Problem 36 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{6}{x}-1\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{6-x}{x}\).

Step 1: Understand the Expression

The given expression is \(\frac{6}{x} - 1\). This is a subtraction of two terms: a rational expression (\(\frac{6}{x}\)) and a whole number (1).

Step 2: Find a Common Denominator

To subtract these terms, they need a common denominator. The rational expression has the denominator \(x\). We can represent 1 as \(\frac{x}{x}\). Thus, the expression becomes \(\frac{6}{x} - \frac{x}{x}\).

Step 3: Perform the Subtraction

Now that both terms have a common denominator, subtract the numerators: \(\frac{6-x}{x}\).

Step 4: Simplify the Expression (if possible)

The expression \(\frac{6-x}{x}\) is already in its simplest form, as there are no common factors in the numerator and the denominator.

Key Concepts

Understanding Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. In our exercise, the rational expression is \(\frac{6}{x}\), where the numerator is the constant 6 and the denominator is the variable \(x\). These types of expressions can sometimes be simplified by factoring polynomials or canceling common factors in the numerator and denominator. However, in this case, since 6 and \(x\) share no common factors, the expression is already as simple as it can get.
The primary challenge with rational expressions, especially those involving variables, is that they are subject to restrictions. For instance, the denominator cannot be zero. In this example, \(x\) must be any real number except zero, because division by zero is undefined.

Subtraction of Fractions

Subtracting fractions involves making sure both fractions have the same denominator. It's much like subtracting usual numbers but requires careful coordination of the denominators. When the denominators are different, as in the example \(\frac{6}{x} - 1\), a common denominator must be determined. Once the denominators match, subtraction proceeds by subtracting the numerators.
In our problem, \(1\) can be rewritten as \(\frac{x}{x}\) to have a denominator of \(x\), making it easier to perform the subtraction: \(\frac{6}{x} - \frac{x}{x}\). Upon subtraction, you focus on the numerators: \(6 - x\). This results in the simplified expression, \(\frac{6-x}{x}\).
Remember to always check if the subtraction can further simplify the numerator or denominator, but if no common factors exist, then the task is complete.

Finding a Common Denominator

A common denominator is crucial when dealing with the addition or subtraction of fractions, including rational expressions. It stands as the unified base which allows numerators to be directly added or subtracted. In our exercise, we began with the expression \(\frac{6}{x} - 1\), where \(x\) acts as the denominator for only the rational expression. To perform subtraction, the whole number 1 needs to be transformed to a rational number with the same denominator.
This is achieved by expressing 1 as a fraction with \(x\) as its denominator: \(\frac{x}{x}\). Now, both fractions \(\frac{6}{x}\) and \(\frac{x}{x}\) share a common denominator of \(x\).
Common denominators simplify calculations and ensure the fractions are on an equal footing, allowing you to subsequently focus on subtracting the numerators. Whether working with simple numbers or complex polynomials, establishing a common denominator is always a critical step!