3.) \( \sqrt{5 x-1}-\sqrt{x+2}=1 \)

To solve the equation \( \sqrt{5x - 1} - \sqrt{x + 2} = 1 \), start by isolating one of the square roots. Rearranging gives us \( \sqrt{5x - 1} = \sqrt{x + 2} + 1 \). Now, square both sides to eliminate the square roots: \((\sqrt{5x - 1})^2 = (\sqrt{x + 2} + 1)^2\), leading to \(5x - 1 = (x + 2) + 2\sqrt{x + 2} + 1\). Simplifying, we have \(5x - 1 = x + 3 + 2\sqrt{x + 2}\). Rearranging gives \(4x - 4 = 2\sqrt{x + 2}\). Divide everything by 2, obtaining \(2x - 2 = \sqrt{x + 2}\). Square again to solve for \(x\): \((2x - 2)^2 = x + 2\). Expanding leads to \(4x^2 - 8x + 4 = x + 2\), which simplifies to \(4x^2 - 9x + 2 = 0\). Using the quadratic formula yields two potential solutions for \(x\). Finally, check both solutions back in the original equation to ensure they are valid. One solution is \(x = 1\), and after checking the second, we find it is not valid, confirming the answer remains \(x = 1\).