Question

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

Short answer

-\frac{2(x+1)}{x+3}

Step-by-step solution

- Combine Fractions in the Numerator

First, combine the two fractions in the numerator \( \frac{x-2}{x+2} + \frac{x-1}{x+1} \). Find a common denominator, which is \( (x+2)(x+1) \). Rewrite each fraction: \[ \frac{(x-2)(x+1)}{(x+2)(x+1)} + \frac{(x-1)(x+2)}{(x+2)(x+1)} \]

- Simplify the Combined Numerator

Expand and combine the terms in the numerator: \[ \frac{(x^2 - x - 2x - 2) + (x^2 - x - 2 + 2)}{(x+2)(x+1)} \] Combine like terms: \[ \frac{x^2 - 3x - 2 + x^2 - x}{(x+2)(x+1)} = \frac{2x^2 - 4x - 2}{(x+2)(x+1)} \]

- Combine Fractions in the Denominator

Combine the two fractions in the denominator \( \frac{x}{x+1} - \frac{2x-3}{x} \). Find a common denominator, which is \( x(x+1) \). Rewrite each fraction: \[ \frac{x \times x}{x(x+1)} - \frac{(2x-3)(x+1)}{x(x+1)} = \frac{x^2 - (2x^2 - x - 3)}{x(x+1)} \]

- Simplify the Combined Denominator

Expand and combine the terms in the denominator: \[ \frac{x^2 - (2x^2 + x - 3)}{x(x+1)} = \frac{x^2 - 2x^2 - x + 3}{x(x+1)} \] Combine like terms: \[ \frac{-x^2 - x + 3}{x(x+1)} \]

- Simplify the Overall Expression

Divide the simplified numerator by the simplified denominator by multiplying by the reciprocal. \[ \frac{\frac{2x^2 - 4x - 2}{(x+2)(x+1)}}{\frac{-x^2 - x + 3}{x(x+1)}} = \frac{2x^2 - 4x - 2}{x+2} \times \frac{x(x+1)}{-x^2 - x + 3} \]

- Simplify the Final Expression

Combine and simplify the final expression to factor the result: \[ \frac{2x^2 - 4x - 2}{x^2 + x - 3} \] This can be factored to \[ \frac{2(x^2 - 2x - 1)}{-(x-1)(x+3)} = \frac{2(x-1)(x+1)}{-(x-1)(x+3)} \] Cancel out common terms: \[ \frac{2(x+1)}{-(x+3)} = -\frac{2(x+1)}{x+3} \]

Key concepts

Factoring

Factoring is an important algebraic process that simplifies expressions by rewriting them as a product of their factors. For example, recognizing that the quadratic expression \(x^2 - 4x + 3\) can be factored into \((x - 1)(x - 3)\) can make solving equations much easier. To factor effectively, look for common factors first. Then, apply techniques such as factoring trinomials, difference of squares, or using the distributive property backward.
Factoring helps in simplifying complex rational expressions, solving polynomial equations, and can make it easier to see solutions or further simplifications in many algebraic contexts.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Examples include expressions like \( \frac{x+2}{x^2-1} \). To work with rational expressions, finding a common denominator is often necessary, just like with regular fractions. You should also simplify these expressions by factoring and canceling common terms.
It's crucial to remember that rational expressions can be undefined for values that make the denominator zero. For instance, \( \frac{x+2}{x^2-1} \) is undefined when \( x = \text{+1 or -1} \). Handling these appropriately involves ensuring meaningful results and avoiding division by zero.

Simplifying Fractions

Simplifying fractions, including those with algebraic terms, involves reducing them to their simplest form. For numerical fractions, this might mean dividing both the numerator and the denominator by their greatest common divisor. For algebraic fractions, factor both the numerator and denominator and cancel any common factors.
Simplifying can significantly reduce the complexity of a problem—making it much easier to understand and solve. For example, simplifying \( \frac{x^2 - 1}{x^2 + 2x + 1} \) involves recognizing that both the numerator and the denominator can be factored and then reduced to \( \frac{(x-1)(x+1)}{(x+1)^2} = \frac{x-1}{x+1} \). Always ensure the final form is as reduced as possible for clarity and simplicity.