Problem 38 - Algebra and Trigonometry

Question

\(35-54\) . 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

To add these fractions, we need a common denominator. The denominators are \(x+1\) and \(x-1\). The common denominator is the product of these two, \((x+1)(x-1)\).

Step 2: Rewrite Each Fraction

Express each fraction with the common denominator \((x+1)(x-1)\).- \(\frac{1}{x+1}\) becomes \(\frac{1 \cdot (x-1)}{(x+1)(x-1)} = \frac{x-1}{(x+1)(x-1)}\).- \(\frac{1}{x-1}\) becomes \(\frac{1 \cdot (x+1)}{(x+1)(x-1)} = \frac{x+1}{(x+1)(x-1)}\).

Step 3: Add the Fractions

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

Step 4: Simplify the Expression

Combine and simplify the terms in the numerator:\( (x-1) + (x+1) = x - 1 + x + 1 = 2x \).Therefore, the expression simplifies to:\[ \frac{2x}{(x+1)(x-1)} \]

Step 5: State the Final Simplified Result

The simplified form of the given expression \( \frac{1}{x+1} + \frac{1}{x-1} \) is:\[ \frac{2x}{(x+1)(x-1)} \]

Key Concepts

Addition of Fractions

When you add fractions, you need to ensure they have the same bottom part, known as the denominator. This is because fractions are like parts of a whole, and we want to make sure we're adding apples to apples, not apples to oranges. Suppose the fractions don't already have the same denominator. In that case, we need to find a way to make them uniform, and this process often requires finding something called a 'common denominator.'

To add numerators, you make sure both fractions have this common denominator. Once they do, you simply add the top parts (the numerators). Remember:

• Keep the denominator the same.
• Make sure to handle the numerators correctly.

Common Denominator

The concept of a common denominator is crucial when dealing with fractions that you want to add or subtract. It is essentially a shared base between multiple fractions, allowing them to be combined easily.

If you think of the denominators as different units, the common denominator is a way to convert those units so they match. In mathematical terms, you often achieve this by multiplying the different denominators together to get a new one that can work for both fractions. For example, in the exercise:

• You have fractions with denominators \(x+1\) and \(x-1\).
• The common denominator is found by multiplying them: \((x+1)(x-1)\).

Once you rewrite each fraction with this new denominator, you make adding or subtracting them straightforward.

Simplifying Expressions

Simplifying expressions is like tidying up a messy room. It's about making things as simple as possible through basic algebra rules. When you have a fraction, simplifying means finding a way to make the numbers smaller or eliminating any redundant parts.

In algebraic expressions, especially with fractions, simplification may involve combining like terms or factoring out constants or variables. For the exercise we looked at, the simplification step included:

• Combining like terms in the numerator, such as adding \((x-1) + (x+1)\) to get \(2x\).
• Leaving the denominator in its factored form, \((x+1)(x-1)\), which is already simplified as these are distinct polynomial terms.

This makes the final expression neat and ready to use or understand at a glance.