Problem 9 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{x+1}{x}+\frac{3}{x}$$

Step-by-Step Solution

Answer

\(\frac{x+4}{x}\)

Step 1: Identify and Understand the Problem

We need to add the fractions \( \frac{x+1}{x} \) and \( \frac{3}{x} \). Both fractions have the same denominator, \( x \).

Step 2: Combine the Numerators

Since the denominators are the same, we can directly add the numerators. Combine the numerators: \((x+1) + 3\), keeping the denominator \(x\).

Step 3: Simplify the Expression

Combine the numerators: \(x + 1 + 3 = x + 4\), resulting in the fraction \(\frac{x+4}{x}\).

Step 4: Check for Further Simplification

Since \(x + 4\) and \(x\) do not have any common factors, this is the simplest form.

Key Concepts

Adding Fractions

When you hear 'adding fractions,' it might seem complex, but it's all about the denominators, which indicate how many parts the whole is divided into. To add fractions, follow these simple steps:

• Ensure both fractions have common denominators. This means they are broken into the same type of parts.
• Add the numerators, which represent the number of parts you have.
• Retain the common denominator.

For example, in the problem \(\frac{x+1}{x} + \frac{3}{x}\), both fractions already have the same denominator, \(x\). This means we can focus on combining the numerators \((x+1)\) and \(3\).

Common Denominators

A common denominator is crucial when working with fractions because it ensures that we are comparing or combining like pieces. Break down the steps further:

• First, determine if the denominators are the same. If so, you're in luck—just focus on the numerators.
• If they aren’t, find a common denominator—typically the least common multiple—so that you can combine fractions.

Using a common denominator simplifies the process, making the operation straightforward. In our case, both fractions such as \(\frac{x+1}{x} \) and \( \frac{3}{x} \) already share the denominator \(x\), allowing direct addition.

Simplifying Expressions

After adding or subtracting fractions, the next step is to see if the resulting expression can be simplified. This involves eliminating any unnecessary complexity. Here's how:

• Combine like terms in the numerator, such as numbers or variable terms.
• Check the new fraction for any common factors in the numerator and denominator.
• If there are common factors, divide them out to simplify.

In the solved problem, the expression \((x+1) + 3\) simplifies to \(x+4\). Since \(x+4\) and \(x\) share no common factors, this fraction, \(\frac{x+4}{x}\), is already simplified.

Rational Expressions

Rational expressions can be intimidating, but think of them as just like numbers. They are ratios or fractions where the numerator and/or the denominator is an algebraic expression. Dealing with rational expressions involves:

• Identifying polynomials in both the numerator and the denominator.
• Recognizing operations are carried out in a similar manner as numerical fractions.
• Simplifying the expressions whenever possible.

In the exercise \(\frac{x+1}{x} + \frac{3}{x}\), both the numerator \(x+1\) and the denominator \(x\) qualify it as a rational expression. By treating them like fractions, we used basic addition rules and simplification techniques to solve for \(\frac{x+4}{x}\).