Problem 7 - Introductory Algebra for College Students

Question

add or subtract as indicated. Simplify the result, if possible. $$\frac{4}{x}+\frac{2}{x}$$

Step-by-Step Solution

Answer

The result is \(\frac{6}{x}\).

Step 1: Identify the Numerators and Denominator

In the given expression, \(\frac{4}{x} + \frac{2}{x}\), the numerators are 4 and 2 and they share a common denominator which is 'x'.

Step 2: Add the Numerators

When two or more fractions share the same denominator, simply add or subtract the numerators. That is \(\frac{4+2}{x}\).

Step 3: Simplify if possible

After Adding the numerators we obtain \(\frac{6}{x}\), which cannot be further simplified as 6 and x have no common factors.

Key Concepts

Addition of Algebraic Fractions

Adding algebraic fractions is much like adding regular fractions. When you add fractions, the first step is to check if they share the same denominator. If they do, you can directly add their numerators. In our example, both fractions have the same denominator 'x'. This means that the fractions can be combined by adding only their numerators. So, for the expression \( \frac{4}{x} + \frac{2}{x} \), the addition is straightforward:

• Add the numerators: 4 + 2 = 6.
• Keep the same denominator: 'x'.

This gives the intermediate result \( \frac{6}{x} \).
Knowing how to add fractions is essential in algebra as it often forms the foundation for solving more complicated equations. With practice, this process becomes quick and intuitive.

Finding a Common Denominator

A common denominator is crucial when adding fractions. It ensures that the fractions are comparable by providing a shared base for comparison. In algebraic fractions, the denominator might contain variables, like 'x' in our example. When fractions have the same denominator, like \( \frac{4}{x} \) and \( \frac{2}{x} \), we can directly perform addition or subtraction:

• Having the same denominator allows direct addition of numerators.
• The result retains the common denominator.

When fractions do not have the same denominator, you need to find what's called the least common denominator (LCD). This is similar to finding the least common multiple (LCM) of the denominators, ensuring each fraction represents the same part of a whole. This is especially important when dealing with algebraic expressions where denominators might be complex polynomials.

Simplifying Fractions

Simplifying fractions is about reducing them to their lowest terms, making them easier to understand and use. After performing operations like addition or subtraction, check if the fraction can be simplified. Simplification involves finding common factors in the numerator and denominator. In our case with \( \frac{6}{x} \), 6 and 'x' don't have common factors (other than 1), so it remains unsimplified.

• Always look for common factors in both numerator and denominator.
• If no common factor exists apart from 1, the fraction is already in simplest form.

Simplification is an important skill that helps make expressions neater and can aid in further calculations. Simplified fractions are easier to interpret, especially as problems become more complex. For learners, mastering this means they can confidently tackle more involved algebraic expressions or equations efficiently.