Problem 42 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(\frac{2x}{(x+1)(x-1)}\)

Step 1: Identify the Common Denominator

For addition of fractions, we need a common denominator. Identify the denominators: \(x+1\) and \(x-1\). Multiply these to create the common denominator: \((x+1)(x-1)\).

Step 2: Rewrite Each Fraction

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

Step 3: Add the Fractions

Now that both fractions have the same denominator, add them:\[\frac{x-1}{(x+1)(x-1)} + \frac{x+1}{(x+1)(x-1)} = \frac{x-1+x+1}{(x+1)(x-1)}\]

Step 4: Simplify the Numerator

Simplify the numerator:\[x-1+x+1 = 2x\]The expression becomes:\[\frac{2x}{(x+1)(x-1)}\]

Step 5: Final Answer

Since there's no further simplification possible, the final simplified expression is:\[\frac{2x}{(x+1)(x-1)}\]

Key Concepts

Common Denominator

When working with fractions, whether they are numerical or algebraic, finding a common denominator is crucial for addition or subtraction. Denominators are the bottom parts of fractions that indicate the "whole" divided amount. For example, in a fraction like \( \frac{1}{x+1} \), "x+1" is the denominator.

To find a common denominator, especially in expressions with different ones, such as \( x+1 \) and \( x-1 \), we multiply them together. This ensures both fractions are expressed in the same terms and can be easily combined. The common denominator in this case is \( (x+1)(x-1) \).

Here's the breakdown:

• Identify individual denominators: "x+1" and "x-1".
• Multiply them: \((x+1)(x-1) = x^2 - 1 \). Use this as the new denominator.

Fraction Addition

Adding fractions requires them to have the same denominator. With the common denominator established, we can rewrite each fraction to reflect it. Consider our fractions:

• Original fractions: \( \frac{1}{x+1} \) and \( \frac{1}{x-1} \).
• Rewrite with common denominator: \( \frac{x-1}{(x+1)(x-1)} \) and \( \frac{x+1}{(x+1)(x-1)} \).

After rewriting, both fractions have the same denominator of \( (x+1)(x-1) \). You can then directly add the numerators:

• Add numerators: \( x-1 + x+1 \).
• This results in the expression: \( \frac{x-1 + x+1}{(x+1)(x-1)} \).

The hardest part is ensuring accuracy in rewriting and adding numerators—simple arithmetic mistakes can lead to errors.

Simplifying Expressions

The final but crucial step is simplifying the combined expression. Simplification makes the mathematical expression easier to understand and work with. Often this involves combining like terms in the numerator, canceling common factors, or restructuring the expression in a simpler form.

For the expression \( \frac{x-1 + x+1}{(x+1)(x-1)} \):

• Combine the numerator terms: \( x - 1 + x + 1 = 2x \).
• This simplifies the fraction to: \( \frac{2x}{(x+1)(x-1)} \).

At this point, the fraction is in its simplest form because no further reduction or cancellation can occur. Always double-check for potential simplifications, such as common factors that might cancel each other out in the numerator and denominator. This careful check helps avoid unnecessary complex expressions while ensuring correctness.