EDGE Math 8

Addition and Subtraction of Rational Expressions with the Same Denominator

Recall how to add fractions like $\frac{3}{10}$ and $\frac{4}{10} .$

\frac{3}{10} + \frac{4}{10} = \frac{3 + 4}{10} = \frac{7}{10}

The two fractions have the numerators 3 and 4 and a common denominator of 10. To get the sum of the fractions, add the numerators and retain the common denominator. This procedure illustrates the following rules:

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} and \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} .

Example

Add or subtract. Then simplify your answer.

1. $\frac{x}{5} + \frac{2 x}{5}$

2. $\frac{4}{3 x} + \frac{5}{3 x}$

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

4. $\frac{3 x}{x + 3} - \frac{x - 6}{x + 3}$

Solution

1. $\frac{x}{5} + \frac{2 x}{5} = \frac{x + 2 x}{5} = \frac{3 x}{5}$

2. $\frac{4}{3 x} + \frac{5}{3 x} = \frac{4 + 5}{3 x} = \frac{9}{3 x} = \frac{3}{x}$

3. $\frac{5 x}{x - 1} - \frac{3 x}{x - 1} = \frac{5 x - 3 x}{x - 1} = \frac{2 x}{x - 1}$

\frac{3 x}{x + 3} - \frac{x - 6}{x + 3} = \frac{3 x - (x - 6)}{x + 3}

= \frac{3 x - x + 6}{x + 3}

= \frac{2 x + 6}{x + 3}

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

= 2