Since the average of 5 and 9 is $\frac{5+9}{2}=7$, then the averages of 5 and $x$ and of $x$ and 9 must be 10 and 12. In other words, $\frac{5+x}{2}$ and $\frac{x+9}{2}$ are equal to 10 and 12 in some order. Adding these, we obtain $\frac{5+x}{2}+\frac{x+9}{2}=10+12$ or $\frac{14+2x}{2}=22$ and so $7+x=22$ or $x=15$.