Problem 10 - Algebra 2

Question

Simplify each sum. \(\frac{1}{2 x}+\frac{1}{2 x}\)

Step-by-Step Solution

Answer

The simplified form of \(\frac{1}{2x}+\frac{1}{2x}\) is \(\frac{1}{x}\)

Step 1: Identify the problem

We first identify that it's asking to simplify the sum of two fractions, which in this case, both fractions have the same denominator. The fractions in question are \(\frac{1}{2x}\) and \(\frac{1}{2x}\)

Step 2: Sum the fractions

Since the denominators of the two fractions are the same, we can sum them by keeping the denominator and just adding up the numerators. This leaves us with \(\frac{1+1}{2x}\)

Step 3: Simplify the sum

Now since 1+1 equals to 2, substitute 1+1 in the numerator spot with 2. This gives us \(\frac{2}{2x}\)

Step 4: Final simplification

Then, we simplify the fraction by dividing the numerator and the denominator by 2. The result of that will be \(\frac{1}{x}\)

Key Concepts

Fraction Addition

When adding fractions, you are essentially combining parts. Think of fractions as pieces of a pie. If you have a fraction like \( \frac{1}{2x} + \frac{1}{2x} \), this is akin to having two identical slices of pie. To add fractions, the denominators (the bottom numbers) must be the same. When they are, as in our exercise, you simply add the numerators (the top numbers) together.

• Step 1: Identify that you have the same denominators.
• Step 2: Add the numerators together.
• Step 3: Keep the denominator the same.

In our example, both fractions have a denominator of \(2x\), and the numerators sum like this: \(1 + 1 = 2\). Thus, you get \( \frac{2}{2x} \) after adding the fractions. By keeping the denominator the same, it ensures that you are adding the fractions correctly.

Common Denominator

A common denominator is essential for fraction addition. It’s like finding a common ground or a shared baseline. Imagine negotiating a deal; you need to agree on the terms first. With fractions, having the same denominator is agreeing on those terms.
The major reason common denominators are so important is because they allow fractions to be added directly. Without a common denominator, you'd first need to convert each fraction so they share one. Thankfully, in our example \( \frac{1}{2x} + \frac{1}{2x} \), the denominators are already common, simplifying the process. It’s simply:

• Keep the denominator the same
• Add the numerators

If fractions have different denominators, you might need to find the Least Common Denominator (LCD) and convert each fraction to have this common baseline before you can add them. But with \( \frac{1}{2x} \) and \( \frac{1}{2x} \), it's straightforward since they already share the term \(2x\).

Algebraic Expressions

Understanding algebraic expressions is vital when simplifying fractions that include variables, not just numbers. These types of expressions combine numbers, variables (like \(x\)), and operations. They're building blocks in algebra, similar to how words form sentences.
In the fraction \( \frac{1}{2x} \), the variable \(x\) adds complexity. But fear not! Working with algebraic fractions follows the same rules as numerical fractions. The operations don’t change.

• You still add, subtract, multiply, and divide as usual.
• Keep an eye on simplifying whenever possible.

In our earlier exercise, simplification happened at two levels: adding the numerators \(1 + 1 = 2\), and then simplifying \( \frac{2}{2x} \) by reducing it to \( \frac{1}{x} \) (since dividing both top and bottom by 2 simplifies the fraction). Remember that simplifying helps make expressions easier to work with and understand. It’s like organizing a messy room – placing everything in its simplest, most understandable order.