(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.
Transform the inequality, using equivalent transformations
> 2
- 2 > 0
- > 0
> 0
> 0
> 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.
ANSWER. The solution set to the given inequality is the union of two intervals (-oo,2) U (4,oo).
Solved.