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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.

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danielmaduroh

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 [tex]a[/tex], so it is true that:

[tex]a+0=a[/tex]

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

This is false, because:

[tex]a+(-a)=a-a=0[/tex]

For any number [tex]a[/tex] there exists a number [tex]-a[/tex] such that [tex]a+(-a)=0[/tex]

  • 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:

[tex]\mid a-b \mid= \mid b-a \mid[/tex]

  • 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:

[tex](a + b)(a + c)=a^2+ac+ab+bc \\ \\ a^2+ab+ac+bc \neq a+(b.c)[/tex]

  • 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:

[tex]\frac{u}{v} \\ \\ v \neq 0[/tex]

Given that [tex]a \ and \ b[/tex] are rational, then the result of dividing them is also a rational number.

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, [tex]|a-b| = |b-a|[/tex]

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

[tex]a + (b.c) = (a + b)(a + c)\\2 + (1.3) \neq (2+1)(2+3)\\5\neq 15[/tex]

E) True

For rational numbers a and b, b is not equal to zero, [tex]\frac{a}{b}[/tex] is always a rational number.

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