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

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



Answer :

To determine the multiplicative inverse of \(4c + 5\), follow these steps:

1. Understand what a multiplicative inverse is:
The multiplicative inverse of a number \(a\) is another number \(b\) such that when \(a\) is multiplied by \(b\), the result is \(1\). Mathematically, \(a \cdot b = 1\).

2. Identify the expression whose inverse is needed:
Here, we need the multiplicative inverse of \(4c + 5\).

3. Set up the equation for the multiplicative inverse:
To find the multiplicative inverse of \(4c + 5\), we look for a term that will satisfy the equation \((4c + 5) \cdot \text{Inverse} = 1\).

4. Solve for the inverse:
Let the multiplicative inverse be \( \frac{1}{4c + 5} \). To verify, check that:
[tex]\[ (4c + 5) \cdot \frac{1}{4c + 5} = 1 \][/tex]

This confirms that \(\frac{1}{4c + 5}\) is indeed the multiplicative inverse of \(4c + 5\).

Therefore, the multiplicative inverse of \(4c + 5\) is \(\frac{1}{4c + 5}\).

Among the given options, the correct one is [tex]\(\frac{1}{4c + 5}\)[/tex].