It helps to know the following:Is √(x-5)² = 5-x?
(1) -x|x| > 0
(2) 5 - x > 0
√ means the POSITIVE ROOT ONLY.
Thus, √(x²) = |x|.
|x-y| = the DISTANCE between x and y.
In the problem at hand:
√(x-5)² = |x-5| = the DISTANCE between x and 5.
A distance must be greater than or equal to 0.
5-x = the DIFFERENCE between 5 and x.
A difference can be negative, 0, or positive.
The DIFFERENCE between two values will be equal to the DISTANCE between the two values whenever the DIFFERENCE is greater than or equal to 0.
Thus, |x-5| = 5-x whenever 5-x≥0.
Question rephrased: Is x≤5?
Statement 1: -x|x| > 0
Since |x| cannot be negative, both factors (-x and |x|) must be positive.
Thus:
-x > 0
x<0.
Since x<0, we know that x≤5.
SUFFICIENT.
Statement 2: 5-x > 0
Thus, x<5.
SUFFICIENT.
The correct answer is D.
