How many values of \(x\) satisfy both of the following conditions?

\((a)\) \(x\) is a multiple of \(7,\) not necessarily positive.

\((b)\) \(x^2\) is less than \(500\).

If \(x^2\) is \(500,\) that means \(-22\le x\le22\). Also, since \(x\) is a multiple of \(7,\) we can list all the multiples of 7 between -22 and 22 out.

-21, -14, -7, 0, 7, 14, 21

Therefore, since there are 7 terms in the list, the answer is \(\boxed{7}\).