To solve for \(r\) in the equation: \(\frac{1}{r - 1} + \frac{2}{r + 1} = \frac{3}{r}\).

Let's start by finding a common denominator for the fractions on the left side. The common denominator is \(r(r - 1)(r + 1)\). 

Multiply each term by this common denominator to eliminate the fractions: \(r(r + 1) + 2r(r - 1) = 3(r - 1)(r + 1)\). 

Simplify each term: \(r^2 + r + 2r^2 - 2r = 3(r^2 - 1)\) Combine like terms: \(3r^2 - r = 3r^2 - 3\) Subtract \(3r^2\) from both sides: -r = -3.

Multiply both sides by -1: r = 3