpoint, each of which consists of all real numbers to the right or to the left of the endpoint, perhaps including the endpoint. The following inequalities represent these types of intervals. x<4 x<4 x>4 xâ&#x2030;Ľ4
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8. The total possible number The total possible number of methods for expressing of methods for expressing an interval with only one an interval with two endpoint endpoints A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: Left: 4. Right: 4. To say that a real number x is between 2 and 3 on the number line means that x > 2 and x < 3, which can also be written as the double inequality 2 < x < 3. The set of all real numbers that are between 2 and 3 is called an interval, and the double inequality 2 < x < 3 is often used to represent that interval. Note that the endpoints of the interval, 2 and 3, are not included in the interval. Sometimes one or both of the endpoints are to be included in an interval. The following inequalities represent four types of intervals, depending on whether the endpoints are included. 2<x<3 2<x<3 2<x<3 2<x<3 There are also four types of intervals with only one endpoint, each of which consists of all real numbers to the right or to the left of the endpoint, perhaps including the endpoint. The following inequalities represent these types of intervals. x<4 x<4 x>4 xâ&#x2030;Ľ4
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9. The distance between a number x and 0 on the number line
The absolute value of x
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information
Correct: C Explanation: The distance between a number x and 0 on the number line is called the absolute value of x, written as |x| . Therefore, 3 = 3 and -3 = 3 becauseeachofthenumbers3and -3 isadistanceof3from0. Notethatif x ispositive,then x = x;if x isnegative,then x = -x;andlastly, 0 = 0. Itfollowsthat the absolute value of any nonzero number is positive. Here are some examples.
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10.
â&#x2C6;Ł0â&#x2C6;Ł
0
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C
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11.
â&#x2C6;Ł!5â&#x2C6;Ł
!5
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C
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12.
â&#x2C6;Ł-23â&#x2C6;Ł
-(-23)
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C
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13.
â&#x2C6;Ł-1101.201â&#x2C6;Ł
1110.1201
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: B Explanation: Read carefully
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14.
(a + b + c) - d
d - (a + b + c)
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: D Explanation: Subtraction is an “order matters” situation. Division is also an “order matters” situation. However, when multiplying or adding order does not matter.
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15.
(a + b) + c - d
a + (b + c) - d
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C
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16.
ab + ac
a + (b + c)
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: D
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17.
ab + ac
a (b + c)
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C
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18.
a+0
(a)(0)
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: D
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19. -2b = 0
-2b
b
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C
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20. a and b are negative
a+b
(a)(b)
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: B
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21. a is positive, b is negative
(a)(b)
a+b
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: D Explanation: (4)(-1) = -4 (1)(-4) = -4 (4) + (-1) = -3 (1) + (-4) = -3 (1/2)(-3) = -1.5 (3)(-1/2) = -1.5 (1/2) + (-3) = -2.5 (3) + (-1/2) = -2.5
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22.
∣a+b∣
∣a∣ + ∣b∣
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: D Explanation: ∣ a + b ∣ < ∣a∣ + ∣b∣ This is known as the triangle inequality.
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23.
∣a∣ x ∣b∣
∣ ab ∣
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: ∣a∣ x ∣b∣ = |ab|
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24.
ab + ac
a + (b + c)
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: D
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25. 0 < b < 1
b"
b
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: B
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26. Which of the following is FALSE ? A.The set of real numbers consists of all rational numbers and all irrational numbers. B. The real numbers include all integers, fractions, and decimals. C. The set of real numbers can be represented by a number line called the real number line. D. Every real number corresponds to a point on the number line, and every point on the number line corresponds to a real number. E. Only the number 0 is either negative or positive. Correct: E Explanation: Only the number 0 is neither negative nor positive.
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27. Which of the following is FALSE ? A.A real number x is less than a real number y if x is to the left of y on the number line, which is written as x < y. B. A real number y is greater than x if y is to the right of x on the number line, which is written as y > x. C. To say that a real number x is between 2 and 3 on the number line means that x > 2 and x < 3, which can also be written as the double inequality 2 < x < 3. D. The set of all real numbers that are between 2 and 3 is called an interval, and the double inequality 2 < x < 3 is used to represent that interval. E. The endpoints are never to be included in an interval. Correct: E
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28. Which of the following is FALSE ? A. The entire real number line is also considered to be an interval.. B. The distance between a number x and 0 on the number line is called the absolute value of x, written as ∣x∣. C. If a and b are real numbers, then a + b = b + a and ab = ba. D. If a, b and c are real numbers, then (a + b) + c = a + (b + c) and (ab)c = a(bc). E. a(b + c) = abc + abc Correct: E Explanation: The entire real number line is also considered to be an interval. True. The endpoints of the number line are infinity and negative infinity. The distance between a number x and 0 on the number line is called the absolute value of x, written as ∣x∣. True. a(b + c) = abc + abc False. a(b + c) = ab + ac True.
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29. Which of the following is FALSE ? A. If a is a real number, then a + 0 = a, (a)(0) = 0, and (a)(1) = a. B. If both a and b are positive, then both a + b and ab are always real and positive . C. If a and b are real numbers, then If both a and b are negative, then a + b is negative and ab is positive. D. If a, b are real numbers, then ∣a+b∣ < ∣a∣ + ∣b∣ .This is known as the triangle inequality. E. If a and b are real numbers, then if a > 1, then a" > a. If 0 < b < 1, then b" < b. Correct: B Explanation: If a is a real number, then a + 0 = a, (a)(0) = 0, and (a)(1) = a. True. If both a and b are positive, then both a + b and ab are positive. True. If both a and b are real and negative, then a + b is negative and ab is positive. True If a is positive and b is negative, then ab is negative. True Any real number, when it is operated to an unreal number, becomes unreal. If both a and b are positive AND REAL, then both a + b and ab are always real and positive. True. * Note: Any real number, when it is operated to an unreal number, becomes unreal
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