Question

Solve the equation. $$\frac{-3 x}{x+1}=\frac{-2}{x-1}$$

Short answer

The solution of the equation is \(x = -1, -\frac{2}{3}\).

Step-by-step solution

Simplify the Equation

Firstly, multiply both sides of the equation by \((x+1)(x-1)\) to get rid of the fractions. This results in \(-3x(x-1) = -2(x+1)\).

Expand Both Sides

Expanding both sides of the equation gives \(-3x^2 + 3x = -2x -2\).

Rearrange the Equation

Next, rearrange the equation to have all terms on one side and zero on the other side. This yields \(3x^2 + 5x + 2 = 0\).

Solve the Quadratic Equation

This is a simple quadratic equation in the form \(ax^2 + bx + c = 0\). We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve this equation. Substituting \(a = 3\), \(b = 5\), and \(c = 2\) yields two possible solutions: \(x = \frac{-5 + \sqrt{25 - 24}}{6} = -1\), and \(x = \frac{-5 - \sqrt{25 -24}}{6} = -\frac{2}{3}\).