1Cademy - Adding and Subtracting Rational Expressions Whose Denominators Are Opposites

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Rational Expression

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Adding and Subtracting Rational Expressions Whose Denominators Are Opposites

When two rational expressions have denominators that are opposites of each other — such as $(a - b)$ and $(b - a)$ — they can be rewritten to share a common denominator by using the algebraic identity:

$b - a = -(a - b)$

This identity shows that reversing the order of a subtraction is equivalent to multiplying by $-1$ . To apply this technique, multiply the numerator and denominator of the fraction whose denominator is in the "opposite" form by $-1$ . This transforms the denominator into the same expression as the other fraction's denominator, creating a common denominator so that the standard addition or subtraction rule can be used.

For instance, to add $\frac{m}{m - 3} + \frac{5}{3 - m}$ , recognize that $(3 - m)$ and $(m - 3)$ are opposites. Multiply the numerator and denominator of the second fraction by $-1$ : $\frac{5}{3 - m} = \frac{5 \cdot (-1)}{(3 - m)\cdot(-1)} = \frac{-5}{m - 3}$ . Now both fractions share the denominator $(m - 3)$ , and they can be combined: $\frac{m + (-5)}{m - 3} = \frac{m - 5}{m - 3}$ .

Updated 2026-04-30

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