Chapter 7 - Section 7.3 - Adding and Subtracting Rational Expressions with the Same Denominator - Exercise Set - Page 508: 80

When one denominator is the opposite, or additive inverse of the other, first multiply either rational expression by $\displaystyle \frac{-1}{-1} =1,$ to obtain a common denominator. Example: $\displaystyle \frac{x}{x-1}-\frac{1}{1-x}=\frac{x}{x-1}-(\frac{1}{1-x}\times\frac{-1}{-1})$ $=\displaystyle \frac{x}{x-1}-\frac{-1}{x-1}$ $=\displaystyle \frac{x}{x-1}+\frac{1}{x-1}$ $=\displaystyle \frac{x+1}{x-1}$

When one denominator is the opposite, or additive inverse of the other, first multiply either rational expression by $\displaystyle \frac{-1}{-1} =1,$ to obtain a common denominator. Example: $\displaystyle \frac{x}{x-1}-\frac{1}{1-x}=\frac{x}{x-1}-(\frac{1}{1-x}\times\frac{-1}{-1})$ $=\displaystyle \frac{x}{x-1}-\frac{-1}{x-1}$ $=\displaystyle \frac{x}{x-1}+\frac{1}{x-1}$ $=\displaystyle \frac{x+1}{x-1}$