Given [tex]$4c + 5 \neq 0$[/tex] and [tex]$c$[/tex] is a real number, what is the multiplicative inverse of [tex][tex]$4c + 5$[/tex][/tex]?

A. 0
B. 1
C. [tex]\frac{1}{4c + 5}[/tex]
D. [tex]\frac{1}{4c} + 5[/tex]



Answer :

To find the multiplicative inverse of [tex]\( 4c + 5 \)[/tex], we need to determine a number that, when multiplied by [tex]\( 4c + 5 \)[/tex], yields the multiplicative identity, which is 1.

In algebraic terms, the multiplicative inverse of a number [tex]\( x \)[/tex] is [tex]\( \frac{1}{x} \)[/tex]. Therefore, the multiplicative inverse of [tex]\( 4c + 5 \)[/tex] should be [tex]\( \frac{1}{4c + 5} \)[/tex], provided that [tex]\( 4c + 5 \neq 0 \)[/tex] to ensure we do not divide by zero.

Let's verify this step-by-step:

1. Consider the expression [tex]\( 4c + 5 \)[/tex].
2. We are looking for a number that, when multiplied by [tex]\( 4c + 5 \)[/tex], gives us 1.
3. By definition, this number is [tex]\( \frac{1}{4c + 5} \)[/tex].
4. To confirm, multiply [tex]\( 4c + 5 \)[/tex] by [tex]\( \frac{1}{4c + 5} \)[/tex]:
[tex]\[ (4c + 5) \times \frac{1}{4c + 5} = 1 \][/tex]

Since this product equals 1, [tex]\( \frac{1}{4c + 5} \)[/tex] is indeed the multiplicative inverse of [tex]\( 4c + 5 \)[/tex].

Thus, the multiplicative inverse of [tex]\( 4c + 5 \)[/tex] is:
[tex]\[ \boxed{\frac{1}{4c + 5}} \][/tex]