Problem 10 - Algebra 1

Question

Simplify the expression. $$\frac{7}{2 x}+\frac{x+2}{2 x}$$

Step-by-Step Solution

Answer

The simplified form of the expression \( \frac{7}{2x} + \frac{x+2}{2x} \) is \( \frac{9+x}{2x} \).

Step 1: Identify the common denominator

Here both terms in the expression, \( \frac{7}{2x} \) and \( \frac{x+2}{2x} \), share the same denominator, 2x. So, 2x is our common denominator.

Step 2: Add the fractions

Adding the two fractions \( \frac{7}{2x} \) and \( \frac{x+2}{2x} \) with the common denominator will look like this: \( \frac{7}{2x} + \frac{x+2}{2x} = \frac{7 + (x + 2)}{2x} \).

Step 3: Simplify the numerator

The numerator becomes \( 7 + x + 2. \)

Step 4: Solve the numerator

Solving the numerator results in \( 9 + x. \)

Step 5: Writing the final expression

The final expression is \( \frac{9+x}{2x} \).

Key Concepts

Common Denominator

When you are working with fractions, especially if you want to add or compare them, finding a common denominator is essential. Let's break this down to make it super simple. A common denominator is a shared multiple of the denominators across all the fractions involved in the expression.

For example, consider the fractions \( \frac{7}{2x} \) and \( \frac{x+2}{2x} \). Both of these fractions already have the same denominator, which is \( 2x \). This makes the process straightforward because we don’t have to adjust the fractions by finding equivalent ones. We can go directly to the next step of adding because they share this common base or denominator.

Remember:

• The common denominator is usually found by observing or calculating the least common multiple (LCM) of the different denominators.
• In simpler cases, such as this, the given denominators are already identical, simplifying the task immensely.

Adding Fractions

Adding fractions can seem intimidating at first, but by following some fundamental principles, it becomes much easier. The first thing to remember is that fractions can only be directly added when they have a common denominator. Once this is the case, you can simply add their numerators together.

In the exercise given, we started with \( \frac{7}{2x} \) and \( \frac{x+2}{2x} \). Since both fractions share the common denominator \( 2x \), we can combine their numerators. This looks like combining all the parts into one numerator: \( 7 + (x + 2) \).

Let's break it down:

• Write the expression over the common denominator: \( \frac{7 + (x + 2)}{2x} \).
• Simplifying the expression inside the numerator happens next.

Doing this allows them to be rewritten as a single fraction which is crucial for simplifying the expression further.

Simplifying Fractions

Simplifying fractions is all about reducing them to the most straightforward expression, where the numerator and denominator have no common factors other than 1. This often makes your final answer more comprehensible and elegant.

In our given example, after combining the fractions, we get \( \frac{9 + x}{2x} \). This expression is at its most simplified because:

• We can't further reduce or factor out anything common between the numerator \( 9 + x \) and the denominator \( 2x \).
• It’s clean and clear without losing any integral part of the fraction.

Why it’s important:

• A simplified fraction is often more useful for further calculations and interpretations.
• It provides a clear and final answer, showcasing your understanding and completeness of the operation.

Never forget to simplify whenever possible, as it represents the essence of most mathematical processes.