Question

fully simplify.\\(\dfrac{\dfrac{1}{2} + \dfrac{1}{x}}{1 - \dfrac{4}{x^2}}\\)

Explanation:

Step1: Simplify numerator and denominator separately

First, simplify the numerator \(\frac{1}{2}+\frac{1}{x}\). Find a common denominator, which is \(2x\). So \(\frac{1}{2}+\frac{1}{x}=\frac{x}{2x}+\frac{2}{2x}=\frac{x + 2}{2x}\).

Next, simplify the denominator \(1-\frac{4}{x^{2}}\). Rewrite \(1\) as \(\frac{x^{2}}{x^{2}}\), then the denominator becomes \(\frac{x^{2}}{x^{2}}-\frac{4}{x^{2}}=\frac{x^{2}-4}{x^{2}}\). Notice that \(x^{2}-4\) is a difference of squares, so \(x^{2}-4=(x + 2)(x - 2)\). So the denominator is \(\frac{(x + 2)(x - 2)}{x^{2}}\).

Step2: Rewrite the complex fraction as a division problem

The original expression \(\frac{\frac{1}{2}+\frac{1}{x}}{1-\frac{4}{x^{2}}}\) is equivalent to \(\frac{x + 2}{2x}\div\frac{(x + 2)(x - 2)}{x^{2}}\).

Step3: Divide the fractions

Dividing by a fraction is the same as multiplying by its reciprocal, so \(\frac{x + 2}{2x}\times\frac{x^{2}}{(x + 2)(x - 2)}\).

Step4: Cancel out common factors

We can cancel out the common factor \((x + 2)\) from the numerator and the denominator. Also, we can cancel out one \(x\) from \(x^{2}\) in the numerator and \(x\) in the denominator \(2x\). So we have \(\frac{x}{2(x - 2)}\).

Answer:

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