Solve $\dfrac{{6x + 1}}{3} + 1 = \dfrac{{x - 3}}{6}$.

Hint: For solving this type of question, where we have to find the value of $x$ from the given equation then we will use the cross multiplication method. By solving it, we can easily get the value for $x$.

Complete step by step answer:
So here in this question we have the equation given as $\dfrac{{6x + 1}}{3} + 1 = \dfrac{{x - 3}}{6}$
First of all we will make one fraction in both the side, so on solving the LHS we will get the equation as
$ \Rightarrow \dfrac{{\left( {6x + 1} \right) + 3}}{3} = \dfrac{{x - 3}}{6}$
And on solving the above equation, we will get the equation as
$ \Rightarrow \dfrac{{6x + 4}}{3} = \dfrac{{x - 3}}{6}$
Now by doing the cross multiplication, we will get the equation as
$ \Rightarrow \left( {6x + 4} \right)6 = \left( {x - 3} \right)3$
Now on solving the multiplication, we will get the equation as
\[ \Rightarrow 36x + 24 = 3x - 9\]
Taking the constant value one side and the variable other side, we will get the equation as,
\[ \Rightarrow 36x - 3x = - 9 - 24\]
And on solving it we get
\[ \Rightarrow 33x = - 33\]
And solving it,
\[ \Rightarrow x = - 1\]

Therefore, on simplifying $\dfrac{{6x + 1}}{3} + 1 = \dfrac{{x - 3}}{6}$. The value of $x$ is $ - 1$.

Note:
As we have got the value of $x$ by solving the whole equation. Now we can check whether we are correct or wrong. By substituting the value of $x$ in both the sides one by one, and if both are the same then the value is correct. So in this way we can check the value.