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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..

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