Question 263390
please help me solve this equation :

{{{ x/2}}} > {{{5/(x+4)}}} + {{{4}}}
~~~~~~~~~~~~~


            The "solutions" from both @CubeyThePenguin and @josgaritmetic are WRONG.


            Below I brought the correct solution.



<pre>
Rewrite it equivalently in this form


    {{{x/2}}} - {{{5/(x+4)}}} - 4 > 0

    {{{x}}} - {{{10/(x+4)}}} - 8 > 0

    {{{x*((x+4)/(x+4))}}} - {{{10/(x+4)}}} - {{{8*((x+4)/(x+4))}}} > 0    <<<--- now we have common denominator (x+4)

    {{{(x*(x+4) - 10 - 8*(x+4))/(x+4)}}} > 0

    {{{(x^2 + 4x - 10 - 8x - 32)/(x+4)}}} > 0

    {{{(x^2 - 4x - 42)/(x+4)}}} > 0


The quadratic polynomial in the numerator has the roots


    {{{x[1,2]}}} = {{{(4 +- sqrt(4^2 + 4*42))/2}}} = {{{(4 +- sqrt(184))/2}}} = {{{2 +- sqrt(46)}}}.


So, the numerator has the roots  {{{x[1]}}} = {{{2 - sqrt(46)}}} = -4.782  and  {{{x[2]}}} = {{{2 + sqrt(46)}}} = 8.782.


There are three critical points on the number line,  

    {{{x[1]}}} = {{{2 - sqrt(46)}}} = -4.782,  -4  and  {{{x[2]}}} = {{{2 + sqrt(46)}}} = 8.782.


They divide the number line in 4 intervals, from left to right

    ({{{-oo}}},{{{x[1]}}}),  ({{{x[1]}}},{{{-4}}}),  ({{{-4}}},{{{x[2]}}})  and  {{{x[2]}}},{{{oo}}}).


The solution set for the given inequality is the union of the second and the fourth intervals.
</pre>

Solved.