We can see from the graph below that there is no real number solution here.

6x + 8/x -3 has no values in the range -16.856 (the value at x = -2√3/3) to 10.856 (the value at x = 2√3/3)

$$4<6x+\frac{8}{x}-3<5$$     

Start by multiplying everything by x²

This will get rid of the fraction.  And, since x² is positive you don't need to turn any signs around.

Is this the question that you meant or was it actually different.  If you want more help make another post asking for more help.