If $-7\le x\le 7$ and $0<y\le 12$, what is the greatest possible value of $(y-x)$?

To maximize the value of the expression $(y-x)$, we need to make $y$ as large as possible and $x$ as small as possible:
$$\text{Maximum value of }(y-x)=y_{\text{max}}-x_{\text{min}}$$

From the given inequalities:

* The maximum value of $y$ is $12$ ($y_{\text{max}}=12$).
* The minimum value of $x$ is $-7$ ($x_{\text{min}}=-7$).

Substituting these values into the expression:

$$\text{Greatest value}=12-(-7)=12+7=19$$