Problem 71 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{4}{n}-\frac{6}{n+4}$$

Step-by-Step Solution

Answer

The simplest form of \(\frac{4}{n} - \frac{6}{n+4}\) is \(\frac{-2(n - 8)}{n(n+4)}\).

Step 1: Identify a common denominator

To subtract fractions, they need to have a common denominator. The denominators here are \(n\) and \(n+4\). The simplest common denominator for these would be \(n(n+4)\).

Step 2: Rewrite the fractions with a common denominator

Transform each fraction so that they both have the common denominator \(n(n+4)\).\\[\frac{4}{n} = \frac{4(n+4)}{n(n+4)}, \quad \frac{6}{n+4} = \frac{6n}{n(n+4)}\]

Step 3: Subtract the numerators

Now that both fractions have a common denominator, subtract the numerators: \\[\frac{4(n+4)}{n(n+4)} - \frac{6n}{n(n+4)} = \frac{4n + 16 - 6n}{n(n+4)}\]

Step 4: Simplify the expression

Simplify the expression obtained from subtracting the numerators: \\[\frac{4n + 16 - 6n}{n(n+4)} = \frac{-2n + 16}{n(n+4)}\]

Step 5: Express in simplest form

Look at the expression \(\frac{-2n + 16}{n(n+4)}\). Factor out \(-2\) from the numerator: \\[\frac{-2(n - 8)}{n(n+4)}\]This is the expression in its simplest form.

Key Concepts

Common Denominator

Understanding how to find a common denominator is crucial when working with fractions, especially in algebra. Denominators are the bottom parts of fractions which indicate into how many parts the whole is divided. In the expression \(\frac{4}{n} - \frac{6}{n+4}\), the denominators \(n\) and \(n+4\) are different, so we need a common denominator to perform subtraction.

To find a common denominator:

• Look at each denominator separately.
• Identify the least common multiple (LCM) or, in algebraic terms, combine the factors.

For these fractions, the common denominator is found by multiplying the two denominators: \(n(n+4)\). This ensures that both fractions are expressed with the same denominator, which simplifies the process of addition or subtraction.

Subtraction of Fractions

Subtracting fractions involves several important steps: achieving a common denominator and then subtracting the numerators. With the fraction expression \(\frac{4}{n} - \frac{6}{n+4}\), once we've rewritten each fraction with the common denominator \(n(n+4)\), subtraction can proceed.

Here's how:

• Each fraction is adjusted: \(\frac{4}{n}\) becomes \(\frac{4(n+4)}{n(n+4)}\) and \(\frac{6}{n+4}\) becomes \(\frac{6n}{n(n+4)}\).
• Once this is done, subtract the numerators of the new fractions.
• This gives the equation: \(\frac{4n + 16 - 6n}{n(n+4)}\).

This subtraction allows us to focus on simplifying the expression by combining like terms in the numerator.

Simplifying Expressions

Simplifying expressions is a critical part of algebra, where the goal is to make the expression as concise as possible. From the subtraction step, we have the expression \(\frac{4n + 16 - 6n}{n(n+4)}\). Let's break down the simplification:

• The expression can be reduced by combining like terms in the numerator: \(4n - 6n\) reduces to \(-2n\).
• This leads to the simplified form \(\frac{-2n + 16}{n(n+4)}\).
• Next, look for common factors in the numerator. Factoring out \(-2\) gives \(\frac{-2(n - 8)}{n(n+4)}\).

This final form is much simpler and is easy to interpret or use in further calculations. Simplifying not only makes the solution neater but often reveals patterns or insights about the structure of the mathematical problem.