Problem 40 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x+4}-\frac{1}{x}}{\frac{1}{x+4}} $$

Step-by-Step Solution

Answer

\frac{-4}{x}

Step 1: Simplify the numerator

First, focus on simplifying the numerator, which is \(\frac{1}{x+4} - \frac{1}{x}\). To simplify this expression, find a common denominator. The common denominator for \(x+4\) and \(x\) is \(x(x+4)\).\[\frac{1}{x+4} = \frac{x}{x(x+4)}\]\[\frac{1}{x} = \frac{x+4}{x(x+4)}\]Now, subtract these fractions:\[\frac{x - (x+4)}{x(x+4)} = \frac{x - x - 4}{x(x+4)} = \frac{-4}{x(x+4)}\]

Step 2: Simplify the given expression

Next, simplify the original expression \(\frac{\frac{1}{x+4} - \frac{1}{x}}{\frac{1}{x+4}}\). We have already simplified the numerator to \(\frac{-4}{x(x+4)}\). The denominator of the given expression is \(\frac{1}{x+4}\).So the new expression becomes: \[\frac{\frac{-4}{x(x+4)}}{\frac{1}{x+4}}\]

Step 3: Divide the fractions

To divide the two fractions, multiply by the reciprocal of the denominator. So, we get:\[\frac{-4}{x(x+4)} \times \frac{x+4}{1}\]The \(x+4\) terms cancel out:\[\frac{-4}{x}\]

Step 4: Simplify the result

After canceling out \(x+4\), we are left with: \[\frac{-4}{x}\]

Key Concepts

Fraction Operations

When dealing with algebra, you'll often come across problems involving fractions. Mastering fraction operations is essential. These operations are primarily: addition, subtraction, multiplication, and division. Let's break it down:

• Addition and Subtraction: When adding or subtracting fractions, you first need a common denominator. This ensures you're working with parts of the same whole.
• Multiplication: For multiplication, simply multiply the numerators together and the denominators together. Simplify if possible.
• Division: Dividing fractions means multiplying by the reciprocal of the divisor. More on reciprocals in the next section!

Applying these principles to our exercise, we first dealt with subtraction of fractions by finding a common denominator. Multiplication and division rules helped us simplify and resolve the final expression. Practice these operations with different examples to get comfortable with these steps.

Common Denominator

To add or subtract fractions, it's vital to have a common denominator. This is essentially the same base, allowing you to easily combine the fractions. Here's the process:

• Identify the Denominators: Look at the denominators of the fractions you need to combine. In the given exercise, the denominators were \(x+4\) and \(x\).
• Find the Least Common Denominator (LCD): The LCD for \(x+4\) and \(x\) is \(x(x+4)\). The LCD should be the smallest expression that each denominator can divide into.
• Rewriting Fractions: Convert each fraction so they share this common denominator. This often involves multiplying the numerator and denominator of each fraction to match the LCD. For example, \[ \frac{1}{x+4} = \frac{x}{x(x+4)} \], \[ \frac{1}{x} = \frac{x+4}{x(x+4)} \].

Achieving a common denominator allows us to perform the subtraction operation easily, giving us the simplified numerator used in later steps.

Reciprocal

The concept of reciprocals is a fundamental one in division of fractions. A reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). To divide fractions, you multiply by the reciprocal of the divisor. Let's understand why:

• Basic Definition: The reciprocal flips the numerator and the denominator of a fraction. For example, the reciprocal of \( \frac{1}{x+4} \) is \((x+4)\frac{1}{1} = x+4\). This transformation aids in simplifying complex fraction divisions.
• Application: In our exercise, we needed to divide \[ \frac{\frac{-4}{x(x+4)}}{\frac{1}{x+4}} \]. By flipping \frac{1}{x+4}\ to \ x+4\ and multiplying, it simplifies our overall expression.

Understanding and using reciprocals properly can make fraction division straightforward, avoiding common pitfalls. Practicing fraction division with reciprocals repeatedly will build your comfort with these operations.