Problem 13 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The simplest form is 1.

Step 1: Identify the Terms

First, observe the expression: \( \frac{x+1}{x} - \frac{1}{x} \). This expression involves two fractions: \( \frac{x+1}{x} \) and \( \frac{1}{x} \). Both fractions have the same denominator \( x \).

Step 2: Combine the Fractions

Since both fractions have the common denominator \( x \), you can combine them directly by subtracting the numerators: \( \frac{(x+1) - 1}{x} \).

Step 3: Simplify the Numerator

In the numerator \( (x+1) - 1 \), simplify by performing the subtraction: \( x + 1 - 1 = x \).

Step 4: Simplify the Expression

Substitute the simplified numerator back into the fraction: \( \frac{x}{x} \). Since \( \frac{x}{x} = 1 \), the expression simplifies to 1.

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. A fraction is considered to be in its simplest form when the numerator and the denominator can no longer be divided by the same number, except for 1. In this context, when we simplified \( \frac{x+1}{x} - \frac{1}{x} \), we were left with \( \frac{x}{x} \). Here, both the numerator and the denominator are the same, which means they can be divided by \( x \), simplifying the fraction to its simplest form: 1.

To simplify fractions effectively:

• Identify any common factors in the numerator and the denominator.
• Divide both the numerator and denominator by their greatest common factor.
• Double-check to ensure no further simplification can be made.

Simplifying fractions not only makes calculations easier but also helps in finding the clearest expression of the fraction.

Subtraction of Fractions

Subtraction of fractions can sometimes be tricky, especially when different denominators are involved.
However, when fractions have the same denominator, the process becomes straightforward. In this exercise, we subtracted \( \frac{1}{x} \) from \( \frac{x+1}{x} \). Both fractions had \( x \) as their common denominator, allowing us to directly subtract the numerators.

Key steps for subtracting fractions with like denominators include:

• Ensure both fractions have the same denominator.
• Subtract the numerators while keeping the denominator the same.
• Simplify the resulting fraction if possible.

When the denominators are different, additional steps, such as finding the least common denominator, are necessary. But in this case, the subtraction was simplified due to the common denominators, focusing only on the numerators.

Common Denominators

The concept of common denominators is crucial when adding or subtracting fractions. A common denominator is a shared multiple of the denominators involved, which allows fractions to be added or subtracted easily.

In our exercise, both fractions, \( \frac{x+1}{x} \) and \( \frac{1}{x} \), shared the denominator \( x \). This common denominator meant we could subtract the fractions directly without additional steps to adjust the denominators.

Here's how to work with common denominators:

• For fractions with the same denominator, simply add or subtract the numerators.
• If denominators differ, find a common denominator; often, this is the least common multiple (LCM).
• Adjust each fraction to share this common denominator before proceeding with the operation.

Working with common denominators simplifies fraction operations, streamlining the process of adding or subtracting them.