Question

quiz: combining rational expressions
hs: algebra 2b m (sequential) / 4:rational equa

1. simplify

\\(\dfrac{\dfrac{1}{x - 2}}{2 + \dfrac{1}{x}}\\)
options:
\\(\dfrac{x - 2}{x}\\)
\\(\dfrac{x}{2x^2 - 3x - 2}\\)
\\(\dfrac{1}{2x^2 - 3x - 1}\\)
\\(2x^2 - 3x - 1\\)

Explanation:

Step1: Simplify the denominator of the complex fraction

First, simplify the expression in the denominator: \(2 + \frac{1}{x}\). To combine these, find a common denominator, which is \(x\). So, \(2=\frac{2x}{x}\), then \(2 + \frac{1}{x}=\frac{2x}{x}+\frac{1}{x}=\frac{2x + 1}{x}\).

Step2: Rewrite the complex fraction

The original complex fraction is \(\frac{\frac{1}{x - 2}}{2+\frac{1}{x}}\), and now that the denominator is \(\frac{2x + 1}{x}\), we can rewrite the division of fractions as multiplication by the reciprocal. So, \(\frac{\frac{1}{x - 2}}{\frac{2x + 1}{x}}=\frac{1}{x - 2}\times\frac{x}{2x + 1}=\frac{x}{(x - 2)(2x + 1)}\).

Step3: Expand the denominator

Expand \((x - 2)(2x + 1)\) using the distributive property (FOIL method): \(x\times2x+x\times1-2\times2x - 2\times1 = 2x^{2}+x - 4x - 2=2x^{2}-3x - 2\).

Step4: Final simplified form

So, the simplified form of the original complex fraction is \(\frac{x}{2x^{2}-3x - 2}\).

Answer:

\(\frac{x}{2x^{2}-3x - 2}\) (corresponding to the option with this expression)