determine the range of value of x for which x* -2x +5 /3x*-2x-5 is greater than 1/2?(here*means power2)

$$x^2 - 2x + 5$$ > 0 for all x since discriminant < 0 and coefficient of x > 0

$$3x^2 - 2x - 5$$ > 0 for x < -1 or x > 5/3

In this range, the given inequality becomes $$2x^2 - 4x + 10 > 3x^2 - 2x - 5$$ => $$x^2 + 2x - 15 < 0$$ => -5 < x < 3
So, the intersection of these two ranges is -5 < x < -1 or 5/3 < x < 3

$$3x^2 - 2x - 5$$ < 0 for -1 < x < 5/3
In this range, the inequality becomes $$2x^2 - 4x + 10 < 3x^2 - 2x - 5$$ => $$x^2 + 2x - 15 > 0$$ => x < -5 or x > 3
The intersection of the two ranges is a null set.

So, the solution set is the range -5 < x < -1 or 5/3 < x < 3