Question

Select True or False for each statement.

Answer 1

Answer:

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

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:

$(a + b)(a + c)=a^2+ac+ab+bc \\ \\ a^2+ab+ac+bc \neq a+(b.c)$

• 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:

$\frac{u}{v} \\ \\ v \neq 0$

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

Answer 2

Answer:

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

$a + (b.c) = (a + b)(a + c)\\2 + (1.3) \neq (2+1)(2+3)\\5\neq 15$

E) True

For rational numbers a and b, b is not equal to zero, $\frac{a}{b}$ is always a rational number.