A rectangle with perimeter of 100 has area at least of 500, within what bounds must the length of the rectangle lie? Problem
The problem states that there is a rectangle that has a perimeter of $100$ and an area of at least $500$ and it asks for the bounds of the length 
which can be given in interval notation or in the <> (greater than or less than) signs
My steps and thought process
So I set some few equations
1)$2x+2y=100$ which becomes $x+y=50$
2)$xy \geq 500$
3)I then made $y = 50-x$
so that I can substitute it into equation #2 to get:
$50x-x^2 \geq 500$
which eventually got me $0 \geq x^2-50x+500$
and this is where I got stuck.

Since $x+y= 100$ and $xy \geq 500$, we have $x(50-x) \geq 500$. Thus $x^2 -50x + 500 \leq 0$. Consequently, $x$ must lie between the roots of the quadratic $x^2-50x+500 = 0$. The roots are $25 \pm 5\sqrt{5}$. Thus $25-5\sqrt{5} \leq x \leq 25+5\sqrt{5}$.
