The Number System: Master the Basics of Numbers

Question 1

Find the values of x that make the following equation true: (x + 3)(x - 2) = 0

Accepted Answer

x = 3 or x = -2

Answer

x = 6

Answer

x = -6

Answer

x = 1

Question 2

Let 'a' and 'b' be integers. Prove that 'a + (-b)' is the additive inverse of 'a'.

Accepted Answer

The additive inverse of 'a' is '-a', so 'a + (-b)' = 'a - b'.

Answer

The additive inverse of 'a' is '-a', so 'a + (-b)' = '-a + (-b)' = '-(a + b)'.

Answer

The additive inverse of 'a' is 'a - 1', so 'a + (-b)' = 'a - 1 + (-b)' = 'a - (b + 1)'.

Question 3

Simplify the expression: (-3) + (2 - 5i)

Accepted Answer

-1 - 5i

Answer

1 - 5i

Answer

-5 - i

Answer

5 - i

Question 4

Which of the following is NOT a property of real numbers?

Accepted Answer

Reflexive property of inequalities

Answer

Commutative property of addition

Answer

Associative property of multiplication

Answer

Distributive property

Question 5

Solve the inequality: |x - 2| > 5

Accepted Answer

x < -3 or x > 7

Answer

x > -3 or x < 7

Answer

x < -5 or x > 3

Answer

x > 5 or x < -3

Question 6

Which of the following is NOT a rational number?

Accepted Answer

π

Answer

1/2

Answer

0.125

Answer

√(4)

Question 7

Which of the following is equivalent to √(16x²)?

Accepted Answer

4x

Answer

8x

Answer

2x

Answer

4|x|

Question 8

Find the greatest common factor (GCF) of 24 and 36

Accepted Answer

6

Answer

2

Answer

4

Answer

12

Question 9

Which of the following is NOT a real number?

Accepted Answer

√-1

Answer

0

Answer

π

Answer

e

Question 10

Simplify the expression: |5 - 7|

Accepted Answer

2

Answer

-2

Answer

-12

Answer

12

Question 11

Which of the following is an irrational number?

Accepted Answer

3.14

Answer

√9

Answer

1/2

Answer

-5

Question 12

Solve for x: 2x - 5 = 15

Accepted Answer

10

Answer

5

Answer

15

Answer

20

Question 13

Which of the following represents the set of all rational numbers?

Accepted Answer

{x | x can be expressed as a fraction}

Answer

{x | x is a real number}

Answer

{x | x is an integer}

Answer

{x | x is a positive number}

Question 14

Identify the greatest common factor (GCF) of 18 and 24.

Accepted Answer

6

Answer

2

Answer

4

Answer

8

Question 15

Given a rational number 'a', prove that its additive inverse, denoted as '-a', is also rational.

Accepted Answer

'-a' is rational because it can be expressed as a fraction with integers for both the numerator and non-zero denominator.

Answer

'-a' is irrational because it cannot be expressed as a fraction with integers for both the numerator and non-zero denominator.

Answer

'-a' is a natural number because it is a positive integer.

Question 16

Are the numbers 1/3 and 0.333... (repeating) equivalent? Explain your reasoning.

Accepted Answer

Yes, they are equal because they represent the same value.

Answer

No, they are not equal because 1/3 is a fraction and 0.333... (repeating) is a decimal.

Answer

Yes, they are equal only if 0.333... (repeating) eventually ends.

Answer

No, they are not equal because 1/3 is greater than 0.333... (repeating).

Question 17

Determine whether the number -√(5) is rational or irrational. Explain your reasoning.

Accepted Answer

Irrational, because -√(5) cannot be expressed as a fraction.

Answer

Rational, because -√(5) can be expressed as a fraction.

Answer

Rational, because -√(5) is a square root.

Answer

Irrational, because -√(5) is a negative number.

Question 18

Prove that the product of any two rational numbers is always a rational number.

Accepted Answer

The product of two rational numbers is always a rational number.

Answer

The product of two rational numbers is always an integer.

Answer

The product of two rational numbers is always an irrational number.

Answer

The product of two rational numbers may be a rational or irrational number.

Question 19

Prove that the sum of any two rational numbers is also a rational number.

Accepted Answer

Let a and b be rational numbers. Then, their sum, a + b, will also be rational because it can be expressed as a fraction of two integers.

Answer

The sum of two rational numbers may not be rational because rational numbers do not have closure under addition.

