Problem 117 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Write the answer as an algebraic expression. $$\frac{11}{x}+\frac{9}{x}$$

Step-by-Step Solution

Answer

The result of the operation is \(\frac{20}{x}.\)

Step 1: Identify the common denominator

The denominators of the two fractions, 11/x and 9/x, are the same, i.e., x.

Step 2: Add the fractions

When two fractions have the same denominator, they can be added directly by adding their numerators. Hence, the sum is given as \(\frac{11 + 9}{x}.\)

Step 3: Simplify the numerator

The sum in the numerator 11 + 9 simplifies to 20. We write the sum as \(\frac{20}{x}.\)

Key Concepts

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When you're working with algebraic fractions, having a common denominator simplifies the process of addition or subtraction. In our exercise, both fractions, \(\frac{11}{x}\) and \(\frac{9}{x}\), already have the same denominator, which is \(x\). This means they can be directly combined.

Why is a common denominator so important? Imagine trying to add fractions with different denominators, like \(\frac{1}{4} + \frac{1}{3}\). You would need to convert each fraction to have a common denominator (in this case, 12) before proceeding. Luckily, when fractions share a common denominator, we can bypass this process and move straight to adding the numerators.

Numerator Addition

Once you have fractions with a common denominator, the next step is to handle the numerators. The rule here is simple: just add up the numerators while keeping the denominator the same. For our fractions, \(\frac{11}{x}\) and \(\frac{9}{x}\), this process is straightforward because they already share the denominator \(x\).

Here are the steps to add the numerators:

• Keep the common denominator, \(x\), unchanged.
• Add the numerators: 11 + 9.

By following these steps, we get \(\frac{11 + 9}{x} = \frac{20}{x}\).

Remember, the denominator remains consistent throughout the process, allowing for an uncomplicated addition of the numerators.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing expressions to their simplest form. In terms of algebraic fractions, once you've added the numerators, your next step might be to simplify the resulting expression if possible. This can involve factoring or combining like terms to make the expression as efficient and comprehensible as possible.

In our exercise, after performing numerator addition, we got \(\frac{20}{x}\). The numerator, 20, is already in its simplest form, meaning there are no further reductions needed for the fraction. If our resultant numerator had been a composite number with a common factor with the denominator, then additional simplification would be requisite.

Simplification not only makes your answer easier to understand but also helps in further operations if needed, ensuring calculations are precise and clear.