How do you solve $\dfrac{{x + 1}}{3} = \dfrac{{x - 1}}{4}?$

Hint: This problem deals with solving the algebraic expressions of the given equation. Here to solve this problem, a cross multiplication method is applied here. This method describes the multiplication of the numerator of one fraction to the denominator of another denominator of the first term to the numerator of another term.

Complete step-by-step solution:
Given an equation with fractional expression on both sides which is given as $\dfrac{{x + 1}}{3} = \dfrac{{x - 1}}{4}$,
Consider the given equation as given below:
$ \Rightarrow \dfrac{{x + 1}}{3} = \dfrac{{x - 1}}{4}$
Now cross-multiply the above equation, which means that the numerator of the left hand side of the equation has to be multiplied with the denominator of the right hand side of the equation.
Whereas the numerator of the right hand side of the equation has to be multiplied by the denominator of the left hand side of the equation.
The cross-multiplication method of the above equation is shown below:
$ \Rightarrow \dfrac{{x + 1}}{3} = \dfrac{{x - 1}}{4}$
The above described method of cross-multiplication is applied to the above equation, as given below:
$ \Rightarrow 4\left( {x + 1} \right) = 3\left( {x - 1} \right)$
Now multiplying the expression $x + 1$ with the number 4, on the left hand side, whereas multiplying the expression $x - 1$ with the number 3, on the right hand side of the equation as shown above
$ \Rightarrow 4x + 4 = 3x - 3$
Now arranging and grouping the like terms and unlike terms together to get the simplified expression as shown below:
$ \Rightarrow 4x - 3x = - 3 - 4$
Here in the above expression taking $x$ common on the left hand side of the above equation and taking the symbol minus common on the right hand side of the above equation as shown below:
$ \Rightarrow \left( {4 - 3} \right)x = - \left( {3 + 4} \right)$
$ \Rightarrow x = - 7$
$\therefore $The solution of the given equation $\dfrac{{x + 1}}{3} = \dfrac{{x - 1}}{4}$ is 7.

The value of the $x$ is -7.

Note: Please note that while solving such kind of problem, we have to be careful while applying the method of cross-multiplication, because we have to multiply the whole expression with the denominator of the other term, remember not just a single term in the expression is multiplied with the denominator of the other term, but the whole expression is multiplied.