x-2
--------- greater than or equal to 6
2(x+3)
First, clear out the fraction. Mult. both sides of this inequality by 2(x+3):
x-2 greater than or equal to 6(2)(x+3)
x-2 greater than or equal to 12x + 36 becomes
-38 greater than or equal to 11x
Thus, -38/11 is greater than or equal to x, or
x is smaller than or equal to -38/11
check: try x = -44/11 = -4 (this is smaller than -38/11)
Substitute -4 into the original inequality:
x-2
--------- greater than or equal to 6
2(x+3)
becomes
-4-2
--------- greater than or equal to 6
2(-4+3)
or
-6
---- = 3. Is this greater than or equal to 6? No.
-2
So, choose a different test value for x: Try one that's on the other side of
-38/11. I'll try 0.
Then
0-2
--------- greater than or equal to 6
2(0+3)
-2
---- = -1/3. Unfortunately, this is not greater than or equal to 6.
6
Thus, this tentative solution fails.
Trying again:
x-2
--------- greater than or equal to 6
2x + 6
Multiply both sides by 2x+6: x-2 is greater than or equal to 6(2x+6)
Then x-2 is greater than or equal to 12x + 36
Add 2 to both sides: x is gr. than or = to 12x + 38
Subtr. 38 from both sides: -38 + x is gr than or equal to 12x, or
-38 is gr than or = to 11x, and -38/11 is gr than or = to x (same as before).
Again, -4 proves NOT to be a solution. Thus, the interval (-infinity, -38/11) is not part of the solution set.
Trying x = 2,
2-2
------------- comes out to 0, which is NOT greater than or equal to 6.
2(2) + 6
Thus, (-38/11, infinity) is not a solution set either.
NO SOLUTION!