Problem 116 - Intermediate Algebra

Question

Write some comments to the student who wrote the following solution, pointing out where she made an error. Subtract: $$\begin{aligned}\frac{1}{x}-\frac{x+1}{x} &=\frac{1-x+1}{x} \\\&=\frac{2-x}{x} \end{aligned}$$

Step-by-Step Solution

Answer

The student's error was in not distributing the negative sign correctly. The correct answer is \( -1 \).

Step 1: Understand the Problem

The student was asked to subtract two fractions: \( \frac{1}{x} - \frac{x+1}{x} \). Both fractions have the same denominator \( x \).

Step 2: Identify the Correct Approach

Since the fractions have a common denominator, we subtract the numerators directly: \( \frac{1 - (x+1)}{x} \). The student's error was in not properly distributing the negative sign to \( (x + 1) \).

Step 3: Apply the Correct Distribution

We need to distribute the negative sign: \( 1 - (x + 1) = 1 - x - 1 \). Simplifying the numerator gives \( 0 - x = -x \).

Step 4: Write the Correct Fraction

The corrected subtraction is: \( \frac{-x}{x} \). This simplifies to \( -1 \) since the numerator \( -x \) and the denominator \( x \) are similar except for the negative sign.

Key Concepts

Common Denominator

When subtracting fractions, having a common denominator is essential. In simpler terms, the denominator allows you to find a shared base so you can focus on only subtracting the numerators, much like subtracting apples from apples.
In the exercise, both fractions \( \frac{1}{x} \) and \( \frac{x+1}{x} \) already share this common denominator of \( x \). This is crucial because it simplifies the process of subtraction to just dealing with the numbers on top, the numerators.
Here’s a quick summary of why a common denominator matters:

• Ensures you are subtracting parts of the same whole.
• Makes operations on fractions simplified, reducing potential errors.

Next time you encounter a fraction subtraction problem, check for a common denominator first, as this will guide how you subtract the numerators.

Distributive Property

The distributive property is an important concept in algebra that allows you to easily distribute and handle negative signs or numbers across terms inside parentheses.
In the given exercise, the mistake was made in not correctly applying the distributive property to \( 1 - (x+1) \). When you subtract \( (x+1) \), it means you have to distribute the negative sign to both terms inside the parentheses:

• \( 1 - (x + 1) = 1 - x - 1 \).
• This simplifies the expression to \( 0 - x \), or just \( -x \).

Using the distributive property ensures that every term within the parentheses gets its sign correct, which is crucial for accurate subtraction in algebra. Always pay attention to the signs, as a small oversight can lead to an incorrect result.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. After performing arithmetic operations on fractions, the final result is often more easily understood if written in its simplest state.
In this exercise, after the subtraction and distribution, you are left with \( \frac{-x}{x} \).
Since both the numerator and the denominator are the same variable (except for the negative sign), the fraction can be simplified. Here's how:

• When you divide \( -x \) by \( x \), the \( x \) terms cancel out.
• This leaves you with \( -1 \).

Simplifying to \( -1 \) makes sense considering the whole expression was about canceling and reducing. It's a crucial step to always check if your final fraction result can be simplified for clarity and correctness in your math solutions.