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Apuntes

To solve the equation \[ \frac{1}{2x - 3} + \frac{3}{2x^2 - 3x} = \frac{5}{x} + \frac{7x - 15}{3x - 2x^2}, \] we first simplify the terms on both sides. ### Step 1: Simplify the Left Side The left side consists of two fractions: 1. \(\frac{1}{2x - 3}\) 2. \(\frac{3}{2x^2 - 3x}\) We can factor the second term: \[ 2x^2 - 3x = x(2x - 3). \] Thus, the left side becomes: \[ \frac{1}{2x - 3} + \frac{3}{x(2x - 3)}. \] To combine these fractions, we find a common denominator, which is \(x(2x - 3)\): \[ \frac{x}{x(2x - 3)} + \frac{3}{x(2x - 3)} = \frac{x + 3}{x(2x - 3)}. \] ### Step 2: Simplify the Right Side Now, we simplify the right side: 1. \(\frac{5}{x}\) 2. \(\frac{7x - 15}{3x - 2x^2}\) We can factor the second term: \[ 3x - 2x^2 = -2x^2 + 3x = -x(2x - 3). \] Thus, the right side becomes: \[ \frac{5}{x} + \frac{7x - 15}{-x(2x - 3)}. \] To combine these fractions, we find a common denominator, which is \(-x(2x - 3)\): \[ -\frac{5(2x - 3)}{x(2x - 3)} + \frac{7x - 15}{-x(2x - 3)} = \frac{-5(2x - 3) + (7x - 15)}{-x(2x - 3)}. \] ### Step 3: Combine and Simplify Now we have: \[ \frac{x + 3}{x(2x - 3)} = \frac{-5(2x - 3) + (7x - 15)}{-x(2x - 3)}. \] We can simplify t...