Let $a\star b = \dfrac{\sqrt{a+b}}{\sqrt{a-b}}$. If $ x \star 24 = 7$, find $x$.
This problem involves a new operation, $\star$, that is defined in terms of the familiar operations of addition, subtraction, and square roots. To solve for $x$, I need to undo the operation $\star$ by reversing the order of the steps and applying the inverse operations. To do that, I can start by cross-multiplying to get rid of the fractions: $7\sqrt{x-24} = \sqrt{x+24}$. Then, I can square both sides to get rid of the square roots: $49(x-24) = x+24$. Expanding and simplifying, I get $49x - 1176 = x + 24$, or $48x = 1200$. Dividing by 48, I get $x = 25$. To check my answer, I can plug it back into the original equation and see if it satisfies it: $25 \star 24 = \dfrac{\sqrt{25+24}}{\sqrt{25-24}} = \dfrac{\sqrt{49}}{\sqrt{1}} = \dfrac{7}{1} = 7$. Indeed, it does, so I am confident that $x = 25$ is the correct solution. 