Question

Please show your workf

Answer 1

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!