x^3 - 2x^2 < 15x


    x^3 - 2x^2 - 15x < 0


Factor x out

    x*(x^2-2x - 15) < 0


Factor further

    x*(x-5)*(x+3) < 0


1)  In the interval  (-oo,-3)  all three factors are negative;

    so, their product is negative.  

    Thus this interval  (-oo,-3)  is the solution (is the part of the solution set).



2)  Next interval is  (-3,0).  In this interval,

        the factor (x+3) is positive; two other factors are negative.

    So, the product of the three factors is positive,

    and hence this interval (-3,0) is not the solution to the inequality.



3)  Next interval is  (0,5).  In this interval,

        the factors (x+3) and x  are positive; the last factor is negative.

    So, the product of the three factors is negative,

    and hence this interval (0,5) is the solution to the inequality.



4)  Last interval is  (5,oo).  In this interval,

        all three factors (x+3) are positive; 

    So, the product of the three factors is positive,

    and hence this interval (5,oo) is not the solution to the inequality.


ANSWER.  The solution set to the given inequality is the union of two intervals  (-oo,-3) U (0,5).