Problem 12 - Introductory Algebra for College Students

Question

add or subtract as indicated. Simplify the result, if possible. $$\frac{8}{x+6}+\frac{10}{x+6}$$

Step-by-Step Solution

Answer

The simplified result of the addition is \(\frac{18}{x+6}\).

Step 1: Identify Like Denominators

The first step is to identify that both fractions have the same denominator, which is \(x + 6\). This means we can directly add the numerators.

Step 2: Add Numerators

We add together \(8 + 10\) which gives us \(18\). So, the result is a new fraction formed by this sum divided by the original denominator, which is \(\frac{18}{x+6}\).

Step 3: Simplify the Result

In this case, the fraction \(\frac{18}{x+6}\) is already simplified to its lowest terms, so no further simplification is needed.

Key Concepts

Adding Fractions

Adding fractions is an important concept in mathematics that involves combining two or more fractions into one. The key aspect here is ensuring that the fractions have a common denominator. When the denominators are the same, the process is straightforward: simply add the numerators together.

For instance, in the given exercise, both fractions have \(x + 6\) as their denominator. This makes adding them quite simple: \(\frac{8}{x+6} + \frac{10}{x+6}\). The numerators are added together just like ordinary numbers: \(8 + 10 = 18\).

Remember, the denominator remains unchanged when adding fractions with like denominators. Therefore, the result is \(\frac{18}{x+6}\).

It's crucial to ensure both fractions have the same denominator before adding; otherwise, you will need to make them equivalent by finding a common denominator, which involves more steps.

Like Denominators

Like denominators mean that the denominators of the fractions you are working with are the same. This simplifies operations such as addition or subtraction because you don’t need to alter either fraction to perform these operations.

In our example with \(\frac{8}{x+6}\) and \(\frac{10}{x+6}\), since both denominators are \(x + 6\), it allows us to seamlessly add the fractions.

This concept is particularly useful as it eliminates the steps involved in finding a common denominator, which can often be complex when dealing with different denominators.

• Check if fractions have like denominators first before adding or subtracting;
• If they do, proceed with adding the numerators;
• If they don't, find a common denominator before performing the operation.

Understanding like denominators can save you time and simplify the addition or subtraction of fractions.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form. This means finding an equivalent fraction where the numerator and denominator have no common factors other than 1. However, in some cases, like in our exercise, the fraction might already be in its simplest form after addition.

In the final result \(\frac{18}{x+6}\), unless \(18\) and \(x+6\) share any common factors, this fraction is considered simplified. Considerations for simplifying include:

• Checking if both the numerator and denominator have any common factors;
• Dividing the numerator and the denominator by their greatest common factor (GCF) if they do;
• Expressing the fraction in the simplest form.

Simplifying a fraction makes it easier to understand and work with, especially in further calculations or comparisons.