Find and correct errors of the following mathematical expressions:
$\dfrac{3}{{4x + 3}} = \dfrac{1}{{4x}}$

Hint: If an expression is defined as a rational number and the numerator is a polynomial term having a monomial denominator then the denominator will not be splitted and neither be shared with the splitted numerator.
i.e. $\dfrac{A}{{Bx + C}} \ne \dfrac{A}{{Bx}} + \dfrac{A}{C}$

Complete step-by-step answer:
Compare the terms after splitting if possible on both sides of the equation and see which term is not matching that will be an error and hence correct it.
1. We are having the expression as:
$\dfrac{3}{{4x + 3}} = \dfrac{1}{{4x}}$
Where,
2. L.H.S = $\dfrac{3}{{4x + 3}}$
And, R.H.S = $\dfrac{1}{{4x}}$
3. Therefore, L.H.S $ \ne $R.H.S
4. The correct expression would be:
$\dfrac{3}{{4x + 3}} = \dfrac{3}{{4x + 3}}$
Hence, L.H.S = R.H.S

Note: The denominator never splits up for the numerator.
Even if we consider the case of a polynomial numerator, the denominator never splits up. But the vice versa is true in the case of numerators.