Plot y =   (red)  and  y = 3x (green).

So, we start from the given inequality

     > 3x.       (1)


Move 3x to the left side

     - 3x > 0


Write with the common denominator

     -  > 0


Make equivalent transformations

     > 0

     > 0

     > 0

     > 0


Multiply both sides by (-1) and change the inequality sign

     < 0.


Divide by 3 both sides

     < 0.      (2)


So, our original inequality is equivalent to the inequality (2).

Now I will solve the inequality (2).


The factors in inequality (2) have three critical points, where linear terms become equal to zero and change the sign.

These points are  -1,    and  0.


The critical points divide the entire number line in four intervals

    1)  (,),   2)  (,),   3)   (,)  and   4)  [,).



In the interval #1, all three linear factors are negative, so the entire rational function is negative.

Thus the entire interval  (,)  IS the solution to inequality (2) and, hence, for the original inequality, too.



In the interval #2, the linear factor  (x+1)  is positive, while two other factors are negative, 
so the entire rational function is positive.

Thus the interval  (,)  is NOT the solution to inequality (2) and, hence, is NOT the solution for the original inequality.



In the interval #3, the linear factors  (x+1)  and    are both positive, while third factor  x   is negative, 
so the entire rational function is negative.

Thus the interval  (,)  IS the solution to inequality (2) and, hence, IS the solution for the original inequality.



In the interval #4, all three linear factors  are positive, so the entire rational function in (2) is positive.

Thus the interval  [,)  is NOT the solution to inequality (2) and, hence, is not the solution for the original inequality.



ANSWER.  The solution set for the given inequality is the union of two intervals  (,) U (,).


Please notice and pay special attention that all endpoints are treated correctly in my solution and in my answer.