Problem 9 - Introductory Algebra for College Students

Question

add or subtract as indicated. Simplify the result, if possible. $$\frac{8}{9 x}+\frac{13}{9 x}$$

Step-by-Step Solution

Answer

The solution is \( \frac{21}{9x} \)

Step 1: Identify the Like Denominators

Given two fractions \( \frac{8}{9x} \) and \( \frac{13}{9x} \). Both fractions have the same denominator which is 9x.

Step 2: Add the Numerators

Since the denominators are the same, just add the numerators. We have \( \frac{8+13}{9x} \)

Step 3: Simplify the Fraction

Simplify the numerator which gives \( \frac{21}{9x} \)

Key Concepts

Addition of Fractions

Understanding the addition of fractions is essential for solving problems where fractions need to be combined. When you add fractions, the most straightforward scenario occurs when the fractions have like denominators. This means the fractions have the same denominator, which simplifies the addition process.
For example, adding \( \frac{8}{9x} + \frac{13}{9x} \) is manageable because both fractions share the denominator \( 9x \). In such cases, you focus solely on adding the numerators, while the denominator remains unchanged. Simply put, you only sum up the numbers on top of the fractions.
This leads to: \( \frac{8+13}{9x} = \frac{21}{9x} \).
If fractions have different denominators, you would first need to find a common denominator, which can sometimes add an extra step to the process.

Common Denominators

Common denominators play a crucial role when adding or subtracting fractions. A common denominator is a shared multiple of the denominators in the fractions you're working with.
When fractions have the same denominator, like in our problem with \( \frac{8}{9x} \) and \( \frac{13}{9x} \), the addition process is simplified. The denominator remains \( 9x \), while the numerators are combined.

• First, ensure the denominators are the same.
• Keep the denominators throughout the addition or subtraction process.
• Focus on adding or subtracting the numerators.

This concept is incredibly important because it sets the foundation for correctly solving fraction problems. When denominators differ, you must find a common denominator before proceeding, which can involve multiplying the denominators or finding the least common multiple.

Fraction Simplification

Simplifying fractions is the process of making them easier to handle. After adding or subtracting fractions, you often need to reduce the result to its simplest form.
Simplification involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number. In the example of \( \frac{21}{9x} \), you can simplify by checking if the numbers share a common factor.
Here, the numerator, 21, and 9 (consider the \(x\) as part of the fraction) have a common factor of 3.

• Divide both 21 and 9 by their greatest common divisor, 3.
• This simplifies the expression to \( \frac{7}{3x} \).

Not all fractions need simplification, but when they do, it ensures the result is in its most concise and accurate form. It's important to practice this skill to handle a wide range of fraction problems efficiently.