Consider two possibly correlated scalar random variables $N$ and $X$. It is known that $1\leq N \leq N_{\max}$. Given that $\mathbb{E}[NX]\leq 0$, does it always hold that $\mathbb{E}[X] \leq 0$?

While it might appear intuitive that positively scaling a random variable should not change the sign of its mean, the possible correlation between N and X might complicate things. So far, I have had no luck proving it. On the other hand, I could not also come up with a counterexample. That is, I could not think of any RV $X$ (with positive mean) which upon multiplying with another positive random variable $N$ with bounded support, makes the mean of the product $NX$ negative.