jmc

The average of $x+6$, $6x+2$, and $2x+7$ is $\frac{1}{3}((x+6)+(6x+2)+(2x+7))$. Simplifying this expression gives $\frac{1}{3}(9x+15)=3x+5$. We know the average is also $4x-7$, so $3x+5=4x-7$. Subtracting $3x-7$ from both sides of this equation gives $x=\boxed{12}$.