Find the rational number $x$ such that: $\dfrac{5}{6}\left(x + \dfrac{3}{5}\right) = \dfrac{5}{6}x + \dfrac{1}{2}$.

Given,

Equation : $\dfrac{5}{6}\left(x + \dfrac{3}{5}\right) = \dfrac{5}{6}x + \dfrac{1}{2}$

Expanding L.H.S. using the distributive property :

$\Rightarrow \dfrac{5}{6}\left(x + \dfrac{3}{5}\right) \\[1em] \Rightarrow \dfrac{5}{6} \times x + \dfrac{5}{6} \times \dfrac{3}{5} \\[1em] \Rightarrow \dfrac{5}{6}x + \dfrac{15}{30} \\[1em] \Rightarrow \dfrac{5}{6}x + \dfrac{1}{2}.$

So, L.H.S. = $\dfrac{5}{6}x + \dfrac{1}{2}$ and R.H.S. = $\dfrac{5}{6}x + \dfrac{1}{2}$.

L.H.S. = R.H.S. for any value of x. Therefore, the given equation is an identity (it is true for all rational numbers x).

Hence, every rational number x satisfies the given equation, i.e., the equation holds true for all rational values of x.