jmc

Since the problem only asks for the difference in the $x$-coordinates, we can ignore the $y$-coordinates. They originally agreed to meet at the midpoint of $(3,5)$ and $(-6,2)$, so the $x$-coordinate of the planned location is $\frac{3+(-6)}{2}=-\frac{3}{2}$. The correct meeting location should be at the midpoint of $(3,5)$ and $(-10,4)$, so the $x$-coordinate should be at $\frac{3+(-10)}{2}=-\frac{7}{2}$. The positive difference is $-\frac{3}{2}-(-\frac{7}{2})=\boxed{2}$. Alternatively, notice that a 4-unit change in the $x$-coordinate of Barbara's location resulted in a 2-unit change in the midpoint since the 4 gets divided by 2. $\frac{3+(-10)}{2}=\frac{3+(-6)}{2}+\frac{-4}{2}=-\frac{3}{2}-\boxed{2}$.