Question 199270
{{{n/(n-4/9)-n/(n+4/9)=1/n}}} Start with the given equation.




{{{n/((1/9)(9n-4))-n/((1/9)(9n+4))=1/n}}} Factor out {{{1/9}}} (to make the inside terms have whole numbers)




{{{(9n)/(9n-4)-(9n)/(9n+4)=1/n}}} Multiply both the numerator and denominator by the reciprocal of {{{1/9}}}




{{{n*cross((9n-4))(9n+4)((9n)/cross((9n-4)))-n(9n-4)cross((9n+4))((9n)/cross((9n+4)))=cross(n)(9n-4)(9n+4)((1)/cross(n))}}} Multiply <font size="4"><b>every</b></font> term on both sides by the LCD {{{n(9n-4)(9n+4)}}}. Doing this will eliminate all of the fractions.



{{{n(9n+4)(9n)-n(9n-4)(9n)=(9n-4)(9n+4)}}} Simplify



{{{9n*n(9n+4)-9n*n(9n-4)=(9n-4)(9n+4)}}} Rearrange the terms.



{{{9n^2(9n+4)-9n^2(9n-4)=(9n-4)(9n+4)}}} Multiply



{{{81n^3+36n^2-81n^3+36n^2=(9n-4)(9n+4)}}} Distribute



{{{72n^2=(9n-4)(9n+4)}}} Combine like terms.



{{{72n^2=81n^2-16}}} FOIL



{{{72n^2-81n^2=-16}}} Subtract {{{81n^2}}} from both sides.



{{{-9n^2=-16}}} Combine like terms.



{{{n^2=-16/(-9)}}} Divide both sides by -9.



{{{n^2=16/9}}} Reduce



{{{n=""+-sqrt(16/9)}}} Take the square root of both sides.



{{{n=sqrt(16/9)}}} or {{{n=-sqrt(16/9)}}} Break up the "plus/minus"



{{{n=4/3}}} or {{{n=-4/3}}} Take the square root of {{{16/9}}} to get {{{4/3}}}




So the solutions are {{{n=4/3}}} or {{{n=-4/3}}}