Problem 51 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{10}{x+3}-\frac{2}{-x-3}$$

Step-by-Step Solution

Answer

\(\frac{8}{x+3}\)

Step 1: Analyze the Fractions

Observe the denominators of the fractions. They are \(x+3\) and \(-x-3\), which are additive inverses of each other. In order to combine fractions, we need a common denominator. Using a negative sign changes \(x+3\) into \(-x-3\).

Step 2: Transforming one of the fractions

Multiply both the numerator and the denominator of the second fraction by -1. So, \(-\frac{2}{-x-3}\) becomes \(\frac{2}{x+3}\). Now, both fractions have a common denominator.

Step 3: Add or subtract fractions

With same denominators, fractions can be easily added or subtracted. Here, subtract the numerators and keep the denominator the same. So, \(\frac{10}{x+3} - \frac{2}{x+3} = \frac{10-2}{x+3} = \frac{8}{x+3}\).

Step 4: Simplifying the resulting fraction

The fraction \(\frac{8}{x+3}\) cannot be simplified any further. This is the final result.

Key Concepts

Common Denominator

When adding or subtracting fractions, one of the most crucial steps is to establish a common denominator. This is important because fractions can only be directly added or subtracted when they share the same denominator. Think of the denominator as the "name" of the fraction, and fractions must have the same "name" to be combined directly.

Why do we need a common denominator? Without a common denominator, we are essentially comparing apples to oranges. When you line up the fractions with the same denominator, it's like making sure you are comparing apples to apples. Then you can easily add or subtract the numerators.

• Find a common denominator by identifying the least common multiple (LCM) of the denominators.
• Adjust each fraction (if necessary) so they share the common denominator.
• This allows for straightforward addition or subtraction of the numerators.

Remember, finding a common denominator is a key algebraic technique whenever fractions need to be combined.

Additive Inverses

Additive inverses are numbers that, when added together, equal zero. In algebraic fractions, recognizing additive inverses is essential for understanding how to manipulate and solve equations. For instance, in the given exercise, the denominators \(x+3\) and \(-x-3\) are additive inverses.

To work with these expressions, we can change one of the fractions to make the operation simpler. By multiplying the numerator and denominator of one fraction by -1, we effectively change its "texture" while keeping its value the same. This is exactly what happens when we convert \(-\frac{2}{-x-3}\) to \(\frac{2}{x+3}\).

• Recognize when terms include additive inverses, such as \(x+3\) and \(-x-3\).
• Multiply by -1 to convert one term to match the other, facilitating easier calculations.
• This simplifies the process of finding a common denominator and combining fractions.

Understanding additive inverses can significantly ease the solving of algebraic problems involving fractions.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This might seem straightforward, but it's crucial to ensure your answers are clear and as easy to interpret as possible. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.

Why simplify? Simplification makes fractions easier to understand and compare. In the exercise, you end up with \(\frac{8}{x+3}\), which is already in its simplest form because 8 and \(x+3\) do not share any common factors.

• Check the numerator and denominator for common factors.
• Divide both by the greatest common factor (GCF) to simplify.
• If no simplification is possible, as in this case, you've reached the end of the problem.

Simplifying fractions is a skill that keeps your results neat and accessible, making it easier for someone else to follow your work.