Problem 73 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{4}{x+1}+\frac{3}{x+1} $$

Step-by-Step Solution

Answer

\( \frac{7}{x+1} \)

Step 1: Identify the Common Denominator

Notice that both fractions have the same denominator \(x+1\). This means we can combine the fractions by keeping the common denominator.

Step 2: Combine the Numerators

Since the denominators are the same, you can add the numerators together: \(4 + 3 = 7\).

Step 3: Write the Combined Fraction

Combine the numerators over the common denominator: \( \frac{7}{x+1} \).

Key Concepts

Common Denominator

When working with fractions, particularly algebraic ones where variables are involved, finding a common denominator is essential. A common denominator is a shared divisor among multiple fractions that allows you to combine or compare them effectively. In simpler terms, it acts as a bridge to bring separate fractions together under a single roof.

In our exercise, both fractions share the same denominator, \(x+1\). This is fortunate because it means we can proceed to the next step without any additional manipulation. When fractions already have a common denominator, you don't need to factor or change anything. This can save a lot of time and potential confusion.

• If the denominators are not the same, you would need to find a common denominator first before combining the fractions.
• The common denominator is the least common multiple of the original denominators when they differ.
• Constructing a common denominator could involve multiplying each fraction by appropriate factors to make the denominators identical.

Once the common denominator is established, it paves the way for combining numerators and simplifies the process significantly.

Combining Numerators

Once you have established a common denominator, the next logical step is to combine the numerators of the fractions. This process is straightforward if you approach it systematically, since the denominators are already the same.

To combine the numerators, you only need to add or subtract them, depending on the operation between the fractions. In this case, our exercise mentions addition. You take the numerators from each fraction, and add them together. The calculation is \(4 + 3 = 7\). This result represents the combined numerator for the new fraction.

• Addition: Simply add the numerators together, as shown in the exercise.
• Subtraction: Instead of adding, simply subtract the second numerator from the first.
• The operation affects only the numerators when the denominators are already common.

Clarity in combining numerators allows you to simplify fractions more directly, yielding neat and tidy solutions without unnecessary complexity.

Algebraic Expressions

Algebraic expressions often include variables, numbers, and operations combined in a particular form. When simplifying fractions involving algebraic expressions, you'll predominantly encounter variables in your work, such as \(x+1\) in the given problem.

Algebraic expressions can take many forms and are a central part of manipulating and simplifying fractions in algebra. Here, the denominator \(x+1\) is treated as a single unit. This combined term simplifies the process since you can deal with it as one entity rather than separate parts.

• Understand each part: Recognize expressions as whole units, especially when they serve as common denominators.
• Simplify where possible: In some cases, algebraic expressions can be simplified further if they share terms or have common factors.
• Stay aware of variable positions: Variables can move or simplify your expressions significantly, depending on their location and function in the overall formula.

Handling algebraic expressions with care ensures the validity of your fraction simplifications, reinforcing your understanding of algebra.