Problem 5 - Elementary and Intermediate Algebra

Question

Fill in the blanks. $$ \text { Consider: } \frac{\frac{x-3}{4}}{\frac{1}{12}-\frac{x}{6}} $$ a. What is the numerator of the complex fraction? Is it a single rational expression? b. What is the denominator of the complex fraction? Is it a single rational expression?

Step-by-Step Solution

Answer

Numerator: \( \frac{x-3}{4} \), single rational expression. Denominator: \( \frac{1}{12} - \frac{x}{6} \), not a single rational expression.

Step 1: Identify the Numerator

The given complex fraction is \( \frac{\frac{x-3}{4}}{\frac{1}{12}-\frac{x}{6}} \). The numerator of this complex fraction is the expression on the top, which is \( \frac{x-3}{4} \). Yes, it is already a single rational expression.

Step 2: Identify the Denominator

The denominator of the complex fraction is the expression on the bottom, which is \( \frac{1}{12} - \frac{x}{6} \). This expression is not a single rational expression yet because it involves the subtraction of two fractions. To make it a single rational expression, we would need to find a common denominator and combine the fractions.

Key Concepts

Understanding Rational Expressions

A rational expression is akin to a fraction in which both the numerator and the denominator are polynomials. In simpler terms, it is a fraction that contains variables. Take, for example, the expression \( \frac{x-3}{4} \). Here, the expression \( x-3 \) in the numerator, and the number 4 in the denominator are both examples of polynomials.

The idea of rational expressions becomes particularly crucial when working with complex fractions, as they can contain multiple layers of these expressions. In our original exercise, we have \( \frac{\frac{x-3}{4}}{\frac{1}{12}-\frac{x}{6}} \), a complex fraction where both top and bottom levels utilize rational expressions. This demonstrates the flexibility and utility of rational expressions in representing relationships between quantities.

• Recognize that rational expressions can be simplified or manipulated just like numerical fractions.
• Always reduce rational expressions to their simplest form when possible.

Numerator and Denominator in Complex Fractions

Within any fraction, including complex fractions, there are two key components: the numerator and the denominator. Let's explore these terms alongside our key example.

The numerator is the top part of the fraction. In a complex fraction like \( \frac{\frac{x-3}{4}}{\frac{1}{12}-\frac{x}{6}} \), determining the numerator involves focusing on the expression that is positioned above the fraction line. In our case, it is \( \frac{x-3}{4} \), which serves as the top level of the entire complex fraction. It is already simplified to a single rational expression.

Meanwhile, the denominator, being the bottom part, is \( \frac{1}{12} - \frac{x}{6} \). Originally, it is not a single rational expression due to the subtraction involved. To express it as a single rational expression, require finding a common denominator and performing the necessary subtraction.

• Always identify the numerator and denominator clearly in any expression.
• Aim to simplify both parts whenever possible to make the expression easier to manipulate.

The Process of Fraction Subtraction

Fraction subtraction is an essential skill when working with rational expressions, especially in the context of complex fractions. The main goal of fraction subtraction is to combine different fractions into a single one.

When subtracting fractions, finding a common denominator is your first step. In our example, \( \frac{1}{12} - \frac{x}{6} \), the denominators 12 and 6 are distinct. We need to find the least common denominator (LCD), which is 12 in this case, to perform the subtraction.

Here’s how:

• Convert \( \frac{x}{6} \) into an equivalent fraction with a denominator of 12, obtaining \( \frac{2x}{12} \).
• Now, subtract: \( \frac{1}{12} - \frac{2x}{12} = \frac{1-2x}{12} \).

Combining these fractions forms a single rational expression, simplifying what initially appears as a complicated process. As you practice this skill, you'll find manipulating rational expressions and complex fractions becomes much more intuitive.