<s>[INST] Help with discrete math proof? I am having trouble proving the following:
If $x\in R$ and $x > 0$, then $x^4+1 \geq x^3+x$. 
Work: I tried to rearrange the equation as $x^4-x^3-x+1 \geq 1$, but that does not really help. I also tried proof by cases where case 1 would be that x is irrational and case 2 would be that x is rational. However, that has not got me far either. I am not really sure how to approach this problem.
 [/INST] Hint: try the rearrangement inequality.
details: if $x>1$:
$$
\frac 1x < 1\\
x<x^4\\
x + x^3 = 1\cdot x + \frac 1x \cdot x^4 \le \frac 1x \cdot x + 1\cdot x^4 = 1+x^4
$$
Otherwise:
$$
\frac 1x \ge 1\\
x\ge x^4\\
x + x^3 = 1\cdot x + \frac 1x \cdot x^4 \le \frac 1x \cdot x + 1\cdot x^4 = 1+x^4
$$
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