Prove (x^2)+1/(x^2 )≥x + (1/x )≥2 for x> 0

First of all, let's prove that

We can rewrite the fraction as

And the inequality as

Again, we can rewrite as

The numerator is a square, so it is always greater than or equal to zero.

The denominator is positive, because we're assuming x>0.

So, this is a fraction between positive numbers, and it is always positive, and the first part is finished.

As for the second part, we have

The numerator is not the product of a square (always greater than or equal to zero) and a quadratic equation with no solutions and positive leading coefficient (always positive), so it's always greater than or equal to zero. The denominator is x^2, and since x>0 by hypothesis, it is positive. This ends the proof.

thats not a system of equations, that is a solution