Solve this inequality with a sign chart
The roots of the denominator
#12y^2+5y-2=0# are
#y_1=(-5+sqrt(5^2-4*12*(-2)))/(2*12)=(-5+11)/(24)=6/24=1/4#
#y_2=(-5-sqrt(5^2-4*12*(-2)))/(2*12)=(-5-11)/(24)=-16/24#
#=-2/3#
The roots of the numerator
#3y^2-9y^3=(3y^2)(1-3y)#
#y_3=0#
and
#y_4=1/3#
Let #f(y)=(3y^2-9y^3)/(12y^2+5y-2)#
Construct the sign chart
#color(white)(aaaa)##y##color(white)(aaaa)##-oo##color(white)(aaaa)##-2/3##color(white)(aaaa)##0##color(white)(aaaa)##1/4##color(white)(aaaa)##1/3##color(white)(aaaa)##+oo#
#color(white)(aaaa)##y+2/3##color(white)(aaaa)##-##color(white)(aaa)##||##color(white)(aa)##+##color(white)(aa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##y^2##color(white)(aaaaaaa)##+##color(white)(aaa)##||##color(white)(aa)##+##color(white)(a)##0##color(white)(a)##+##color(white)(aaaa)##+##color(white)(aaa)##+#
#color(white)(aaaa)##y-1/4##color(white)(aaaa)##-##color(white)(aaa)##||##color(white)(aa)##-##color(white)(a)##0##color(white)(a)##-##color(white)(a)##||##color(white)(aa)##+##color(white)(aaa)##+#
#color(white)(aaaa)##1-3y##color(white)(aaaa)##+##color(white)(aaa)##||##color(white)(aa)##+##color(white)(a)##0##color(white)(a)##+##color(white)(a)##||##color(white)(aa)##+##color(white)(a)##0##color(white)(a)##-#
#color(white)(aaaa)##f(y)##color(white)(aaaaaa)##+##color(white)(aaa)##||##color(white)(aa)##-##color(white)(a)##0##color(white)(a)##-##color(white)(a)##||##color(white)(aa)##+##color(white)(a)##0##color(white)(a)##-#
Therefore,
#f(y)>=0# when #y in(-oo,-2/3)uu {0}uu(1/4,1/3]#
graph{(3x^2-9x^3)/(12x^2+5x-2) [-8.02, 6.03, -2.8, 4.23]}