jmc

First, we note that $x$ must be positive, since otherwise $\lceil x\rceil +x$ is nonpositive. Next, we know that the decimal part of $x$ must be $\frac{2}{7}$. We write $x$ as $n+\frac{2}{7}$, where $n$ is the greatest integer less than $x.$ Then, $\lceil x\rceil =n+1.$ Therefore, we can write $\lceil x\rceil +x$ as $n+1+n+\frac{2}{7}=\frac{23}{7}$. Solving, we get $n=1$. Therefore, the only value $x$ that satisfies the equation is $1+\frac{2}{7}=\boxed{\frac{9}{7}}$.