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]
