Problem 62 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{x}{x-2}-\frac{x+2}{x+3} $$

Step-by-Step Solution

Answer

\( \frac{3x + 4}{(x-2)(x+3)} \)

Step 1: Identify the LCD

To perform the subtraction, first identify the lowest common denominator (LCD) for the fractions. The denominators are \( x-2 \) and \( x+3 \). Therefore, the LCD is \((x-2)(x+3)\).

Step 2: Rewrite Fractions with Common Denominator

Rewrite each fraction with the common denominator: \[ \frac{x}{x-2} = \frac{x(x+3)}{(x-2)(x+3)} \] and \[ \frac{x+2}{x+3} = \frac{(x+2)(x-2)}{(x-3)(x+3)} \].

Step 3: Perform the Subtraction

Subtract the two fractions: \[ \frac{x(x+3) - (x+2)(x-2)}{(x-2)(x+3)} \]. Expand both numerators: \[ x(x+3) = x^2 + 3x \] and \[ (x+2)(x-2) = x^2 - 4 \].

Step 4: Simplify the Numerator

Subtract the expanded forms: \[ (x^2 + 3x) - (x^2 - 4) = x^2 + 3x - x^2 + 4 = 3x + 4 \].

Step 5: Express the Result

The result of the simplification is: \[ \frac{3x + 4}{(x-2)(x+3)} \]. This is the simplest form since \( 3x + 4 \) cannot be factored further to cancel with the denominator.

Key Concepts

Common Denominator

When working with algebraic fractions, especially during operations like addition or subtraction, identifying a common denominator is essential. The common denominator aligns the fractions by providing a shared base for the fractions' denominators. This shared base facilitates easy computation.

For example, consider the fractions \( \frac{x}{x-2} \) and \( \frac{x+2}{x+3} \). To subtract these, you need a denominator common to both. This is termed the lowest common denominator (LCD).

• The LCD should be a multiple of each denominator.
• Here, the denominators are \( x-2 \) and \( x+3 \), so their product \((x-2)(x+3)\) serves as the LCD.

Using the LCD simplifies the subtraction process, enabling straightforward arithmetic with the numerators. This crucial step prepares the fractions for subsequent computations like subtraction or addition.

Fraction Subtraction

Subtracting fractions involves a few steps that begin after establishing a common denominator for each fraction. With a common denominator, you can easily perform operations on the numerators.

In the given problem, the fractions are aligned onto a common base \((x-2)(x+3)\), allowing us to subtract the numerators directly:

• The original expression is \( \frac{x(x+3)}{(x-2)(x+3)} - \frac{(x+2)(x-2)}{(x-2)(x+3)} \).
• With a common denominator, the focus shifts to the numerators: subtract \( (x+2)(x-2) \) from \( x(x+3) \).

This process strips off the complexity of different bases (denominators) and highlights the interaction purely between the numerators. It simplifies algebraic manipulation and ensures calculations are precise. The final subtraction simplifies to a single expression that combines our initial fractions into one.

Simplifying Expressions

Once you perform the subtraction of the numerators, the next task is simplification. This is critical because simplified expressions are more elegant, easier to interpret, and often reveal insights into potential factors and asymptotes.

In our example, after subtracting \( (x+2)(x-2) \) from \( x(x+3) \), we simplify the result. The expanded numerators become:

• \( x(x+3) = x^2 + 3x \)
• \( (x+2)(x-2) = x^2 - 4 \)

Subtracting these expanded forms yields \( x^2 + 3x - x^2 + 4 = 3x + 4 \).
Finally, check for common factors in the numerator \( 3x + 4 \) and the denominator \((x-2)(x+3)\). Here, there are none to simplify further.

This leaves you with the final simplified expression \( \frac{3x + 4}{(x-2)(x+3)} \). Simplification is a critical final step in ensuring your algebraic expressions are as concise as possible.