College Algebra, Chapter 1, 1.1, Section 1.1, Problem 40

The equation $\displaystyle \frac{4}{x-1} + \frac{2}{x+1} = \frac{35}{x^2 -1}$ is either linear or equivalent to a linear equation. Solve the equation

$\begin{equation} \begin{aligned} \frac{4}{x-1} + \frac{2}{x+1} &= \frac{35}{x^2 -1} && \text{Get the LCD}\\ \\ \frac{4(x+1)+2(x-1)}{(x-1)(x+1)} &= \frac{35}{x^2-1} && \text{Apply Distributive property}\\ \\ \frac{4x+4+2x-2}{x^2-1} &= \frac{35}{x^2-1} && \text{Simplify}\\ \\ \frac{6x+2}{x^2-1} &= \frac{35}{x^2-1} && \text{Multiply both sides by }(x^2-1)\\ \\ \cancel{(x^2-1)} & \left[ \frac{6x+2}{\cancel{x^2-1}} = \frac{35}{\cancel{x^2-1}} \right] \cancel{(x^2-1)} && \text{Simplify}\\ \\ 6x + 2 &= 35 && \text{Subtract both sides by } 2\\ \\ 6x + 2 - 2 &= 35 - 2 && \text{Combine like terms}\\ \\ 6x &= 33 && \text{Divide both sides by 6}\\ \\ \frac{\cancel{6}x}{\cancel{6}} &= \frac{33}{6} && \text{Simplify}\\ \\ x &= \frac{33}{6} && \text{Reduce to lowest term}\\ \\ x &= \frac{11}{2} \end{aligned} \end{equation}$