Problem 24 - Intermediate Algebra

Question

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{3 x}{2 x+2}+\frac{x+4}{2 x+2}$$

Step-by-Step Solution

Answer

The solution is 2.

Step 1: Identify the Problem

The problem is to add two fractions: \(\frac{3x}{2x+2}\) and \(\frac{x+4}{2x+2}\), which have the same denominator.

Step 2: Combine the Fractions

Since both fractions have the same denominator (\(2x+2\)), we can add their numerators directly:\[\frac{3x}{2x+2} + \frac{x+4}{2x+2} = \frac{3x + (x+4)}{2x+2}\]

Step 3: Simplify the Numerator

Combine like terms in the numerator:\[3x + x + 4 = 4x + 4\]Therefore, the expression becomes:\[\frac{4x + 4}{2x+2}\]

Step 4: Factor and Simplify the Fraction

Factor the numerator and the denominator:\[\frac{4(x + 1)}{2(x + 1)}\]Cancel out the common factor \((x + 1)\):\[\frac{4}{2} = 2\]

Step 5: Conclude with the Simplified Expression

The simplified expression is \(2\). This is the final simplified form after adding the fractions and simplifying.

Key Concepts

Adding Fractions

Adding fractions is a fundamental concept in algebra, especially when dealing with algebraic fractions where variables are involved. When we're tasked with adding two fractions, the primary goal is to create a single fraction. This requires finding a common denominator. In the exercise given, both fractions, \( \frac{3x}{2x+2} \) and \( \frac{x+4}{2x+2} \), already share the same denominator, \( 2x + 2 \). This makes our job easier as we can directly add the numerators while keeping the denominator the same. When working with fractions that do not initially have the same denominator, you need to find the least common denominator (LCD) that both denominators can divide into. Thankfully, when fractions share a denominator, as they do here, it saves us the often laborious task of determining the LCD. Instead, we can jump directly to adding the numerators. Remember:

• The denominator in addition of fractions remains unchanged.
• Focus on finding common denominators when they're different.
• Add the numerators together to combine the fractions.

Understanding this foundational step helps to simplify the entire process of adding algebraic fractions.

Simplifying Expressions

Simplifying expressions is about making them easier to understand or solve. After adding fractions, the next task is to simplify the expression, if possible, to reach a more straightforward form. Following the combination of the numerators in the initial exercise, we have \( \frac{4x + 4}{2x+2} \).Here, the aim is to check if further reduction can be achieved. This can be done by:

• Combining or canceling out like terms.
• Looking for common factors or expressions in the numerator and denominator.

In our example, we combined like terms from the original expression by simplifying \( 3x + x + 4 \) into \( 4x + 4 \). This reduction paves the way to potentially factor out similar terms afterward. Simplifying makes the expression less cumbersome and manageable, preparing it for the next step - factoring.

Factoring

Factoring is a crucial skill in algebra that helps to reduce expressions and solve equations. Once we reach the expression \( \frac{4x + 4}{2x+2} \), we apply factoring to both the numerator and the denominator to simplify further. Factoring is about breaking down a complex expression into simpler multiplicative components. Here's how it works in our example:

• Factor out the greatest common factor from the numerator \( 4x + 4 \), which is 4, to get \( 4(x + 1) \).
• Similarly, factor out the greatest common factor from the denominator \( 2x + 2 \), which is 2, to get \( 2(x + 1) \).

This yields the expression \( \frac{4(x + 1)}{2(x + 1)} \). Notice the factor \( (x + 1) \) appears in both the numerator and denominator. We can cancel these common factors, simplifying the fraction to \( \frac{4}{2} = 2 \). Factoring not only simplifies an expression but can often be crucial in solving larger algebraic problems. Recognizing common factors and utilizing them allows you to reduce expressions efficiently, leading to neat and simplified final answers.