Solve the given inequality for real x:

$\frac{x}{3}>\frac{x}{2}+1$

$\Rightarrow \frac{x}{3}-\frac{x}{2}>1$

$\Rightarrow \frac{2 x-3 x}{6}>1$

$\Rightarrow-\frac{x}{6}>1$

$\Rightarrow-x>6$

$\Rightarrow x<-6$

Thus, all real numbers $x$, which are less than $-6$, are the solutions of the given inequality.

Hence, the solution set of the given inequality is $(-\infty,-6)$.