Problem 74 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{2}{x}-\frac{6}{3 x-2}$$

Step-by-Step Solution

Answer

\(\frac{-4}{x(3x-2)}\)

Step 1: Identify a Common Denominator

To add or subtract fractions, they must have a common denominator. The denominators in this problem are \(x\) and \(3x - 2\). A common denominator can be found by multiplying the two denominators together, resulting in \(x(3x - 2)\).

Step 2: Rewrite Each Fraction with the Common Denominator

Convert \(\frac{2}{x}\) and \(\frac{6}{3x-2}\) to have \(x(3x-2)\) as their denominator. The fraction \(\frac{2}{x}\) becomes \(\frac{2(3x-2)}{x(3x-2)}\), and \(\frac{6}{3x-2}\) becomes \(\frac{6x}{x(3x-2)}\).

Step 3: Perform the Subtraction

Subtract the numerators of the converted fractions: \(\frac{2(3x-2)}{x(3x-2)} - \frac{6x}{x(3x-2)} = \frac{6x - 4 - 6x}{x(3x-2)}\).

Step 4: Simplify the Resulting Expression

Combine like terms in the numerator: \(6x - 4 - 6x = -4\). So the expression becomes \(\frac{-4}{x(3x-2)}\).

Step 5: Finalize the Simplified Expression

Ensure the expression \(\frac{-4}{x(3x-2)}\) is in its simplest form. Since there are no common factors in the numerator and denominator other than 1, this is the simplest form.

Key Concepts

Common Denominator

When adding or subtracting rational expressions, finding a common denominator is essential. This common denominator acts like a baseline which allows us to combine the fractions seamlessly.

• The common denominator should be a multiple of all the original denominators involved in the operation.
• It often involves finding the least common denominator (LCD) which is optimal but not always necessary for the solution.

In our problem, we have two denominators: \(x\) and \(3x - 2\). To determine a suitable common denominator, we multiply these terms together. This gives us the common denominator: \(x(3x - 2)\). This step ensures both fractions are comparable, setting the stage for addition or subtraction.

By using this approach, you can efficiently deal with fractions or rational expressions and simplify the process of adding or subtracting them. It's important to remember that the terms in the denominator must be multiplied, making it critical to handle these expressions carefully to avoid errors.

Simplifying Fractions

Simplifying fractions is a crucial part of working with rational expressions. It involves reducing the fraction to its simplest form by ensuring the greatest common factor between the numerator and the denominator is reduced to 1.

• Always start by simplifying numerators and denominators separately.
• If any common factors exist, divide both the numerator and denominator by these factors.

In our exercise, after finding a common denominator and performing the subtraction, we achieved a new expression: \(\frac{-4}{x(3x-2)}\). Here, it is vital to inspect both the numerator and the denominator:

• The numerator is \(-4\), which has no common factors with \(x(3x - 2)\) other than 1.
• The denominator \(x(3x - 2)\) is a polynomial, and further simplification is not feasible.

Thus, our fraction is already in its simplest form, highlighting the importance of analyzing each component to determine its reducibility.

Numerator and Denominator

Understanding the roles of the numerator and denominator is fundamental when dealing with fractions. Each part represents a crucial element in rational expressions.

• Numerator: The top part of a fraction. It shows how many parts of the whole are considered.
• Denominator: The bottom part. It indicates the total number of equal parts the whole is divided into.

In expressions like \(\frac{2}{x}\) and \(\frac{6}{3x-2}\), both terms carry specific meanings in context:

• For \(\frac{2}{x}\), the numerator (2) specifies the amount in question defined by \(x\).
• For \(\frac{6}{3x-2}\), the numerator (6) relates to the polynomial expression in the denominator \((3x-2)\).

This separation helps in combining, subtracting, or even simplifying rational expressions. It requires adjusting the numerators once both fractions have the same denominator, allowing for seamless mathematical operations.