Answer:
4)
is equivalent to
for all
. (Answer: A)
5)
is equivalent to
for all
. (Answer: B)
6)
is equivalent to
for all
. (Answer: None)
7)
is equivalent to
. (Answer: None)
8)
is equivalent to
for all
. (Answer: A)
Step-by-step explanation:
We proceed to simplify each expression below:
4) 
(i)
Given
(ii)
Distributive property
(iii)
Distributive property
(iv)
Existence of multiplicative inverse/Modulative property/Result
Rational functions are undefined when denominator equals 0. That is:


Hence, we conclude that
is equivalent to
for all
. (Answer: A)
5) 
(i)
Given
(ii)
Distributive property
(iii)
Distributive property
(iv)
Commutative property/Existence of multiplicative inverse/Modulative property/Result
Rational functions are undefined when denominator equals 0. That is:



Hence, we conclude that
is equivalent to
for all
. (Answer: B)
6) 
(i)
Given
(ii)

(iii)
Commutative and distributive properties.
(iv)
Existence of multiplicative inverse/Modulative property/Result
Rational functions are undefined when denominator equals 0. That is:


Hence, we conclude that
is equivalent to
for all
. (Answer: None)
7) 
(i)
Given
(ii)

(iii)
Commutative and distributive properties.
(iv)
Existence of additive inverse/Modulative property/Result
Polynomic function are defined for all value of
.
is equivalent to
. (Answer: None)
8) 
(i) 
(ii)
/Result
Rational functions are undefined when denominator equals 0. That is:


Hence,
is equivalent to
for all
. (Answer: A)