Proving x^4 + 1 > x^9 + x with x <= 1

[KeynesianDude]

Homework Statement

Let x$$\in$$$$\textbf{R}$$. Prove that if x⁴+1 > x⁹+x , then x$$\leq$$1

Homework Equations

[above]

The Attempt at a Solution

Sort of clueless on this one. My intuition tells me I should algebraically rearrange the equation such that 1 is on one side of the inequality. I also considered factoring out an x.

 

[Mark44]

Maybe this will help you get started.
x⁴ + 1 > x⁹ + x
<==> x⁹ - x⁴ + x - 1 < 0
<==> (x - 1)(x⁸ + x⁷ + x⁶ +x⁵ + x⁴ + 1) < 0

I got this factorization by using synthetic division.

If x = 1, the product is zero, so the inequality doesn't hold. If x > 1, both factors are positive, so the inequality again doesn't hold.

Can you say anything about the product if 0 < x < 1?

The inequality holds if x = 0. Can you say something about the product if x < 0?