Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ \frac{\frac{x-2}{x+1}-\frac{x}{x-2}}{x+3} $$

Short answer

\ \[ \frac{-5x + 4}{(x+1)(x-2)(x+3)} \]

Step-by-step solution

Simplify the Numerator

Simplify the numerator by combining the fractions. The numerator is \ \ \[ \frac{x-2}{x+1} - \frac{x}{x-2} \] \ \To combine these, find a common denominator: \ \ \[ \text{Common denominator} = (x+1)(x-2) \ \ \frac{(x-2)^2}{(x+1)(x-2)} - \frac{x(x+1)}{(x+1)(x-2)} \ \ \frac{(x-2)^2 - x(x+1)}{(x+1)(x-2)} \]

Expand and Simplify the Numerator

Expand and simplify the numerator: \ \ \[ \frac{(x^2 - 4x + 4) - (x^2 + x)}{(x+1)(x-2)} \ \ = \frac{x^2 - 4x + 4 - x^2 - x}{(x+1)(x-2)} \ \ = \frac{-5x + 4}{(x+1)(x-2)} \]

Divide by the Denominator

Divide the simplified numerator by the denominator \ \ \[ \frac{-5x + 4}{(x+1)(x-2)} \bigg/ (x+3) \] \ \This is equivalent to multiplying by the reciprocal: \ \ \[ \frac{-5x + 4}{(x+1)(x-2)} \times \frac{1}{x+3} = \frac{-5x + 4}{(x+1)(x-2)(x+3)} \]

Factor the Simplified Expression

Factor the expression if possible and leave the answer in factored form. However, \ \ \[ \frac{-5x + 4}{(x+1)(x-2)(x+3)} \] cannot be factored any further.

Key concepts

Common Denominators

When working with algebraic fractions, it is important to understand how to find common denominators.
This is necessary to combine fractions through addition or subtraction.
Common denominators represent a shared base, allowing the fractions to be merged into one.
In cases where fractions have different denominators, we need to create a common denominator before performing any arithmetic operations.
This is usually done by multiplying the denominators together.

• Identify the individual denominators of the fractions.
• Find the least common multiple (LCM) of these denominators.
• Adjust the numerators accordingly to ensure the fractions have the same denominator.

For example, in the original exercise's numerator, the fractions are:
\( \frac{x-2}{x+1} \)-\( \frac{x}{x-2} \).
Their common denominator is (x+1)(x-2).
This ensures that both fractions can be combined into one expression.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing them to their simplest form.
The goal is to make the expressions more manageable without changing their value.
Simplification generally includes combining like terms and reducing fractions by canceling out common factors.

• Combine similar terms in the numerator or denominator.
• Factorize both the numerator and denominator if possible.
• Cancel out any common factors shared between the numerator and the denominator.

In our exercise, once we have a common denominator, we expand and simplify the numerator:
\[ \frac{(x^2 - 4x + 4) - (x^2 + x)}{(x+1)(x-2)} \]= \[\frac{x^2 - 4x + 4 - x^2 - x}{(x+1)(x-2)}\]= \[ \frac{-5x + 4}{(x+1)(x-2)} \].
This step-by-step reduction helps in understanding how to carefully simplify without losing the essence of the equation.

Factoring Algebraic Expressions

Factoring algebraic expressions is essential as it allows further simplification and solution of equations.
To factor an expression:

• Look for common factors in each term of the expression.
• Apply factoring techniques like grouping, using the distributive property, or identifying special products (such as difference of squares).
• Break down the expression into product form, making simplifications easier.

In the given exercise, once simplified, we need to factor if possible: \[ \frac{-5x + 4}{(x+1)(x-2)(x+3)} \].
Here, \(-5x + 4\) cannot be factored further, and it remains in its simplest form.
Understanding factoring helps in both simplification and solving more complex algebraic problems efficiently.