(x-3) / (x+2) >= 0
There are two points to consider:
x - 3 = 0 >> x = 3
x + 2 = 0 >> x = -2
Number line:
<-----(-2)-----(+3)----->
Let's try something to the left of -2:
(x-3)/(x+2) when x = -3
(-3-3)/(-3+2)
(-6)/(-1)
+1
1 is >= 0, so that section is in.
Let's try something between -2 and +3:
(x-3)/(x+2) when x = 0
(-3)/(+2)
-3/2
-3/2 is not >=0, so that section is out.
Let's try something to the right of +3:
(x-3)/(x+2) when x = 4
(4-3)/(4+2)
1/6
1/6 is >= 0, so that section is in.
Answer: x <= -2 or x >= +3