Question

Select True or False for each statement.

For a real number a, a + 0 = a.
For a real number a, a + (-a) = 1.
For a real numbers a and b, | a - b | = | b - a |.
For real numbers a, b, and c, a + (b ∙ c) = (a + b)(a + c).
For rational numbers a and b when b ≠ 0, is always a rational number.

Answer 1

A) True

B) False

C) True

D) False

E) True

Step-by-step explanation:

We are given the following statements in the question:

A) True

For every real number, a, a + 0 = a. 0 is known as the additive identity.

B) False

For a real number a, a + (-a) = 0.

C) True

For a real numbers a and b,
`|a-b| = |b-a|`

D) False

For real numbers a, b, and c, a + (b ∙ c) = (a + b)(a + c).

Counter example: For a = 2, b = 1, c = 3

`a + (b.c) = (a + b)(a + c)\\2 + (1.3) \\eq (2+1)(2+3)\\5\\eq 15`

E) True

For rational numbers a and b, b is not equal to zero,
`(a)/(b)` is always a rational number.

Answer 2

For a real number a, a + 0 = a. TRUE

For a real number a, a + (-a) = 1. FALSE

For a real numbers a and b, | a - b | = | b - a |. TRUE

For real numbers a, b, and c, a + (b ∙ c) = (a + b)(a + c). FALSE

For rational numbers a and b when b ≠ 0, is always a rational number. TRUE

Step-by-step explanation:

• For a real number a, a + 0 = a. TRUE

This comes from the identity property for addition that tells us that zero added to any number is the number itself. So the number in this case is
`a`, so it is true that:

`a+0=a`

• For a real number a, a + (-a) = 1. FALSE

This is false, because:

`a+(-a)=a-a=0`

For any number
`a` there exists a number
`-a` such that
`a+(-a)=0`

• For a real numbers a and b, | a - b | = | b - a |. TRUE

This is a property of absolute value. The absolute value means remove the negative for the number, so it is true that:

`\mid a-b \mid= \mid b-a \mid`

• For real numbers a, b, and c, a + (b ∙ c) = (a + b)(a + c). FALSE

This is false. By using distributive property we get that:

`(a + b)(a + c)=a^2+ac+ab+bc \\ \\ a^2+ab+ac+bc \\eq a+(b.c)`

• For rational numbers a and b when b ≠ 0, is always a rational number. TRUE

A rational number is a number made by two integers and written in the form:

`(u)/(v) \\ \\ v \\eq 0`

Given that
`a \ and \ b` are rational, then the result of dividing them is also a rational number.