Problem 43 - College Algebra

Question

Add or subtract as indicated. $$ \frac{3}{x+1}-\frac{3}{x} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( -3 /(x(x+1)) \)

Step 1: Finding the Common Denominator

The first task is to find a common denominator for both fractions. Since denominators are \(x+1\) and \(x\), the least common denominator (LCD) will be the product of the two, which is \(x(x+1)\).

Step 2: Rewrite the Fractions

Next, rewrite each fraction as an equivalent fraction with the common denominator. The first fraction is multiplied by \(x/x\) and the second by \((x+1)/(x+1)\) to obtain equivalent fractions:\[\frac{3}{x+1} \cdot \frac{x}{x} - \frac{3}{x} \cdot \frac{(x+1)}{(x+1)} \]This simplifies to:\[\frac{3x}{x(x+1)} - \frac{3(x+1)}{x(x+1)}\]

Step 3: Simplifying the Expression

Subtract the numerators of these equivalent fractions:\[\frac{3x - 3(x+1)}{x(x+1)} = \frac{3x - 3x - 3}{x(x+1)} \]After further simplifying this equation, get:\[\frac{-3}{x(x+1)}\]

Key Concepts

Least Common Denominator

Rational expressions, similar to regular fractions, need a common denominator to add or subtract them effectively. This denominator is termed the Least Common Denominator (LCD). The LCD is crucial because it allows us to combine fractions by creating a shared base. In the exercise, the two denominators are \(x+1\) and \(x\). The LCD for these expressions is the product of both, which results in \(x(x+1)\). This step ensures each fraction speaks the same 'language', providing a basis for direct subtraction.Finding the LCD involves:

• Listing each denominator's factors
• Identifying the highest power of each factor
• Calculating their product

By ensuring both fractions have this common denominator, you set the stage for straightforward arithmetic operations, just like adding apples to apples rather than apples to oranges.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions (such as variables and constants). They follow the same set of rules applied to numerical fractions, which means operations like addition, subtraction, multiplication, and division require finding a common denominator.The exercise involves the fractions \( \frac{3}{x+1} \) and \( \frac{3}{x} \). These fractions contain variables in the denominators, which makes them algebraic. Handling them requires first finding the Least Common Denominator to combine them easily.When working with algebraic fractions:

• Pay attention to variable expressions in the denominators
• Follow rules like those for numeric fractions, especially for finding LCDs
• Transform each fraction to have a uniform denominator before performing arithmetic operations

Converting each fraction to have the same denominator, as seen in the solution, transforms our expressions into recognizable and simpler equations that are easily manageable.

Simplifying Expressions

Simplifying algebraic expressions is about breaking down an expression to its most concise form, making it easier to understand and work with.In the solution, after ensuring a common denominator, each fraction's numerators are adjusted and combined: \(\frac{3x}{x(x+1)} - \frac{3(x+1)}{x(x+1)}\). The numerators are combined by distributing any constants or variables across like terms. This results in \(3x - 3(x+1)\), which simplifies to \(3x - 3x - 3\). Finally, the solution simplifies further to \(\frac{-3}{x(x+1)}\).Simplifying involves:

• Combining like terms wherever possible
• Factoring out common factors
• Seeking to cancel terms in the numerator against those in the denominator

The goal is always to arrive at the simplest form of the expression for ease in interpretation and further calculations.