jainrahul1985 wrote:Is sqrt(x-5)^2 = 5 - x
(1) -x|x| > 0
(2) 5 - x > 0

OA D

Try a couple of plug ins for the question stem to figure out what's going on:

If x is greater than 5 (for example x=6), then sqrt(x-5)^2 = sqrt(6-5)^2 = sqrt (1)^2 = 1, while 5 - x = 5-6=-1.
This will be true for any x greater than 5(e.g. x=7, 8, or 10,000): the left side will be positive (the ^2 makes sure of that), while the right side will be negative. The answer to the question stem is "no".

If x is smaller than 5 (for example x=4), then sqrt(x-5)^2 = sqrt(4-5)^2 = sqrt (-1)^2 = sqrt (1) = 1, while 5 - x = 5-4=1.
This will be true for any x smaller than 5 (e.g. x=3, 2, or even -1): both the left and right side will be positive and equal to each other. The answer to the question stem is "Yes".

So the question stem is really asking "is x smaller than 5?"

Once you realize that, the statements are easy:

Stat. (1) -x|x|>0

|x| is positive, so you can divide by |x| without changing the direction of the sign:

-x>0 --> x<0.
If x is smaller than zero, it is definitely smaller than 5. The answer is "yes", and the stat. is sufficient.

Stat. (2): 5-x>0
5>x
The answer is "yes", and the stat. is sufficient.

Answer is D.