How do you solve #32p^2-2p^6<0# using a sign chart?

1 Answer
May 30, 2018

Answer:

The solution is #p in (-oo,-2) uu(2,+oo)#

Explanation:

The inequality is

#32p^2-2p^6<0#

Factorising

#2p^2(16-p^4)<0#

#2p^2(4-p^2)(4+p^2)<0#

#2p^2(2+p)(2-p)(4+p^2)<0#

Let's build the sign chart

#(4+p^2)>0#

Let #f(p)=2p^2(2+p)(2-p)(4+p^2)#

#color(white)(aaaa)##p##color(white)(aaaa)##-oo##color(white)(aaaa)##-2##color(white)(aaaaaaa)##0##color(white)(aaaaaaaa)##2##color(white)(aaaa)##+oo#

#color(white)(aaaa)##p^2##color(white)(aaaaaaaa)##+##color(white)(aaaaa)##+##color(white)(aaa)##0##color(white)(aaaa)##+##color(white)(aaaa)##+#

#color(white)(aaaa)##2+p##color(white)(aaaaaa)##-##color(white)(aa)##0##color(white)(aa)##+##color(white)(aaaaaaaa)##+##color(white)(aaaa)##+#

#color(white)(aaaa)##2-p##color(white)(aaaaaa)##+##color(white)(aaaaa)##+##color(white)(aa)####color(white)(aaaaaa)##+##color(white)(a)##0##color(white)(aa)##-#

#color(white)(aaaa)##f(p)##color(white)(aaaaaaa)##-##color(white)(aa)##0##color(white)(aa)##+##color(white)(aaa)##0##color(white)(aaaa)##+##color(white)(a)##0##color(white)(aa)##-#

Therefore,

#f(p)<0# when #p in (-oo,-2) uu(2,+oo)#

graph{32x^2-2x^6 [-7.554, 6.49, -3.425, 3.595]}