jmc

We know that $x\star 24=\frac{\sqrt{x+24}}{\sqrt{x-24}}=7$. Because we cannot take the square root of a negative number and because the denominator of a fraction cannot be zero, we know that $x-24>0$. Thus, a reasonable guess for $x$ would be $x=25$. $\frac{\sqrt{25+24}}{\sqrt{25-24}}=\frac{\sqrt{49}}{\sqrt{1}}=7$, as desired, so our answer is indeed $x=\boxed{25}$.