Problem 49 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{1}{a+1}-\frac{1}{a-1}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Common Denominator

To subtract the fractions \( \frac{1}{a+1} - \frac{1}{a-1} \), we need to find a common denominator. The least common denominator (LCD) of \( a+1 \) and \( a-1 \) is \( (a+1)(a-1) \), which is the product of both denominators.

Step 2: Rewrite Each Fraction

Rewrite each fraction with the common denominator. This transforms \( \frac{1}{a+1} \) into \( \frac{1(a-1)}{(a+1)(a-1)} = \frac{a-1}{(a+1)(a-1)} \) and \( \frac{1}{a-1} \) into \( \frac{1(a+1)}{(a-1)(a+1)} = \frac{a+1}{(a+1)(a-1)} \).

Step 3: Subtract the Numerators

With a common denominator, you can subtract the numerators: \( \frac{a-1}{(a+1)(a-1)} - \frac{a+1}{(a+1)(a-1)} = \frac{(a-1) - (a+1)}{(a+1)(a-1)} \).

Step 4: Simplify the Expression

Simplify the numerator: \((a-1) - (a+1) = a - 1 - a - 1 = -2\). Thus, the expression becomes \( \frac{-2}{(a+1)(a-1)} \).

Step 5: Final Answer

The difference of the fractions is \( \frac{-2}{(a+1)(a-1)} \).

Key Concepts

Understanding Fractions

Fractions are ways to represent parts of a whole, expressed as a ratio where the numerator (top number) shows how many parts we have and the denominator (bottom number) indicates the total parts. For example, in the fraction \( \frac{1}{a+1} \), \( 1 \) is the numerator and \( a+1 \) is the denominator.
Understanding fractions means being aware of their roles in mathematics, such as showing proportions or performing operations like addition and subtraction. When operating with fractions, the denominators must be considered carefully to ensure the fractions are comparable or combinable.
It is crucial to get comfortable with manipulating fractions. Especially, identifying things like common denominators, which play a vital role when adding, subtracting, or comparing fractions.

Finding a Common Denominator

When dealing with fractions, particularly addition or subtraction, finding a common denominator is essential. A common denominator allows different fractions to be expressed with the same denominator, thus making calculations straightforward.
Here's why it's important:

• It provides a uniform base for comparison or calculation.
• It ensures that we are working with pieces of the same size.

A common denominator doesn't always have to be the least one, but finding the least common denominator (LCD) is efficient and simplifies the arithmetic. In our example, the denominators are \( a+1 \) and \( a-1 \), so the least common denominator is the product \((a+1)(a-1)\).
Including a common denominator allows us to transform the fractions so that they share this denominator, making the subtraction operation possible and way simpler.

Simplification of Fractions

Simplification is the process of making fractions easier to understand or solve by reducing them to their simplest form. It involves removing unnecessary parts or breaking down complex expressions into simpler forms. In subtraction and addition of fractions, this typically means subtracting or adding numerators first, while keeping the common denominator intact.
In our subtraction example:

• After transforming both fractions to have the same denominator, we subtracted the numerators: \( \frac{a-1}{(a+1)(a-1)} - \frac{a+1}{(a+1)(a-1)} \) provides \( \frac{(a-1) - (a+1)}{(a+1)(a-1)} \).
• Then we simplify the expression in the numerator: \((a-1)-(a+1)\) simplifies to \(-2\).

The result, \( \frac{-2}{(a+1)(a-1)} \), is the simplest form of the difference. Simplification not only makes the expression clearer but also allows us to see the real value or relationship the expression represents.