That was easy because it was clear that the denominator must e positive to satisfy the inequality but usually we cannot be sure of that. More generally, the simplest way to solve complicated inequalities is to first solve the associated equality- here, . Solving that is also easy: multiplying both sides by 2x- 3, 1= 5(2x- 3). 10x- 15= 1 gives 10x= 16 so x= 16/10= 8/5.
The point of that is that such an inequality can "swap" directions only where the two sides are equal or where there is a "discontinuity". Rational functions are discontinuous only where the denominator is 0:here where 2x-3= 0 or x= 3/2. The two points at which the inquality can reverse are 3/2 and 8/5. They divide the real number line into three intervals. Checking a single point in each interval will tell you which is ">" and which is "<".