Expert: Scott A Wilson - 12/1/2009

Question
Good Day -

Having trouble with the following:
(1/2)x -2 less than or equal to 2/3x -6

In solving for x, I went as far as 1/3x is less than or equal to 2/3x -4.  How to move the 2/3x to left side of 'less than/equal to' Sign? Do I multiply both sides by reciprocal 3/2 or subtract 2/3x?

Please clarify.

Thank you in advance  

Answer
The equation is x/2 - 2 <= 2x/3 - 6, right?
Add 6 to both sides and get x/2 + 4 <= 2x/3.
Subtract x/2 from both sides and get 4 <= (4x-3x)/6.
Now (4x-3x)/6 = x/6, so that is really 4 <= x/6.
Multiply both sides by 6 and get 24 <= x.

To check, put x=24 back in the original equation.
This gives 12 - 2 vs. 16 - 6, and we get 10 vs. 10, so at x=24, both sides are equal.
If 24 <= x, try x=30.  We get 15 - 2 vs. 20 - 6, and that is 13 vs. 14, and 13 < 14.  Check.