Problem 39 - Precalculus Mathematics for Calculus

Question

Perform the addition or subtraction and simplify. $$2+\frac{x}{x+3}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{3x + 6}{x+3}\).

Step 1: Identify the Terms

In the expression \(2 + \frac{x}{x+3}\), the two terms involved are the integer 2, and the fraction \(\frac{x}{x+3}\). We need to combine them into one coherent expression.

Step 2: Find a Common Denominator

To add these terms, we must have a common denominator. The integer 2 can be rewritten as \(\frac{2(x+3)}{x+3}\) to have the same denominator as \(\frac{x}{x+3}\).

Step 3: Rewrite the Expression

Now we rewrite 2 as \(\frac{2(x+3)}{x+3}\). Thus, the expression becomes \(\frac{2(x+3)}{x+3} + \frac{x}{x+3}\).

Step 4: Combine the Fractions

Since both fractions now share the same denominator, we can combine them: \[\frac{2(x+3) + x}{x+3}\]

Step 5: Simplify the Numerator

Simplify the numerator: First, distribute 2 in \(2(x+3)\), resulting in \(2x + 6\).Combine the terms: \[2x + 6 + x = 3x + 6\]

Step 6: Write Final Simplified Fraction

Now, write the simplified expression using the combined numerator: \[\frac{3x + 6}{x+3}\], which is the simplified form of the original expression.

Key Concepts

Addition of Fractions

Adding fractions can be tricky when they don't initially share the same denominator. To perform addition, the fractions need to "speak the same language," or in other words, have the same denominator. When adding fractions like \( \frac{x}{x+3} \) to an integer such as 2, it's crucial to convert the integer into a fraction with a denominator that matches the given fraction.

Here's how it works:

• Start by expressing the integer as a fraction with the new common denominator.
• In the exercise, 2 is converted into \( \frac{2(x+3)}{x+3} \) so that it can be added to \( \frac{x}{x+3} \).
• This aligns both terms under the same denominator, making addition straightforward.

Once prepared in this way, fractions can be easily added by simply combining their numerators and keeping the common denominator intact.

Common Denominator

Understanding the common denominator is crucial for adding fractions. Without it, the fractions speak different languages, like converting between completely different systems. In math, you must ensure the denominators are the same. This enables you to add the numerators directly.

In our case, when adding \( \frac{x}{x+3} \) to the number 2, the step involves:

• Transforming 2 into \( \frac{2(x+3)}{x+3} \) so it shares the same base or denominator \( x+3 \) with \( \frac{x}{x+3} \).
• Now both terms are set to combine since they share \( x+3 \) in their denominators.

Common denominators are fundamental in making expressions ready for further calculations such as addition or subtraction, ensuring the process is smooth and manageable.

Simplifying Expressions

Simplifying expressions involves making them as straightforward and compact as possible. After the fractions are combined, simplifying focuses on the numerator primarily. Let's break this down step by step:

• After combining the fractions \( \frac{2(x+3) + x}{x+3} \), tackle the numerator \( 2(x+3) + x \).
• Distribute the 2 across \( (x+3) \) to expand it into \( 2x + 6 \).
• Combine like terms: here, \( 2x + x \) results in \( 3x \), and the expression turns into \( 3x + 6 \).

Finally, the simplified form of the whole expression is \( \frac{3x+6}{x+3} \). The goal is to make the expression simpler, easier to work with, and ready for further analysis or evaluation. Continuous practice in simplifying ensures confidence and accuracy in handling more complex algebraic expressions.