Problem 50 - Algebra A Combined Function

Question

Perform the indicated operations. $$ \frac{x+3}{4}-\frac{2 x-1}{4} $$

Step-by-Step Solution

Answer

The result is \( \frac{-x + 4}{4} \).

Step 1: Identify common denominators

First, notice that both fractions have the same denominator, which is 4. This allows us to easily combine the fractions into one.

Step 2: Combine the numerators

Since the denominators are the same, combine the numerators: \[ (x+3) - (2x - 1) \] Remember that you need to distribute the negative sign in front of the second fraction across the numerator inside the parentheses.

Step 3: Simplify the expression inside the numerator

Simplify the expression inside the parentheses: 1. Subtract \( 2x \) from \( x \), which gives \(-x\).2. Subtract \(-1\) from \(+3\), which gives \(+4\).Thus, the numerator becomes \(-x + 4\).

Step 4: Write the final simplified expression

Combine the simplified numerator over the common denominator:\[ \frac{-x + 4}{4} \]

Step 5: Verify the result

Ensure each step was calculated correctly and confirm that no further simplification is possible, as \(-x + 4\) does not have common factors other than 1.

Key Concepts

Common Denominators

To perform operations involving algebraic fractions efficiently, it's crucial to understand the concept of common denominators. A common denominator is a shared multiple of the denominators of the fractions you want to combine or compare. Here, both fractions in the problem \( \frac{x+3}{4} - \frac{2x-1}{4} \) have the same denominator, 4.

Common denominators make it easy to subtract, add, or compare fractions without altering the original values. It’s like having a common language for the fractions to communicate. If the denominators are not the same, you would need to find a least common denominator (LCD) first. However, since we have the same denominator in this exercise, we can focus on working with the numerators immediately.

Combining Fractions

Once you have a common denominator, the next logical step is to combine the fractions by focusing on the numerators. The operation you are performing (addition or subtraction) will dictate how these numerators are combined.

In the original exercise, you subtract:

• Take the first numerator, \(x + 3\), and the second numerator, \(2x - 1\).
• Subtract the second numerator from the first, keeping an eye on distributing the negative sign.

This process turns the operation into a single fraction with one numerator and a common denominator. Think of it as condensing the fraction into a more manageable form before any simplification. Combining fractions is a fundamental skill that lays the groundwork for simplifying and interpreting complex expressions.

Simplifying Expressions

After combining the fractions, it’s time to simplify the resulting expression. Simplifying makes the expression easier to understand and use. The key steps to simplify include:

• Distributing any factors such as negative signs or coefficients across terms within parentheses.
• Combining like terms in the numerator.

In our case, the combined numerator \((x+3) - (2x-1)\) requires us to distribute the negative sign, yielding \(-x + 4\).

Simplifying helps you see the core components of the expression clearly and ensures that there are no unnecessary complexities. It's like cleaning up your workspace to focus on what’s really important.

Subtraction of Fractions

Subtraction of fractions, especially algebraic fractions, can seem daunting at first. However, the concept becomes manageable with a systematic approach. Fractions with like denominators are straightforward to subtract since you mainly focus on subtracting the numerators.

Here's a simple process for subtracting fractions that share a denominator:

• Keep the denominator the same.
• Focus on the numerators: Subtract the second numerator from the first.
• Ensure that any negative signs are distributed correctly across the terms.

The subtraction in the exercise combines all three steps, as both fractions share a common denominator. This means you only need to subtract the numerators, enabling you to focus on simplifying the resulting expression. It's like peeling back the layers to see the substance underneath. This method keeps the operation straightforward and effective.