EDGE Math 8

Addition and Subtraction of Rational Expressions with the Same Denominator

To add or subtract two rational expressions with unlike denominators, follow the same steps that you are doing when you are adding or subtracting unlike fractions.

Example 1

Add

\frac{5 x}{x - 1} + \frac{3 x}{x + 1} .

Solution

Step 1:

Find the least common denominator (LCD) of the two rational expressions.
The denominators are

x - 1

and

x + 1;

hence, the LCD is their product

(x - 1) (x + 1) .

Step 2:

Rewrite each rational expression to its equivalent rational expression using the LCD.
To do this, you need to multiply both the numerator and the denominator by the same factor that will result to the LCD. Thus,

\frac{5 x}{x - 1} = \frac{5 x (x + 1)}{(x - 1) (x + 1)}

= \frac{5 x^{2} + 5 x}{(x - 1) (x + 1)}

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

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

Step 3:

Add the resulting rational expressions with the same denominator. Simplify the sum, if possible.

\frac{5 x^{2} + 5 x}{(x - 1) (x + 1)} + \frac{3 x^{2} - 3 x}{(x + 1) (x - 1)}

= \frac{5 x^{2} + 5 x + 3 x^{2} - 3 x}{(x + 1) (x - 1)}

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

Hence, $\frac{5 x}{x - 1} + \frac{3 x}{x + 1} = \frac{8 x^{2} + 2 x}{(x + 1) (x - 1)} .$