Question 707026
We do not know if (2x-3) is positive or negative. 


Find critical points or point, treating first the equality.


{{{(x + 1)/(2x -3) > 2}}}
Change to "=", and solve to find critical point:
{{{((x+1)/(2x-3) )*(2x-3)=2(2x-3)}}}
{{{x+1=2(2x-3)}}}
{{{x+1=4x-6}}}
.
x=7/3, critical point.


2 is less than the critical point, and 3 is greater than the critical point.  They are just examples.  The values are not too important; just they are on specific intervals of the real line based on the found critical  point.


Test using 2, for x<7/3
{{{(2+1)/(2*2-3)=3}}}, and {{{3>2}}}, so 
{{{x<7/3}}} is solution.


Test using 3, for x>7/3
{{{(3+1)/(2*3-3)=4/3>2}}}, FALSE.  This means x>7/3 are NOT solutions.


The solution is {{{x<7/3}}}