Does $A:=\{x\in [0,\infty)\,|\,0\leq \{x\},\{x\log x\}\leq 1/2\}$ have positive density?

For a subset $S\subseteq [0,\infty)$, we say that $S$ has positive density if $$\liminf_{T\to\infty}\frac{m(S\cap[0,T])}{T}>0.$$Let $A:=\{x\in [0,\infty)\,|\,0\leq \{x\},\{x\log x\}\leq 1/2\}$, where $\{x\}$ be the fractional part of $x$. Is it true that $A$ has positive density?

Clearly, if we only require that $0\leq \{x\}\leq 1/2$ or $0\leq \{x\log x\}\leq 1/2$, then the resulting sets have positive densities. But I fail to prove their intersection does the same.