Answer
$\dfrac{\dfrac{2}{x}+\dfrac{x}{2}}{\dfrac{2}{x}-\dfrac{x}{2}}=\dfrac{x^{2}+4}{(4-x)(4+x)}$
Work Step by Step
$\dfrac{\dfrac{2}{x}+\dfrac{x}{2}}{\dfrac{2}{x}-\dfrac{x}{2}}$
Evaluate the sum indicated in the numerator and the substraction indicated in the denominator:
$\dfrac{\dfrac{2}{x}+\dfrac{x}{2}}{\dfrac{2}{x}-\dfrac{x}{2}}=\dfrac{\dfrac{4+x^{2}}{2x}}{\dfrac{4-x^{2}}{2x}}=...$
Evaluate the division and simplify:
$...=\dfrac{4+x^{2}}{2x}\div\dfrac{4-x^{2}}{2x}=\dfrac{2x(4+x^{2})}{2x(4-x^{2})}=\dfrac{4+x^{2}}{4-x^{2}}=...$
Factor the denominator to provide a more simplified answer:
$...=\dfrac{x^{2}+4}{(4-x)(4+x)}$
