Question 693272
(3x - 10) / (x - 4) > 2
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            The solution in the post by  @CubeyThePenguin is  ABSOLUTELY  WRONG.


            I came to bring the correct solution.



<pre>
Transform the inequality, using equivalent transformations

    {{{(3x-10)/(x-4)}}} > 2

    {{{(3x-10)/(x-4)}}} - 2 > 0

    {{{(3x-10)/(x-4)}}} - {{{2*((x-4)/(x-4))}}} > 0

    {{{((3x-10) - 2*(x-4))/(x-4)}}} > 0

    {{{(3x - 10 - 2x + 8)/(x-4)}}} > 0

    {{{(x -2)/(x-4)}}} > 0    (*)


Inequality (*) is equivalent to the given inequality.


There are two critical points x= 2 and x= 4, that divide the number line in three non-intersecting intervals

          (-oo,2), (2,4) and (4,oo).


In the first interval, both the numerator and denominator of (*) are negative;  so the inequality (*) is valid; 
so the interval (-oo,2)  is the part of the solution set.


In the second interval, the numerator of (*) is positive, while the denominator of (*) is negative;  
so the inequality (*) is not valid; thus the interval (2,4)  is NOT the part of the solution set.


In the third interval, both the numerator and denominator of (*) are positive;  so the inequality (*) is valid; 
so the interval (4,oo)  is the part of the solution set.


<U>ANSWER</U>.  The solution set to the given inequality is the union of two intervals  (-oo,2) U (4,oo).
</pre>

Solved.