Problem 50 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{10}{x-2}-\frac{6}{2-x}$$

Step-by-Step Solution

Answer

\(\frac{4}{x-2}\)

Step 1: Recognize the Relationship between the Denominators

The denominators \(x-2\) and \(2-x\) are additive inverses of each other. This is because the expression \(2-x\) can be rewritten as \(-(x-2)\). Therefore, the denominators are actually the same when we take into account the negative symbol.

Step 2: Adjust the Fractions

To align the fractions, let's write the expression as: \(\frac{10}{x-2} + \frac{-6}{x-2}\). This makes it easier to perform the addition operation.

Step 3: Add the Fractions

We now have two fractions with the same denominators. As such, they can be added directly: \(\frac{10 - 6}{x - 2} = \frac{4}{x-2}\).

Step 4: Finalize the Answer

The solution, simplified as much as possible, to the expression is \(\frac{4}{x-2}\). There are no further operations that can simplify this expression.

Key Concepts

Understanding Additive Inverses

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