SOLUTION: <pre>Solve the inequality and graph the solution on a real number line 1/x <u>></u> 1/(x+3)</pre>

Solve the inequality and graph the 
solution on a real number line 

1/x > 1/(x+3)

1/x - 1/(x+3) > 0

(x+3) - x
--------- > 0
  x(x+3)

    3
-------- > 0
 x(x+3)


The critical values are x=0 and x=-3

Mark those on a number line:

<----------o--------o----------->
          -3        0

Choose a value left of (less than) -3, say -4
and substitute it into 

    3
-------- 
 x(x+3)

    3          3
---------- = ---- which is positive 
 -4(-4+3)      4

so our number line becomes

 ++++++++++
<----------o--------o----------->
          -3        0

Choose a value between -3 and 0, say -1
and substitute it in 

    3
-------- 
 x(x+3)

    3          3
---------- = ---- which is negative 
 -1(-1+3)     -2

so our number line becomes

 ++++++++++ --------
<----------o--------o----------->
          -3        0

Choose a value right of 0, that is , 
greater than 0, say 1, and substitute
it into 

    3
-------- 
 x(x+3)

    3          3
---------- = ---- which is positive 
 1(1+3)        4

so our number line becomes

 ++++++++++ -------- +++++++++++
<----------o--------o----------->
          -3        0

The expression

    3
-------- 
 x(x+3)

is not defined at the endpoints
0 or -3, so the solution, since

    3
-------- > 0
 x(x+3)

requires the left side to be non-negative,
the solution set in interval notation is

(-oo, -3) U (0, oo)

Edwin
AnlytcPhil@aol.com