Solve the nonlinear inequality that has the variable in the denominator. 2/x+3=

We have to solve
2/x+3=<1/x-3

2/(x+3)=<1/(x-3)

=>
2/(x+3) - 1/(x-3) =< 0

=> [2(x-3) - (x+3)]/(x
- 3)(x + 3) =< 0

=> (2x - 6 - x-3)/(x - 3)(x
+ 3) =< 0

=> (x-9)/(x - 3)(x + 3) =<
0

This is less than or equal to 0 if one of the terms is
less than 0 or all the three terms are less than or equal to
0.

If all the three terms are less than or equal to
0:

• (x-9)=< 0 , (x - 3) =< 0 and (x
  + 3) =< 0

=> x=< 9 ,
x=< 3, x=< -3

x=< -3 satisfies all the
three conditions.

If one of the terms is less than or equal
to 0:

• x=< 9 , x > 3 and x >
  -3

=> 9 >= x >
3

• x=< 3 , x > 9 and x > -3
  , gives no solutions

• x=< -3 ,
  x > 3 and x > 9, gives no
  solutions.

Therefore the values
of x are (-inf , -3] U (3 ; 9]