Let $x+1$ be any prime greater than $3$.
By Bertrand's Postulate, there is at least one prime between $\frac{x}{2}$ and $x$.
Let $\{p_1,p_2,\dots, p_n\}$ be the primes between $\frac{x}{2}$ and $x$
In all cases that I've checked, there exists $p_i$ in this set where either $2p_i-1$ or $2p_i+1$ is a prime.
Is this always true?