Problem 50 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem Statement

You are given two algebraic fractions, \( \frac{1}{x+z} \) and \( \frac{1}{x-z} \), and need to find their sum.

Step 2: Identify the Formula for Adding Algebraic Fractions

To add fractions with different denominators, we use the formula: \( \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \). This involves finding a common denominator.

Step 3: Determine the Common Denominator

The common denominator for \( \frac{1}{x+z} \) and \( \frac{1}{x-z} \) is the product of the two denominators: \((x+z)(x-z)\).

Step 4: Rewrite Each Fraction with the Common Denominator

Rewrite \( \frac{1}{x+z} \) as \( \frac{x-z}{(x+z)(x-z)} \) and \( \frac{1}{x-z} \) as \( \frac{x+z}{(x+z)(x-z)} \).

Step 5: Combine the Fractions

Add the two fractions: \( \frac{x-z}{(x+z)(x-z)} + \frac{x+z}{(x+z)(x-z)} = \frac{(x-z) + (x+z)}{(x+z)(x-z)} \).

Step 6: Simplify the Numerator

Simplify the numerator \((x-z) + (x+z)\) to \(2x\).

Step 7: Write the Final Expression

The final simplified expression is \( \frac{2x}{(x+z)(x-z)} \). This is your answer.

Key Concepts

Common Denominator

When dealing with fractions, one of the most important steps is finding a common denominator. This allows us to combine fractions into a single expression.

• The common denominator is essentially a shared multiple of the individual denominators.
• For the fractions \( \frac{1}{x+z} \) and \( \frac{1}{x-z} \), their denominators are \( x+z \) and \( x-z \).
• To combine these fractions, we multiply these denominators: \((x+z)(x-z)\).

This becomes the standard denominator for both fractions, enabling easier addition or subtraction.
Finding this common ground simplifies the process of working with algebraic fractions significantly.

Addition of Fractions

Adding fractions, especially with different denominators, can seem tricky at first.

• The key is to rewrite each fraction so that they share the same denominator.
• Once they have a common denominator, the fractions can be added together by combining their numerators.

In our exercise:

• We rewrite \( \frac{1}{x+z} \) as \( \frac{x-z}{(x+z)(x-z)} \).
• The other fraction \( \frac{1}{x-z} \) becomes \( \frac{x+z}{(x+z)(x-z)} \).
• This makes the addition straightforward: \( \frac{x-z + x+z}{(x+z)(x-z)} \).

Understanding this method ensures we can handle any fraction addition with ease.

Simplifying Fractions

Once fractions are added, simplifying the resulting expression is the next step. This involves reducing the fraction to its simplest form.

• For \( \frac{(x-z) + (x+z)}{(x+z)(x-z)} \), we first simplify the numerator.
• The expression \((x-z) + (x+z)\) simplifies to \(2x\).

Our fraction now looks like \( \frac{2x}{(x+z)(x-z)} \).
At this point, it’s important to check if further simplification is possible, although in this example, \(2x\) is already the simplest form.
This reduces complexity and makes the final answer easier to interpret.