Answer

All rational numbers are positive, so their sum must also be positive, which implies that the sum is rational.

Answer

The sum of two rational numbers can be an irrational number.

Question 20

Solve the inequality |2x - 5| > 3 and show the solution on a number line.

Accepted Answer

x < 1 or x > 4

Answer

x > 1 or x < 4

Answer

x > 2 or x < 3

Answer

x < -1 or x > -4

Question 21

Prove that the sum of any two rational numbers is always a rational number. Explain your reasoning.

Accepted Answer

The sum of two rational numbers is always rational because rational numbers are closed under addition. This means that when you add two rational numbers, the result will also be a rational number.

Answer

The sum of two rational numbers can be irrational if the numbers are very close to each other.

Answer

Rational numbers are only closed under multiplication, not addition.

Answer

The sum of two rational numbers is always an integer.

Question 22

Solve the inequality: |x + 3| < 5. Show your work and graph the solution on a number line.

Accepted Answer

-3 < x < 2

Answer

-2 < x < 3

Answer

|x + 3| = 5

Answer

-5 < x < 5

Question 23

What are irrational numbers? Give examples and explain their significance and applications in different fields.

Accepted Answer

Irrational numbers are non-terminating, non-repeating decimals, like √2 and π.

Answer

Irrational numbers are rational numbers rounded off to an infinite number of decimal places.

Answer

Irrational numbers are complex numbers that don't exist on the real number line.

Answer

Irrational numbers have no practical applications.

Question 24

Find the least common multiple (LCM) of 4 and 6.

Accepted Answer

12

Answer

2

Answer

4

Answer

24

Question 25

Simplify the expression: (2x²y)³

Accepted Answer

6x⁶y³

Answer

8x⁶y³

Answer

6x⁵y²

Answer

8x⁵y²

Question 26

Which of the following statements correctly describes the number 0?

Accepted Answer

It is neither positive nor negative.

Answer

It is both positive and negative.

Answer

It is positive but not negative.

Answer

It is negative but not positive.

Question 27

Find the number that is opposite to -5.

Accepted Answer

5

Answer

-5

Answer

10

Answer

-10

Question 28

Which of the following is not a real number?

Accepted Answer

∞

Answer

0

Answer

√2

Answer

π

Question 29

Which of the following numbers is a rational number?

Accepted Answer

0.123

Answer

√3

Answer

π

Answer

e

Question 30

Which of the following is equivalent to 2^5?

Accepted Answer

2^10

Answer

5^2

Answer

10^2

Answer

5^10

Question 31

Prove that the product of two irrational numbers is always irrational.

Accepted Answer

Assume the product of two irrational numbers is rational and derive a contradiction.

Answer

Irrational numbers are not closed under multiplication.

Answer

The product of any two irrational numbers is always rational.

Answer

It's impossible to determine the product's nature without knowing the specific irrational numbers.

Question 32

Classify the number as rational or irrational: 3.14159265358979...

Accepted Answer

Irrational

Answer

Rational

Question 33

Are the numbers 1/3 and 0.333... (repeating) equal? Explain your reasoning.

Accepted Answer

Yes, they are equal because they represent the same value.

Answer

No, they are not equal because 1/3 is a rational number and 0.333... (repeating) is an irrational number.

Answer

Yes, they are equal because they have the same decimal representation.

Answer

No, they are not equal because 1/3 is a fraction and 0.333... (repeating) is a decimal.

Question 34

When x and y are rational numbers, why must their sum, x + y, also be rational?

Accepted Answer

The sum of two rational numbers is always rational.

Answer

The sum of two rational numbers can be irrational.

Answer

The sum of two rational numbers is always an integer.

Question 35

Which of the following is NOT a real number?

Accepted Answer

√(-1)

Answer

0

Answer

π

Answer

1/2

Question 36

Compare the following numbers: √3 and 2

Accepted Answer

√3 > 2

Answer

√3 < 2

Answer

√3 = 2

Answer

Cannot be determined

Question 37

Solve for x: |x - 3| = 5

Accepted Answer

x = -2 or x = 8

Answer

x = 2 or x = 8

Answer

x = -8 or x = 2

Answer

x = -2 or x = -8

Question 38

Which of the following inequalities is true for all real numbers?

Accepted Answer

x² ≥ 0

Answer

x² > 0

Answer

x² < 0

Answer

x² > 0 for all positive x

Question 39

Which of the following is NOT a property of real numbers?

