Question

For which values of x is the expression x4 – 10x2 less than -9?

Answer 1

$x^4-10x^2\ \textless \ -9\\ x^4-10x^2+9\ \textless \ 0\\ x^4-x^2-9x^2+9\ \textless \ 0\\ x^2(x^2-1)-9(x^2-1)\ \textless \ 0\\ (x^2-9)(x^2-1)\ \textless \ 0\\ (x-3)(x+3)(x-1)(x+1)\ \textless \ 0\\ x\in(-3,-1)\cup(1,3)$

Answer 2

x^4-10x^2<-9

x^4-10x^2+9<0

x^4-x^2-9x^2+9<0

x^2(x^2-1)-9(x^2-1)<0

(x^2-9)(x^2-1)<0  now for this to be true, the expressions must have different signs, so that the product is negative...

x<±3 and x>±1

±1<x<±3

x=(-3, -1)U(1, 3)