<s>[INST] Proving that if $x^4 + 5x + 1 < 27$ then $x < 2$ I need help trying to figure out how I can prove a statement like this. So looking at this I can conclude that this statement is of the form $P \Rightarrow Q$
$$
P: x^4 + 5x + 1 < 27
$$
$$
Q: x < 2
$$
I wanted to try and prove this by contrapositive , so this state would become

If $X \geq 2$ then $x^4 + 5x + 1 \geq 27$

Over here I was not sure if I could plug in the value 2 or anythin greater than 2 to see if this is true. plugging in 2 gets me $2^4 + 5 \times 2 + 1 = 27$ and since $27 \geq 27$ this statement is True.
Am I allowed to prove it like this? Is there a different way to prove a question like this?
 [/INST] If $x\ge 2$ is it true that $x^4 \ge 16$?
If $x \ge 2$ is it true that $5x \ge 10$?
So if $x \ge 2$ is it true that $x^4 + 5x + 1 \ge 16 + 10 + 1 = 27$?
...
It could get tedious to go to axioms and prove that if $x\ge 2 > 0$ then $x^4 \ge 2x^3 \ge 4x^2 \ge 8x \ge 16$ via the axiom: if $a > 0$ and $m < n$ then $am < an$ (applied three times); and $5 > 0$ so $x\ge 2 \implies 5x \ge 2*5 =10$ by the same axiom; and by the axiom $a + c > b + c \iff a > b$ then $x^4 + 5x + 1 \ge x^4 + 10 + 1 \ge 16+ 10 + 1=27$.  But I think it is safe to assume if we had to prove those we could.

This may be tedious over kill but:
If $x \ge 2$ then let $d = x - 2 \ge 0$.  Then $x = 2 + d$ 
Now because $d \ge 0$ then $d^k \ge 0$. 
So $x^4 = (2 + d)^4 = 2^4 + 4*2^3*d + 6*2^2*d^2 + 4*2*d^3 + d^4 \ge 2^4$
And $5x = 5*(2+d) = 5*2 + 5*d \ge 5*2$
So $x^4 + 5x + 1\ge 2^4 + 5*2 + 1 = 27$.
That should do it.....
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