Problem 48 - Algebra and Trigonometry

Question

Perform the addition or subtraction and simplify. $$ \frac{1}{x+1}+\frac{1}{x-1} $$

Step-by-Step Solution

Answer

\( \frac{2x}{(x+1)(x-1)} \) is the simplified expression.

Step 1: Find a Common Denominator

To add these two fractions, we need a common denominator. The common denominator for \(\frac{1}{x+1}\) and \(\frac{1}{x-1}\) is \((x+1)(x-1)\).

Step 2: Rewrite Fractions with Common Denominator

Rewrite each fraction with the common denominator \((x+1)(x-1)\):\[\frac{1}{x+1} = \frac{x-1}{(x+1)(x-1)}and \frac{1}{x-1} = \frac{x+1}{(x+1)(x-1)}.\]

Step 3: Add the Fractions

Now add the fractions:\[\frac{x-1}{(x+1)(x-1)} + \frac{x+1}{(x+1)(x-1)} = \frac{(x-1) + (x+1)}{(x+1)(x-1)}.\]Combine the numerators: \[(x-1) + (x+1) = 2x.\]

Step 4: Simplify the Expression

The expression becomes:\[\frac{2x}{(x+1)(x-1)}.\]There are no common factors between the numerator and the denominator to simplify it further, so this is the simplified expression.

Key Concepts

Common Denominator

When dealing with an addition of fractions, finding a common denominator is crucial. The common denominator is essentially a shared base that both fractions can align with, like finding a common language for two speakers. In our exercise, each fraction had a different denominator: \( x+1 \) and \( x-1 \). To add them, we must change them so they share the same denominator.
The quickest way to do this is by multiplying their denominators together: \((x+1)(x-1)\). This forms our common denominator. The process involves:

• Identifying the distinct parts of each denominator.
• Multiplying these parts to form a new denominator that accommodates both original fractions.

Choosing the common denominator this way ensures that the two fractions can be rewritten with a unified base, which is key for performing the addition or subtraction.

Addition of Fractions

Once you have the common denominator established, the next step is adding the fractions. Addition of fractions requires more than just adding numerators because the denominators indicate different parts of the whole. Here's how it works:

Rewriting the Fractions

With our fractions \( \frac{1}{x+1} \) and \( \frac{1}{x-1} \), using the common denominator \((x+1)(x-1)\), we rewrite:

• \( \frac{1}{x+1} \) becomes \( \frac{x-1}{(x+1)(x-1)} \).
• \( \frac{1}{x-1} \) becomes \( \frac{x+1}{(x+1)(x-1)} \).

In this adjusted form, both fractions now operate over the same base.

Numerator Addition

Now we simply add the numerators together while keeping the denominator unchanged:

• \((x-1) + (x+1)\) simplifies to \(2x\).

The result is a new fraction \( \frac{2x}{(x+1)(x-1)} \), which represents the combined value of the two original fractions over their shared whole.

Simplifying Fractions

Simplifying fractions makes them easier to understand and often uses smaller numbers. It essentially involves breaking down the fraction to its most basic form. However, not all fractions can be simplified if there are no common factors shared by the numerator and denominator.
In our example, after adding and reaching the fraction \( \frac{2x}{(x+1)(x-1)} \), the numerator is just \(2x\) and the denominator \((x+1)(x-1)\).

• Check for common factors: \(2x\) has factors of \(2\) and \(x\), but neither match the denominator factors.
• Since there are no shared common factors between numerator and denominator, \( \frac{2x}{(x+1)(x-1)} \) is already simplified.

This understanding reveals that the operation has been fully executed and optimized, with no extra steps necessary for further simplification.