Okay, let's see. I need to solve the equation √(2x)/√(3x - 1) = 3/2. Hmm, radicals in the equation. Alright, first step, maybe I can eliminate the square roots by squaring both sides. That usually helps. Let me try that.

So, if I square both sides, the left side becomes (2x)/(3x - 1) because the square of the numerator over the square of the denominator. And the right side becomes (3/2) squared, which is 9/4. So the equation now is (2x)/(3x - 1) = 9/4.

Now, cross-multiplying to solve for x. Multiply both sides by (3x - 1) and by 4 to get rid of the denominators. So, 4*2x = 9*(3x - 1). Let's compute that. 8x = 27x - 9. Hmm, now subtract 8x from both sides: 0 = 19x - 9. Then, add 9 to both sides: 19x = 9. So x = 9/19. Wait, is that right?

Wait, let me check my steps again. Starting from the beginning.

Original equation: √(2x)/√(3x - 1) = 3/2. Squaring both sides gives (2x)/(3x - 1) = 9/4. Cross-multiplying: 8x = 27x - 9. Subtract 8x: 0 = 19x - 9. Then 19x = 9, so x = 9/19. Hmm. That seems straightforward. But let me check if this solution is valid in the original equation. Because sometimes squaring can introduce extraneous solutions.

So, plug x = 9/19 into the original equation. Let's compute numerator and denominator separately.

Numerator: √(2x) = √(2*(9/19)) = √(18/19). Denominator: √(3x - 1) = √(3*(9/19) - 1) = √(27/19 - 19/19) = √(8/19). So the left side is √(18/19)/√(8/19) = √(18/19 ÷ 8/19) = √(18/8) = √(9/4) = 3/2. Which matches the right side. So yes, x = 9/19 is a valid solution. Therefore, the answer is 9/19. Seems correct.

But wait, let me double-check the algebra steps again because sometimes I might make a mistake in cross-multiplication. Let's redo that.

Original equation after squaring: (2x)/(3x - 1) = 9/4. Cross-multiplying: 2x * 4 = 9*(3x - 1). So 8x = 27x - 9. Subtract 27x: 8x - 27x = -9 => -19x = -9. Multiply both sides by -1: 19x = 9. So x = 9/19. Yep, same result. So that's correct.

I think that's all. The steps are straightforward, and the check confirms the solution. So the answer is 9/19.
Therefore, the solution to the equation is $\boxed{\frac{9}{19}}$.
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