Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{x-3}{x+2}-\frac{x+4}{x-2} $$

Short answer

The simplified form is \ \frac{-(11x + 2)}{(x+2)(x-2)} \.

Step-by-step solution

- Find a Common Denominator

In order to subtract these fractions, determine the common denominator for the fractions. The denominators are \(x+2\) and \(x-2\). The common denominator will be the product of these two expressions: \( (x+2)(x-2) \).

- Rewrite Each Fraction with the Common Denominator

Rewrite each fraction with the common denominator \( (x+2)(x-2) \). Multiply the numerator and the denominator of the first fraction by \( (x-2) \) and the numerator and the denominator of the second fraction by \( (x+2) \): \[ \frac{x-3}{x+2} \cdot \frac{x-2}{x-2} - \frac{x+4}{x-2} \cdot \frac{x+2}{x+2} \]This results in: \[ \frac{(x-3)(x-2)}{(x+2)(x-2)} - \frac{(x+4)(x+2)}{(x+2)(x-2)} \]

- Expand the Numerators

Expand the expressions in the numerators: \[ \frac{x^2 - 2x - 3x + 6}{(x+2)(x-2)} - \frac{x^2 + 2x + 4x + 8}{(x+2)(x-2)} \]This simplifies to: \[ \frac{x^2 - 5x + 6}{(x+2)(x-2)} - \frac{x^2 + 6x + 8}{(x+2)(x-2)} \]

- Subtract the Numerators

Combine the fractions by subtracting the numerators: \[ \frac{(x^2 - 5x + 6) - (x^2 + 6x + 8)}{(x+2)(x-2)} \]

- Simplify the Numerator

Simplify the resulting expression in the numerator: \[ x^2 - 5x + 6 - x^2 - 6x - 8 \] Combine like terms: \[ -11x - 2 \]This simplifies to: \[ \frac{-11x - 2}{(x+2)(x-2)} \]

- Factor the Numerator (if possible)

Notice that \ -11x - 2 \ does not factor further. Thus, the expression is already in its simplest form. The final answer is: \[ \frac{-(11x + 2)}{(x+2)(x-2)} \]

Key concepts

Common Denominator

When working with algebraic fractions, especially if you need to add or subtract them, it's crucial to have a common denominator. The common denominator is a shared multiple of the individual denominators of each fraction. In the given exercise, the denominators are \(x+2\) and \(x-2\), and their common denominator is the product \( (x+2)(x-2) \). This allows us to rewrite both fractions with the same denominator, which makes the subtraction process possible.

Numerator and Denominator

These are the two main parts of a fraction. The numerator is the top part, showing how many parts of the whole are being considered. The denominator is the bottom part, indicating how many equal parts the whole is divided into. In our exercise, the numerators are \(x-3\) and \(x+4\), while the denominators are \(x+2\) and \(x-2\). To perform operations, we must manipulate both numerators and denominators accordingly.

Simplifying Expressions

Simplifying expressions helps make our work more manageable and understandable. It involves reducing expressions to their simplest form. In the exercise, simplification comes after we rewrite fractions with a common denominator. This involves expanding and then combining the numerators. For example, we expand \( (x-3)(x-2) \) to \( x^2 - 5x + 6 \), and \( (x+4)(x+2) \) to \( x^2 + 6x + 8 \). After expanding, the expressions are combined and like terms are simplified to get \( -11x - 2 \).

Factoring

Factoring identifies parts that can be multiplied together to get the original expression. It's a crucial step in simplifying fractions. In the provided exercise, we needed to see if the numerator \( -11x - 2 \) could be factored further after simplifying. Since it couldn't, the fraction was left in its simplified form: \( \frac{-(11x + 2)}{(x+2)(x-2)} \). Factoring helps us break down complex problems and, in this case, confirmed we had the simplest form.