The theorem mention is very similar to Markov's Inequality, however the integral on the right-hand-side has a lower-bound of x instead of zero.

I think it might be interesting to see if the theorem is actually true so maybe test some probability density function on it such as exponential or beta distribution.

Edit:

The theorem is true just take a look at the proof of Markov Inequality, "Probability Theoretic Approach": "Method 1". You should be able to use part of the proof.

https://en.wikipedia.org/wiki/Markov's_inequality