Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ \frac{\frac{x+4}{x-2}-\frac{x-3}{x+1}}{x+1} $$

Short answer

\(\frac{2(5x - 1)}{(x-2)(x+1)^2} \)

Step-by-step solution

Simplify the Numerator Expressions

First, we need to subtract the two fractions in the numerator. The common denominator of the fractions \(\frac{x+4}{x-2}\) and \(\frac{x-3}{x+1}\) is \((x-2)(x+1)\). Rewrite each fraction with this common denominator. \ \ \ \(\frac{x+4}{x-2} = \frac{(x+4)(x+1)}{(x-2)(x+1)}\) \ \ \(\frac{x-3}{x+1} = \frac{(x-3)(x-2)}{(x-2)(x+1)}\)

Subtract the Fractions in the Numerator

Combine the fractions under a common denominator: \ \(\frac{(x+4)(x+1) - (x-3)(x-2)}{(x-2)(x+1)}\) \ Now expand and simplify the numerator:\ \ \(\frac{(x^2 + x + 4x + 4) - (x^2 -2x - 3x + 6)}{(x-2)(x+1)}\) \(\frac{x^2 + 5x + 4 - x^2 + 5x - 6}{(x-2)(x+1)}\) \ Combine like terms: \(\frac{10x - 2}{(x-2)(x+1)}\)

Factor the Simplified Numerator

The simplified numerator \(10x - 2\) can be factored out as: \(2(5x - 1)\). So the expression becomes: \(\frac{2(5x - 1)}{(x-2)(x+1)}\)

Divide by the Denominator

Now, divide the result by \(x + 1\): \ \(\frac{2(5x - 1)}{(x-2)(x+1)} \times \frac{1}{x+1}\) This simplifies to \(\frac{2(5x - 1)}{(x-2)(x+1) \times (x+1)}\)

Final Simplification

Simplify further by canceling common factor: \ \(\frac{2(5x - 1)}{(x-2)(x+1)^2}\)

Key concepts

Simplification

Simplification in algebra involves reducing expressions to their simplest form. In this problem, we start by focusing on simplifying the numerator which contains fractions. When we simplify, our goal is to make the expression more manageable.

Here, we subtract the two fractions in the numerator \(\frac{x+4}{x-2} - \frac{x-3}{x+1}\). To do this, we need a common denominator. The common denominator for \(x-2\) and \(x+1\) is their product, \((x-2)(x+1)\). With this common denominator in hand, we can rewrite each fraction so that they can be subtracted.

Steps:

• Rewrite fractions with a common denominator.
• Combine like terms.
• Simplify the resulting expression.

Simplification is key as it helps in reducing expression complications and making further operations easier.

Common Denominator

Finding a common denominator is an essential technique when working with algebraic fractions. The common denominator allows fractions with different denominators to be combined by converting them to equivalent fractions with the same denominator.

In this exercise, the two fractions in the numerator are \(\frac{x+4}{x-2}\) and \(\frac{x-3}{x+1}\). The least common denominator (LCD) is \((x-2)(x+1)\).

Here's how you can handle it:

• Identify the denominators of each fraction.
• Find the product of these denominators to determine the LCD.
• Rewrite each fraction with the LCD as the new denominator while adjusting the numerators accordingly.

Using the LCD allows us to straightforwardly subtract the two fractions by combing the numerators over the common denominator. This process simplifies the algebraic manipulation.

Factoring

Factoring is the process of breaking down an expression into products of simpler expressions. In many algebraic problems, factoring makes it easier to simplify or solve equations.

In this exercise, we start with the simplified numerator \(10x - 2\). Factoring involves rewriting this expression by finding common factors.

Steps to factor \(10x - 2\):

• Identify common factors in each term. Here, both terms \(10x\) and \-2\ share a common factor of \2\.
• Factor out this common term: \2(5x - 1)\.
• Incorporate this factored form into the larger mathematical expression.

Factoring is particularly helpful as it often reveals common terms that can be canceled out, further simplifying the algebraic expression. As seen in the final step, noticing that \2(5x - 1)\ can be refactored allows for easier division and ultimately a more straightforward solution.