Problem 55 - Precalculus Mathematics for Calculus

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

The simplified expression is \(\frac{5x - 6}{x(x-1)}\).

Step 1: Identify Common Denominator

The first step is to find a common denominator for all three fractions. The denominators are \(x\), \(x-1\), and \(x^2-x\). Rewrite \(x^2-x\) as \(x(x-1)\). Thus, the common denominator is \(x(x-1)\).

Step 2: Rewrite Each Fraction with Common Denominator

Convert each fraction to have the common denominator \(x(x-1)\): - For \(\frac{2}{x}\), multiply both numerator and denominator by \((x-1)\) to get \(\frac{2(x-1)}{x(x-1)}\).- For \(\frac{3}{x-1}\), multiply both numerator and denominator by \(x\) to get \(\frac{3x}{x(x-1)}\).- The fraction \(\frac{4}{x^2-x}\) is already with the denominator \(x(x-1)\).

Step 3: Combine the Fractions

Now that all fractions have the same denominator, combine them into a single fraction:\[ \frac{2(x-1) + 3x - 4}{x(x-1)} \]

Step 4: Simplify the Numerator

Expand and simplify the numerator:- Distribute the \(2\) in \(2(x-1)\) to get \(2x - 2\).- Add like terms in the numerator: \[ 2x - 2 + 3x - 4 = 5x - 6 \]

Step 5: Write the Final Simplified Expression

Now the fraction is:\[ \frac{5x - 6}{x(x-1)} \]This is the simplified form since no common factors appear in the numerator and the denominator.

Key Concepts

Common Denominator

When dealing with fraction addition and subtraction, finding a common denominator is crucial. This is similar to finding a common language that all fractions can understand. Without a common denominator, it's like trying to add apples and oranges - it just doesn't work easily.

• Each original fraction might have a different denominator, making direct addition or subtraction impossible.
• A common denominator is found by identifying the least common multiple (LCM) of the original denominators.
• In this exercise, the denominators were \(x\), \(x-1\), and \(x^2-x\), where \(x^2-x\) was rewritten as \(x(x-1)\).

By rewriting the fractions with a common denominator, it becomes possible to add or subtract them directly. This step lays a solid foundation for combining fractions.

Simplifying Expressions

After working the fractions into a single expression with a common denominator, the next challenge is simplifying the expression. Simplification requires careful handling of algebraic expressions to make them more digestible.

• Begin by expanding the expressions, meaning you'll open up terms like \(2(x-1)\) into \(2x - 2\).
• Combine like terms, which are terms that have the same variables raised to the same power. Here, you combine terms like \(2x\), \(3x\), and \(-4\) to get \(5x - 6\).
• Write this expression over the common denominator.

Simplifying expressions makes them less cluttered and can often reveal more insights into the nature of the problem, preparing them for further algebraic manipulation if necessary.

Algebraic Fractions

Algebraic fractions behave like numerical fractions but involve variables, introducing a layer of complexity. Understanding them is crucial for a firm grasp of many algebraic techniques and methods.

• Algebraic fractions can include any polynomial expressions in their numerators and denominators.
• Operations like addition and subtraction require a special focus on finding a common denominator, unlike straightforward numerical fractions.
• During calculations, always check that any simplification steps maintain the conditions for non-zero denominators to avoid undefined expressions.

Working with algebraic fractions builds a deeper understanding of both individual and combined algebraic expressions, enhancing one's ability to tackle complex math problems with confidence.