Problem 65 - Algebra A Combined Function

Question

For Exercises 65 and \(66,\) an algebra student approaches you with each incorrect solution. Find the error and correct the work shown below. $$ \begin{array}{l} \frac{2 x-6}{x-5}-\frac{x+4}{x-5} \\ =\frac{2 x-6-x+4}{x-5} \\ =\frac{x-2}{x-5} \end{array} $$

Step-by-Step Solution

Answer

The correct answer is \( \frac{x-10}{x-5} \).

Step 1: Analyze the Original Problem

The problem given is the subtraction of two fractions: \( \frac{2x-6}{x-5} - \frac{x+4}{x-5} \). Both fractions have a common denominator \(x-5\), so the numerators can be combined for subtraction.

Step 2: Identify the Incorrect Step

The student combined the numerators as follows: \( \frac{2x-6-x+4}{x-5} \). The student subtracted incorrectly here as they did not consider distributing the negative sign into the second fraction's numerator.

Step 3: Correct the Subtraction

The numerators should be properly subtracted by distributing the negative sign: \( 2x-6 - (x+4) = 2x - 6 - x - 4 \). This simplifies to \( x - 10 \).

Step 4: Write the Correct Expression

Combine the new numerator with the common denominator: \( \frac{x-10}{x-5} \).

Step 5: Final Answer

The correct result of the subtraction is \( \frac{x-10}{x-5} \).

Key Concepts

Algebraic Expressions

Algebraic expressions form the building blocks of algebra. They consist of variables, constants, and operational symbols such as addition, subtraction, multiplication, and division. In this exercise, we dealt with expressions like \(2x-6\) and \(x+4\). These are parts of the fractions that we needed to subtract from one another.
Expressions can be as simple as a single number or letter, or far more complex with multiple terms. Here are some key points to consider:

• Variables represent unknowns and can be added, subtracted, multiplied, or divided.
• Constants are fixed values, like the numbers 6 and 4 in our example.
• Operational symbols tell us what to do with the constants and variables.

Understanding the parts of an algebraic expression is crucial for performing operations like addition and subtraction correctly.

Distributing Negative Signs

When subtracting fractions or any algebraic expressions, distributing a negative sign is pivotal. It ensures that all terms you're subtracting behave correctly. In our problem, this wasn't initially done correctly, leading to mistakes.
Let's explore what this means:

• The negative sign affects all terms inside the parentheses following it.
• This change is because subtracting an expression is the same as adding the opposite of that expression.
• For example, subtracting \((x+4)\) means we must change it to \(-x - 4\).

By redistributing the negative sign in the numerator, the expression became \(2x - 6 - x - 4\) instead of \(2x - 6 - x + 4\). This correctly leads us towards a simplified result, demonstrating why paying attention to these small details matters so much.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. After adjusting our numerators correctly and ensuring our negative signs were properly distributed, simplifying made more sense. Here’s how:

• First, combine like terms to make the expression more straightforward.
• The like terms were \(2x - x\) and \(-6 - 4\), which simplifies to \(x - 10\).
• The simplified fraction then becomes \(\frac{x-10}{x-5}\).

Unlike arithmetic fractions, algebraic fractions require not only simplification but also the consideration of undefined points. In this case, it’s essential to remember that the original denominator \(x-5\) cannot be zero. Therefore, including the restriction \(x eq 5\) ensures that our algebraic logic holds true in every scenario. Simplifying is not just about cleaner expressions; it’s about maintaining mathematical integrity.