1Cademy - Adding $$\frac{7}{d} + \frac{5}{-d}$$

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Adding and Subtracting Rational Expressions Whose Denominators Are Opposites

Example

Adding $\frac{7}{d} + \frac{5}{-d}$

To add two rational expressions whose denominators are opposites, we can multiply one of the fractions by $\frac{-1}{-1}$ to create a common denominator. For example, consider the expression $\frac{7}{d} + \frac{5}{-d}$ . Since $d$ and $-d$ are opposites, we multiply the second fraction by $\frac{-1}{-1}$ :

$\frac{5}{-d} \cdot \frac{-1}{-1} = \frac{-5}{d}$

Now the denominators are the same, and we can add the numerators:

$\frac{7}{d} + \frac{-5}{d} = \frac{7 + (-5)}{d} = \frac{2}{d}$

This simple example illustrates the general principle that when denominators are opposites, multiplying one fraction by $\frac{-1}{-1}$ yields a common denominator, allowing for addition or subtraction.

Updated 2026-06-03

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OpenStax Intermediate Algebra

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