# X^4 + 1 > x^9 + x, then x<=1

## Homework Statement

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

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## 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
Mentor
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?