Problem 75 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{2}{x^{2}-1}-\frac{x+1}{x^{2}-1} $$

Step-by-Step Solution

Answer

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

Step 1: Identify a Common Denominator

Observe that both fractions share the same denominator, which is \(x^2 - 1\). The expression is \(\frac{2}{x^2 - 1} - \frac{x + 1}{x^2 - 1}\). So, in this case, the common denominator is already present.

Step 2: Combine the Fractions

Since the denominators are the same, we can combine the numerators. The subtraction of fractions becomes: \(\frac{2 - (x + 1)}{x^2 - 1}\).

Step 3: Simplify the Numerator

Distribute the negative sign across the terms in the numerator. This becomes \(\frac{2 - x - 1}{x^2 - 1}\). Simplifying the numerator yields \(\frac{1 - x}{x^2 - 1}\).

Step 4: Review the Result

Now that we have simplified the expression to \(\frac{1-x}{x^2-1}\), confirm that there are no further simplifications or common factors.

Key Concepts

Simplification

When working with algebraic fractions, simplification is the main goal. It results in an expression that is easier to interpret and use. Simplification involves reducing complex expressions to their simplest form.

In the given exercise, we simplify the expression \( \frac{2}{x^2 - 1} - \frac{x + 1}{x^2 - 1} \) by focusing on reducing the complexity of the numerators and, sometimes, identifying potential common factors in the denominator and numerator. This makes the fraction easier to understand and solve further if required.

• Aim to make numerators and denominators as simple as possible.
• Look for opportunities to factor and cancel common factors.
• Always double-check if further simplification is possible.

Common Denominator

A common denominator is essential to combine fractions seamlessly. It is the shared or same denominator between two or more fractions, allowing you to either add or subtract them without changing the values.

For the expression \( \frac{2}{x^2 - 1} - \frac{x+1}{x^2 - 1} \), the denominator \( x^2 - 1 \) is already common. By recognizing this common denominator, we can proceed to directly subtract the numerators.

• Ensure that the denominators are the same to combine fractions.
• If not shared, find a least common denominator (LCD) to reconcile differences.
• This step is crucial for effective simplification and combination of fractions.

Numerator Combination

The core action in combining the numerators when subtracting fractions is straightforward. With a shared denominator, these components can be subtracted directly.

In our example, the numerators \( 2 \) and \( x+1 \) are combined by performing subtraction. This means taking \( 2 - (x + 1) \). It is crucial to correctly distribute signs during this step.

• Subtract the second numerator from the first.
• Ensure the subtraction is applied to the entire second term.
• Consider using distributive properties to simplify signs.

Fraction Subtraction

Subtracting algebraic fractions is similar to subtracting numerical ones, provided you handle all elements carefully, especially signs.

Once the numerators have been combined, as seen in \( \frac{2 - x - 1}{x^2 - 1} \), simplifying by combining like terms yields \( \frac{1 - x}{x^2 - 1} \).

Key steps in fraction subtraction include:

• Ensuring one shared common denominator.
• Carefully managing and distributing the negative sign over all terms in the subtracted numerator.
• Simplifying the result to ensure no further reduction is possible.

Fraction subtraction can often lead to simpler forms, making it easier to analyze or solve further.