Question

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

Short answer

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

Step-by-step solution

Simplify the Numerator

Simplify the expression in the numerator: \(2 - \frac{x+1}{x}\). Find a common denominator to combine the terms. The common denominator is \(x\).\[2 - \frac{x+1}{x} = \frac{2x}{x} - \frac{x+1}{x} = \frac{2x - (x+1)}{x} = \frac{2x - x - 1}{x} = \frac{x - 1}{x}.\]

Simplify the Denominator

Simplify the expression in the denominator: \(3 + \frac{x-1}{x+1}\). Find a common denominator to combine the terms. The common denominator is \(x+1\).\[3 + \frac{x-1}{x+1} = \frac{3(x+1)}{x+1} + \frac{x-1}{x+1} = \frac{3x + 3 + x - 1}{x+1} = \frac{4x + 2}{x+1}.\]

Combine the Simplified Numerator and Denominator

Divide the simplified numerator by the simplified denominator.\[\frac{\frac{x-1}{x}}{\frac{4x+2}{x+1}} = \frac{x-1}{x} \cdot \frac{x+1}{4x+2} = \frac{(x-1)(x+1)}{x(4x+2)}. \]

Simplify Further

Factor the terms in the denominator.\[4x + 2 = 2(2x+1),\]so the expression becomes\[\frac{(x-1)(x+1)}{x \cdot 2(2x + 1)}.\]

Write in Factored Form

Final simplified and factored expression:\[\frac{(x-1)(x+1)}{2x(2x+1)}.\]

Key concepts

Factoring

Factoring is a mathematical method used to break down expressions into simpler components, or factors, that can be multiplied together to get the original expression. Factoring is essential in simplifying rational expressions. For instance, consider the polynomial in the denominator: \(4x + 2\). By factoring out the greatest common factor (GCF), which is 2 in this case, we can rewrite it as \(2(2x + 1)\). This step helps in reducing complex fractions by canceling out common terms.

Common Denominator

A common denominator is necessary when adding or subtracting fractions. It is the least common multiple (LCM) of the denominators involved. For example, in the expression \(2 - \frac{x+1}{x}\), the common denominator is \(x\). Similarly, for \(3 + \frac{x-1}{x+1}\), the common denominator is \(x+1\). By expressing each term with the same denominator, you can combine the fractions. This step converts the expression into simpler forms that enable further simplification.

Numerator and Denominator Simplification

Simplifying the numerators and denominators separately is a critical step before combining fractions. For the numerator, \(2 - \frac{x+1}{x}\) gets simplified to \(\frac{x-1}{x}\). Likewise, the denominator \(3 + \frac{x-1}{x+1}\) becomes \(\frac{4x+2}{x+1}\). This includes performing operations like addition, subtraction, and finding the common factors. By reducing the expression to its simplest form, solving or further manipulating the fraction becomes easier.

Fraction Multiplication

Multiplying fractions involves multiplying the numerators together and the denominators together. For example, when transforming \(\frac{\frac{x-1}{x}}{\frac{4x+2}{x+1}}\), first write it as \(\frac{x-1}{x} \times \frac{x+1}{4x+2}\). Here, we multiply the numerators \((x-1) \times (x+1)\) and the denominators \(x \times (4x+2)\). The final rational expression can then be simplified or factored further. This process ensures that the resultant fraction is in its simplest and most usable form.