What is the result of adding any real number x and 0?

• x + 0 = x

• x + 0 = 0

• x + 0 = 1

Answer explanation

Adding any real number x and 0 results in x itself, as 0 is the additive identity. Therefore, the correct answer is x + 0 = x.

What is the result of multiplying any real number x and 1?

• x × 1 = x

• x × 1 = 1

• x × 1 = 0

• x × 1 = x + 1

Answer explanation

Multiplying any real number x by 1 results in the same number x. This is due to the identity property of multiplication, which states that any number multiplied by 1 remains unchanged.

For any real number x, what is the additive inverse property?

• For any real number x, there is a unique number, denoted by –x, such that x + (–x) = 0.

• For any real number x, there is a unique number, denoted by x, such that x - x = 1.

• For any real number x, there is a unique number, denoted by –x, such that x × (–x) = 0.

Answer explanation

The additive inverse property states that for any real number x, there exists a unique number –x such that their sum equals zero: x + (–x) = 0. This is correctly described in the first answer choice.

For any real number x different from 0, what is the multiplicative inverse property?

• For any real number x different from 0, there is a unique number, denoted by x⁻¹, such that x × x⁻¹ = 1.

• For any real number x different from 0, x × x = 1.

• For any real number x different from 0, there is a unique number, denoted by x⁻¹, such that x + x⁻¹ = 1.

• For any real number x different from 0, x × 0 = 1.

Answer explanation

The multiplicative inverse property states that for any non-zero real number x, there exists a unique number x⁻¹ such that their product equals 1, i.e., x × x⁻¹ = 1. This makes the first choice the correct answer.

What is the associative law of addition for real numbers x, y, and z?

• x + (y + z) = (x + y) + z

• x + y = z + x

Answer explanation

The associative law of addition states that the way in which numbers are grouped does not affect their sum. Thus, x + (y + z) = (x + y) + z is the correct representation of this law.

What is the commutative law of addition for real numbers x and y?

• x + y = y + x

• x + y = x - y

• x + y = x \times y

• x + y = y - x

Answer explanation

The commutative law of addition states that the order of adding two real numbers does not affect the sum. Therefore, x + y equals y + x, making the first option the correct choice.

What is the associative law of multiplication for real numbers x, y, and z?

• x × (y × z) = (x × y) × z

• x × (y × z) = x × y + z

• x × (y × z) = (x + y) × z

• x × (y × z) = x × y × z + 1

Answer explanation

The associative law of multiplication states that the way in which numbers are grouped does not affect the product. Thus, x × (y × z) = (x × y) × z is correct, while the other options do not represent this law.

What is the commutative law of multiplication for real numbers x and y?

• x × y = y × x

• x + y = x - y

• x × y = x + y

• x × y = y - x

Answer explanation

The commutative law of multiplication states that the order of multiplication does not affect the product. Therefore, for real numbers x and y, x × y = y × x is the correct expression.

What is the distributive law for real numbers x, y, and z?

• x × (y + z) = x × y + x × z

• x + (y × z) = (x + y) × (x + z)

• x × (y + z) = x × y × z

• x + (y × z) = x + y + z

Answer explanation

The distributive law states that multiplying a number by a sum is the same as multiplying each addend separately and then adding the results. Thus, x × (y + z) = x × y + x × z is the correct choice.

Which of the following sets includes all integers?

• ℚ (rational numbers)

• ℝ (real numbers)

• ℤ (integers)

• ℤ+ (positive integers)

Answer explanation

The set R (real numbers) includes all integers, as it encompasses both rational and irrational numbers. The set Z (integers) specifically contains only integers, while Z+ (positive integers) includes only positive integers.