Accepted Answer

Identity property of division

Answer

Associative property of addition

Answer

Inverse property of multiplication

Answer

Distributive property of multiplication over subtraction

Question 40

Which property of the real number system ensures that every set of real numbers with an upper bound has a least upper bound?

Accepted Answer

Completeness property

Answer

Closure under addition

Answer

Associativity of multiplication

Answer

Distributive property

Question 41

Identify the option that does not belong to the subset of real numbers:

Accepted Answer

Imaginary numbers

Answer

Integers

Answer

Rational numbers

Answer

Irrational numbers

Question 42

Which basic arithmetic operation is undefined for some real numbers?

Accepted Answer

Division

Answer

Addition

Answer

Subtraction

Answer

Multiplication

Question 43

Using the order of operations, simplify the expression: 2 + 3 × (4 - 1)

Accepted Answer

9

Answer

5

Answer

11

Answer

13

Question 44

Which of the following is not a rational number?

Accepted Answer

√2

Answer

-3/7

Answer

0

Answer

0.5

Question 45

Which of the following is equivalent to the decimal 0.625?

Accepted Answer

5/8

Answer

1/4

Answer

1/16

Answer

3/5

Question 46

Find the value of x if |x| = 5:

Accepted Answer

±5

Answer

0

Answer

5

Answer

-5

Question 47

Simplify the expression √(16x^4y^2):

Accepted Answer

4x^2y

Answer

2x^2y

Answer

16x^2y^2

Answer

8x^2y

Question 48

Which of the following is NOT considered a rational number?

Accepted Answer

√2

Answer

0.5

Answer

8.333

Answer

-3/4

Question 49

Which of the following is NOT a subset of rational numbers?

Accepted Answer

Irrational numbers

Answer

Decimals

Answer

Terminating decimals

Answer

Integers

Question 50

Solve the absolute value equation for x: |x + 3| = 5

Accepted Answer

x = 2 or x = -8

Answer

x = 2

Answer

x = 5 or x = -5

Answer

x = -8

Question 51

Identify the quadratic inequality from the given options:

Accepted Answer

x² - 3x + 2 > 0

Answer

√(x + 1) > 2

Answer

x / 2 - 3 < 1

Answer

x³ + 2x² - 1 = 0

Question 52

Which of the following is NOT a subset of the real numbers?

Accepted Answer

Natural numbers

Answer

Rational numbers

Answer

Irrational numbers

Answer

Integers

Question 53

Which of the following is an irrational number?

Accepted Answer

π

Answer

0.25

Answer

1/2

Answer

-5

Question 54

Simplify the expression: |3 - (-4)| + |-2|

Accepted Answer

9

Answer

1

Answer

-9

Answer

5

Question 55

Which of the following is equivalent to the decimal 0.75?

Accepted Answer

3/4

Answer

1/2

Answer

5/8

Answer

1/4

Question 56

Which of the following is an example of a rational number? Rational numbers can be expressed as a fraction of two integers.

Accepted Answer

-1/3

Answer

√3

Answer

π

Question 57

Which of the following is NOT a subset of the real number system? The real number system includes all rational and irrational numbers.

Accepted Answer

Complex numbers

Answer

Integers

Answer

Rational numbers

Answer

Irrational numbers

Question 58

Which of the following is an example of an irrational number? Irrational numbers cannot be expressed as a fraction of two integers.

Accepted Answer

√2

Answer

1/3

Answer

-7

Answer

0.5

Question 59

Which of the following is the smallest number? Real numbers can be compared using their value on the number line.

Accepted Answer

-3.14

Answer

-2

Answer

0

Answer

2

Question 60

Which of the following is the greatest common factor (GCF) of 12 and 18? The GCF is the largest factor that divides both numbers without leaving a remainder.

Accepted Answer

6

Answer

2

Answer

3

Answer

12

Question 61

Which of the following is the least common multiple (LCM) of 6 and 8? The LCM is the smallest number that is divisible by both numbers.

Accepted Answer

24

Answer

12

Answer

16

Question 62

Which of these is NOT a real number?

Accepted Answer

√-1

Answer

0

Answer

-7

Answer

3.14

Question 63

Which of these is a rational number?

Accepted Answer

-3/4

Answer

√2

Answer

π

Answer

0.1010010001...

Question 64

Which of these is the least number?

Accepted Answer

-√9

Answer

10

Answer

1

Answer

0