Problem 31 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 3 or \(5 .\) $$ \frac{\frac{1}{x}-3}{\frac{5}{x}+2} $$

Step-by-Step Solution

Answer

\( \frac{1 - 3x}{5 + 2x} \)

Step 1: Understanding the Problem

The given expression is a complex fraction with another fraction in its numerator and denominator. It is of the form: \( \frac{\frac{1}{x} - 3}{\frac{5}{x} + 2} \). We need to simplify this to a single fraction.

Step 2: Finding the LCD of the mini fractions

Identify the least common denominator (LCD) of the fractions within the complex fraction. Both the numerator \(\frac{1}{x} - 3\) and the denominator \(\frac{5}{x} + 2\) contain terms with \(x\), so the LCD is \(x\).

Step 3: Rewriting Numerator and Denominator

Rewrite both the numerator and denominator so they are single fractions over the common denominator \(x\). For the numerator, rearrange to get: \(\frac{1}{x} - \frac{3x}{x} = \frac{1 - 3x}{x}\). For the denominator, rearrange to get: \(\frac{5}{x} + \frac{2x}{x} = \frac{5 + 2x}{x}\).

Step 4: Simplifying the Complex Fraction

The expression is now: \( \frac{\frac{1 - 3x}{x}}{\frac{5 + 2x}{x}} \). Since both the numerator and the denominator are fractions over the same \(x\), divide the two fractions: \(\frac{1 - 3x}{x} \div \frac{5 + 2x}{x} = \frac{1 - 3x}{5 + 2x}\).

Step 5: Simplified Result

The complex fraction simplifies to a simple fraction \( \frac{1 - 3x}{5 + 2x} \). This is the simplest form of the given complex fraction.

Key Concepts

Least Common Denominator

When simplifying complex fractions, one important task is identifying a common baseline for comparison, known as the least common denominator (LCD). The LCD is the smallest number or expression into which all the denominators in a given set can be divided without leaving a remainder. This shared denominator helps in combining fractions, making it easier to simplify them.

In the complex fraction \( \frac{\frac{1}{x} - 3}{\frac{5}{x} + 2} \), the denominators within the fractions in both the numerator and the denominator involve the variable \(x\). Therefore, \(x\) is the least common denominator. By using this LCD, the mini-fractions in both the numerator and denominator can be efficiently rewritten over a common denominator \(x\).

Using the LCD not only simplifies the understanding of the fraction's parts, but it also lays the foundation for the simplification process by creating a single, unified denominator across all components.

Numerator Simplification

After determining the least common denominator, the next step is to simplify the numerator of the complex fraction. Our goal is to express the numerator as a single fraction over the given LCD.

Take the numerator from \( \frac{1}{x} - 3 \). The denominator here is \(x\), and we need to transform each term so they share this common denominator. The number 3 can be rewritten with a denominator of \(x\) by expressing it as \( \frac{3x}{x} \). Thus, the numerator becomes:

• \( \frac{1}{x} - \frac{3x}{x} = \frac{1 - 3x}{x} \)

This new form, \( \frac{1 - 3x}{x} \), represents the entire numerator of the complex fraction as a single fraction, making it ready for further simplification.

Denominator Simplification

Simplifying the denominator follows the same principle as simplifying the numerator: rewrite terms over the same least common denominator.

For the denominator fraction \( \frac{5}{x} + 2 \), the process involves rewriting each term to have \(x\) as the denominator. We can write 2 as \( \frac{2x}{x} \) to match the denominator \(x\). The expression would then look like this:

• \( \frac{5}{x} + \frac{2x}{x} = \frac{5 + 2x}{x} \)

Now, the denominator is consolidated into a single fraction, \( \frac{5 + 2x}{x} \). This format is essential for solving the complex fraction, as it aligns the components for an easier division process.