Mathematics » Rational Expressions and Equations » Add and Subtract Rational Expressions with a Common Denominator

# Adding and Subtracting Rational Expressions Whose Denominators Are Opposites

## Adding and Subtracting Rational Expressions Whose Denominators Are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by $$\frac{-1}{-1}$$.

Let’s see how this works.

 Multiply the second fraction by $$\frac{-1}{-1}$$. The denominators are the same. Simplify.

## Example

Add: $$\frac{4u-1}{3u-1}+\frac{u}{1-3u}.$$

### Solution

 The denominators are opposites, so multiply the second fraction by $$\frac{-1}{-1}$$. Simplify the second fraction. The denominators are the same. Add the numerators. Simplify. Simplify.

## Example

Subtract: $$\frac{{m}^{2}-6m}{{m}^{2}-1}-\frac{3m+2}{1-{m}^{2}}.$$

### Solution

 The denominators are opposites, so multiply the second fraction by $$\frac{-1}{-1}$$. Simplify the second fraction. The denominators are the same. Subtract the numerators. Distribute. $$\frac{{m}^{2}-6m+3m+2}{{m}^{2}-1}$$ Combine like terms. Factor the numerator and denominator. Simplify by removing common factors. Simplify.