Before we try to solve the inequality #x^3-36x<=0#, let us factorize #x^3-36x#.
#x^3-36x=x(x^2-36)=x(x-6)(x+6)#
We may write this as #(x+6)x(x-6)#
Hence, the term #x^3-36x# is a multiplication of #(x+6)#, #x# and #(x-6)# and for equality part, we have solutions #x=-6#, #x=0# and #x=6#.
These three points divide number line in four parts.
(1) In segment #x<-6#, all the three terms of are negative and hence product is negative i.e. less than zero and this is a solution.
(2) In segment #-6 < x < 0#, while first term #(x+6)# is positive, other two terms are negative. Hence product is positive and this is not a solution.
(3) In segment #0 < x < 6# while first two terms are positive, the third term #(x-6)# is negative, hence product is negative and this is a solution.
(4) In segment #x>6# all the terms are positive, hence product is positive and hence this is not a solution.
Hence solution set is #x<=-6# and #0 <= x <= 6#
