Find the value of $x$ that satisfies $\frac{\sqrt{5x}}{\sqrt{3(x-1)}}=2$. Express your answer in simplest fractional form.
To solve for $x$, I want to isolate it on one side of the equation by using inverse operations. First, I can multiply both sides by the denominator $\sqrt{3(x-1)}$ to clear the fraction. This gives me $\sqrt{5x} = 2\sqrt{3(x-1)}$. Next, I can square both sides to get rid of the square roots. This gives me $5x = 4(3)(x-1)$. Expanding the right-hand side, I get $5x = 12x - 12$. Subtracting $12x$ from both sides, I get $-7x = -12$. Dividing both sides by $-7$, I get $x = \frac{12}{7}$. 