How do you solve #x^3<=4x^2# using a sign chart?

1 Answer
Jan 9, 2018

Answer:

The solution is #x in (-oo, 4]#

Explanation:

The inequality is

#x^3<=4x^2#

Rearranging

#x^3-4x^2<=0#

#x^2(x-4)<=0#

Let #f(x)=x^2(x-4)#

Let's build the sign chart

#color(white)(aaaa)##x##color(white)(aaaaa)##-oo##color(white)(aaaaa)##0##color(white)(aaaaaaaaa)##4##color(white)(aaaaaa)##+oo#

#color(white)(aaaa)##x^2##color(white)(aaaaaaa)##+##color(white)(aaa)##0##color(white)(aaaa)##+##color(white)(aaaaaa)##+#

#color(white)(aaaa)##x-4##color(white)(aaaaa)##-##color(white)(aaa)####color(white)(aaaa)##-##color(white)(aaa)##0##color(white)(aaa)##+#

#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##-##color(white)(aaa)##0##color(white)(aaa)##-##color(white)(aaa)##0##color(white)(aaa)##+#

Therefore,

#f(x)<=0# when #x in (-oo, 4]#

graph{x^3-4x^2 [-14.29, 14.18, -10.41, 3.83]}