Need help with a tricky inequality problem I need help with this inequality:

Given that $y \geq 1$ and $x > 0$ and knowing that $y \geq 4 x \ln(x)$ show that this implies $y \geq x \ln(2y)$.


I saw this problem while reading a theoretical machine learning paper and it says this inequality follows but I have attempted many things including going directly from the assumption to working backwards. Seeing on desmos the inequality does seem to hold for the values of x,y listed. Also it may be important to note that y is a sample size (so natural numbers). Thank you!

If $x^4>2y$ finish because $y> x ln(x^4)$, how $x^4> 2y$ then $y> x ln(x^4)>xln(2y)$.
Now just prove $x^4>2y$.
$y> 4xln(x)$ then when you calculate $x^4 > 2(4xln(x))$ you will see $x>0$ is the solution.
For x=0 this inequality don't exist
