# Adding and Subtracting Fractions With Variables

• You must have a common denominator.
• To find the Least Common Denominator (LCD), take the least common multiple of the individual denominators.
• Express each fraction as a new fraction with the common denominator, by multiplying by one in an appropriate form.
• To add fractions with the same denominator: add the numerators, and keep the denominator the same. That is, use the rule:
$$\frac{A}{C} + \frac{B}{C} = \frac{A+B}{C}$$

## Example

Question: Combine into a single fraction: $$\cssId{s12}{\frac{2}{x+3} - \frac{3x}{x-1}}$$
Solution: Note that the LCD is $\,(x+3)(x-1)\,.$
 $\displaystyle\frac{2}{x+3} - \frac{3x}{x-1}$ original expression $\displaystyle = \frac{2}{x+3}\cdot\frac{x-1}{x-1}$ $\displaystyle \qquad -\ \frac{3x}{x-1}\cdot\frac{x+3}{x+3}$ get a common denominator by multiplying by $\,1\,$ $\displaystyle = \frac{2(x-1)-3x(x+3)}{(x+3)(x-1)}$ keep the denominator the same; add the numerators $\displaystyle = \frac{2x-2-3x^2 - 9x}{(x+3)(x-1)}$ multiply out the numerator $\displaystyle = \frac{-3x^2 - 7x - 2}{(x+3)(x-1)}$ combine like terms; write numerator in standard form