Solve the rational equation: \(\frac{2x}{x - 1} - \frac{3}{x} = 1\)

Step 1 ：Step 1: To solve the given equation, we need to clear the fraction. We first find the least common denominator (LCD) of the fractions which is \(x(x - 1)\).

Step 2 ：Step 2: Multiply each term in the equation by the LCD, we get: \(2x^2 - 3(x - 1) = x(x - 1)\).

Step 3 ：Step 3: Simplify the equation, we get: \(2x^2 - 3x + 3 = x^2 - x\).

Step 4 ：Step 4: Rearrange the equation so that all terms are on one side of the equation. We get: \(x^2 - 2x + 3 = 0\).

Step 5 ：Step 5: Use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), to solve for x. Here, a = 1, b = -2, and c = 3. Substituting these values into the quadratic formula, we get: \(x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(3)}}{2(1)}\).

Step 6 ：Step 6: Simplify the expression. We get that \(x = 1 \pm \sqrt{2}\iota\).