Problem 39 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{4}{x-3}+\frac{2}{3-x}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2}{x-3}\)

Step 1: Recognize and Rewrite the Denominators

The denominators are additive inverses, which means they are the same if we change the sign. We can rewrite the denominator \(3 - x\) as \(-(x - 3)\). The expression now looks as follows: \(\frac{4}{x-3} + \frac{2}{-(x-3)}\).

Step 2: Simplify the Fraction

The second fraction now has a negative denominator. To further simplify we can move this negative sign to the numerator: \(\frac{4}{x-3} - \frac{2}{x-3}\)

Step 3: Common Denominator

With the same denominators in both fractions, we can simply combine the numerators: \(\frac{4 - 2}{x-3} = \frac{2}{x-3}\).

Key Concepts

Additive Inverses

Additive inverses are a core concept in mathematics. They refer to pairs of numbers that, when added together, equate to zero. This is crucial when dealing with fractions. In rational expressions, if you have denominators like \(x - 3\) and \(3 - x\), you are facing additive inverses. The expression \(3 - x\) can be rewritten as \(-(x - 3)\).

By understanding this, you identify and manipulate expressions more fluidly. Recognizing an additive inverse allows for simplifying expressions. This aligns the denominators, making subsequent calculations much simpler.

Here's what to remember:

• Additive inverses sum to zero.
• Use them to simplify expressions.
• Adjust the expression to match the forms entirely.

Knowing that additive inverses exist in a problem can significantly change your approach, leading to quicker solutions.

Common Denominator

Finding a common denominator is vital when adding or subtracting fractions. It lets you combine fractions in a coherent manner. In the given problem, after recognizing the additive inverse, both denominators \(x-3\), were aligned, leading to a straightforward approach to finding the common denominator.

With a common denominator, fractions can be combined simply by adding or subtracting their numerators. This is crucial because fractions must have the same base to allow operations across them.

Key Points about common denominators:

• Essential for combining fractions.
• Simplifies calculation steps.
• Must be established before adding or subtracting fractions.

Unifying denominators upfront makes solving the rational expression straightforward and reduces complexity in the overall process.

Fraction Simplification

Simplifying fractions is the art of reducing them to their simplest form. After aligning denominators as in the exercise, the operation shifts focus to the numerators, simplifying the expression further.

In the example, once fractions \( \frac{4}{x-3} \) and \( \frac{-2}{x-3} \) share the same denominator, the numerators are combined to yield \( \frac{2}{x-3} \). The fraction is now in its simplest form, meaning it can no longer be reduced.

Steps to simplify fractions:

• Combine numerators after matching denominators.
• Reduce the result, if possible.
• Check that the final form cannot be simplified further.

By mastering fraction simplification, mathematical expressions become clearer, making the overall calculus much more manageable.