Have you ever noticed that when you add a number to its reciprocal, it is always greater than or equal to 2?
In this article I want to prove it in a very simple way
We all know that the square of any real number is positive, so:
(x - 1)² ≥ 0
You may ask why I put "≥".
Well, if you put 1 instead of x, it becomes zero
Now let's expand the inequality:
x² + 1 - 2x ≥ 0
x² + 1 ≥ 2x
Now I divide all terms by x:
x + 1/x ≥ 2
Yes, we have proven that the sum of a number and its reciprocal is always greater than or equal to 2.
Interesting, right?
But what if our x is negative?
I just multiply the terms by -1
-x - 1/x ≤ -2
We found that it is always less than or equal to -2!
