Problem 61 - Algebra and Trigonometry

Question

Perform the addition or subtraction and simplify. $$ \frac{2}{x}+\frac{3}{x-1}-\frac{4}{x^{2}-x} $$

Step-by-Step Solution

Answer

\( \frac{5x - 6}{x(x-1)} \)

Step 1: Simplify the Denominator

First, take the expression \( \frac{4}{x^2-x} \). Notice that the denominator can be factored as \( x(x-1) \). This gives us the term \( \frac{4}{x(x-1)} \). This factorization will help us combine the fractions later.

Step 2: Identify the Least Common Denominator (LCD)

We want to perform addition and subtraction on the fractions: \( \frac{2}{x} \), \( \frac{3}{x-1} \), and \( \frac{4}{x(x-1)} \). Find the least common denominator (LCD), which is the product of all different factors: \( x(x-1) \). This is because \( x \) and \( x-1 \) are the individual denominators, while \( x(x-1) \) combines all factors.

Step 3: Rewrite Each Fraction with the LCD

Convert each fraction to have the common denominator \( x(x-1) \). - For \( \frac{2}{x} \), multiply the numerator and denominator by \( (x-1) \) to get \( \frac{2(x-1)}{x(x-1)} \).- For \( \frac{3}{x-1} \), multiply the numerator and denominator by \( x \) to get \( \frac{3x}{x(x-1)} \).- \( \frac{4}{x(x-1)} \) is already in the correct form.

Step 4: Combine the Fractions

Now that all fractions have the common denominator \( x(x-1) \), combine them into a single fraction:\[\frac{2(x-1) + 3x - 4}{x(x-1)}\] Simplify the numerator: - Distribute and combine like terms: \[2(x-1) = 2x - 2\]Thus, the numerator is: \[(2x - 2) + 3x - 4 = 5x - 6\]

Step 5: Simplify the Expression

The combined and simplified expression is:\[\frac{5x - 6}{x(x-1)}\]This is the simplified form of the sum and difference of the given fractions.

Key Concepts

Addition and Subtraction of Fractions

When dealing with fractions, whether in simple arithmetic or more complex algebraic expressions, the operations of addition and subtraction follow the same principle: you can't directly add or subtract fractions unless they have the same denominator.
For example, with the expression \( \frac{2}{x} + \frac{3}{x-1} - \frac{4}{x^2-x} \), the denominators \( x \), \( x-1 \), and \( x(x-1) \) are different.
Therefore, the initial task involves finding a common denominator for all three fractions. Once this is achieved, the fractions can easily be added or subtracted by operating on their numerators. Remember, it's crucial to keep the denominator the same throughout the operation.

Factoring Denominators

Factoring is an essential step when working with algebraic fractions because it helps reveal the structure and commonality of the expressions involved.
In the problem \( \frac{4}{x^2-x} \), the denominator \( x^2-x \) can be factored into \( x(x-1) \).
This step simplifies the expression and aligns it with the denominators of other fractions involved.
Factoring makes it easier to identify the least common denominator since it exposes the basic components of the algebraic expressions.

• Identify terms that share common factors.
• Apply the distributive property to break down complex terms.

Understanding and performing factorization benefits not only rational expressions but also many other areas of algebra.

Least Common Denominator

Finding the least common denominator (LCD) is crucial when you want to add or subtract fractions with unlike denominators. The LCD is the smallest expression that includes each factor from the denominators of the fractions involved.
In our case, the denominators are \( x \), \( x-1 \), and \( x(x-1) \), and the least common denominator is \( x(x-1) \).
This choice covers all the factors present in each denominator, allowing you to rewrite each fraction with the common denominator.
Why is this important?

• Ensures that all fractions represent equivalent values using a uniform denominator.
• Makes it possible to combine the fractions by directly adding or subtracting their numerators.

Proper use of the least common denominator facilitates clean and efficient problem-solving.

Simplifying Expressions

Once the fractions are combined, the resulting expression often involves simplification to reach its most basic form.
Simplifying expressions involves a few systematic steps: replacing complex expressions with their simplest equivalent, both in the numerator and denominator.
In the combined expression \( \frac{2(x-1) + 3x - 4}{x(x-1)} \), the process involves:

• Distributing numbers and variables inside parentheses.
• Combining like terms to consolidate and simplify the numerator.

In this scenario, it leads to \((2x - 2) + 3x - 4 = 5x - 6\).
Simplifying expressions ensures that the solution is both accurate and presented in its clearest form. It allows for better interpretation and further operations if needed.