User:Curtis Autery/Sandbox

$$\frac{x}{x + 1} + \frac{1}{x - 1} + 1$$

$$\mathbf{\color{Blue}\frac{x}{x + 1}} + \frac{1}{x - 1} + 1$$

$$\frac{x}{x + 1} = \frac{x}{x + 1} \cdot {\color{Red}\frac{x - 1}{x - 1}} = \frac{x^2 - x}{x^2 - 1}$$

$$\mathbf{\color{Blue}\frac{x^2 - x}{x^2 - 1}} + \frac{1}{x - 1} + 1$$

$$\frac{x^2 - x}{x^2 - 1} + \mathbf{\color{Blue}\frac{1}{x - 1}} + 1$$

$$\frac{1}{x - 1} = \frac{1}{x - 1} \cdot {\color{Red}\frac{x + 1}{x + 1}} = \frac{x + 1}{x^2 - 1}$$

$$\frac{x^2 - x}{x^2 - 1} + \mathbf{\color{Blue}\frac{x + 1}{x^2 - 1}} + 1$$

$$\frac{x^2 - x}{x^2 - 1} + \frac{x + 1}{x^2 - 1} + \mathbf{\color{Blue}1}$$

$$1 = \frac{x^2 - 1}{x^2 - 1}$$

$$\frac{x^2 - x}{x^2 - 1} + \frac{x + 1}{x^2 - 1} + \mathbf{\color{Blue}\frac{x^2 - 1}{x^2 - 1}}$$

$$\frac{(x^2 - x) + (x + 1) + (x^2 - 1)}{x^2 - 1}$$

$$\frac{x^2 \mathbf{\color{Red}- x + x} + 1 + x^2 - 1}{x^2 - 1}$$

$$\frac{x^2 \mathbf{\color{Red}+ 1} + x^2 \mathbf{\color{Red}- 1}}{x^2 - 1}$$

$$\frac{x^2 + x^2}{x^2 - 1}$$

$$\frac{2x^2}{x^2 - 1}$$