Problem 80 - Introductory Algebra for College Students

Question

Explain how to add rational expressions when denominators are opposites. Use an example to support your explanation.

Step-by-Step Solution

Answer

The addition of two rational expressions with opposite denominators \(\frac{1}{a}\) and \(\frac{2}{-a}\) will be \(\frac{-1}{a}\) after simplifying.

Step 1: Introduction

Consider two rational expressions with opposite denominators. For instance, \(\frac{1}{a}\) and \(\frac{2}{-a}\), where a is not equal to zero.

Step 2: Add the Rational Expressions

Addition of rational expressions with opposite denominators can be done by changing the sign of the second rational expression and treating it as addition. This can be achieved by multiplying the numerator and denominator of the second fraction by -1; yielded as such \(\frac{1}{a} + (\frac{2}{-a} * -\frac{1}{1})\). The expression will now be \(\frac{1}{a} + \frac{-2}{a}\).

Step 3: Simplify the Result

Now treat it as addition of fractions with like denominators, which is very straight-forward: add the numerators and write it over the common denominator. So, boil it down as \(\frac{1-2}{a}\) which simplifies to \(\frac{-1}{a}\). Hence the sum of the initial fractions.