EDGE Math 8

Example 2

Perform the indicated operations in $\frac{x - 4}{x - 2} - \frac{5}{x + 2} + \frac{4 x}{x^{2} - 4} .$

Solution

The denominators are $x - 2,$ $x + 2,$ and $x^{2} - 4.$ Since $x^{2} - 4 = (x + 2) (x - 2),$ the LCD is $(x + 2) (x - 2) .$

\frac{x - 4}{x - 2} - \frac{5}{x + 2} + \frac{4 x}{x^{2} - 4}

= \frac{(x - 4) (x + 2)}{(x - 2) (x + 2)} - \frac{5 (x - 2)}{(x + 2) (x - 2)}

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

= \frac{x^{2} - 2 x - 8}{(x - 2) (x + 2)} - \frac{5 x - 10}{(x + 2) (x - 2)}

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

= \frac{x^{2} - 2 x - 8 - (5 x - 10) + 4 x}{(x - 2) (x + 2)}

= \frac{x^{2} - 2 x - 8 - 5 x + 10 + 4 x}{(x - 2) (x + 2)}

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

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

= \frac{x - 1}{x + 2}

Hence, $\frac{x - 4}{x - 2} - \frac{5}{x + 2} + \frac{4 x}{x^{2} - 4} = \frac{x - 1}{x + 2} .$