Answer
$\delta=\frac{1}{18}$
Work Step by Step
See graph; we can find $|c-a|=|1/2-4/9|=\frac{1}{18}$ and $|c-b|=|1/2-4/7|=\frac{1}{14}$. We choose $\delta=\frac{1}{18}$ (smaller of the two values) so that for all $x$, when $0\lt|x-c|\lt\delta$, we have $a\lt x\lt b$ as can be seen from the figure.
