EDGE Math 8

Example 3

Add $\frac{3 x - 2}{x^{2} + x - 2} + \frac{4}{1 - x} .$

Solution

First, factor $x^{2} + x - 2.$

x^{2} + x - 2 = (x - 1) (x + 2)

The denominator of the second addend is $1 - x,$ which is the same as $- (x - 1) .$ Thus,

\frac{3 x - 2}{x^{2} + x - 2} + \frac{4}{1 - x}

= \frac{3 x - 2}{(x - 1) (x + 2)} - \frac{4}{x - 1}

The LCD of the rational expressions is $(x + 2) (x - 1) .$ Hence, the rational expressions may be subtracted as follows:

\frac{3 x - 2}{(x - 1) (x + 2)} - \frac{4}{x - 1}

= \frac{3 x - 2}{(x - 1) (x + 2)} - \frac{4 (x + 2)}{(x - 1) (x + 2)}

= \frac{3 x - 2}{(x - 1) (x + 2)} - \frac{4 x + 8}{(x - 1) (x + 2)}

= \frac{3 x - 2 - 4 x - 8}{(x - 1) (x + 2)}

= \frac{- x - 10}{(x - 1) (x + 2)}

Thus, $\frac{3 x - 2}{x^{2} + x - 2} + \frac{4}{1 - x} = \frac{- x - 10}{(x - 1) (x + 2)} .$