mrcrescentfresh wrote:
I am having a hard time grasping why we cannot simplify the problem as:
((X-3)^2)^1/2 = (3 - X) to (X - 3)^2 = (3 - X)^2?
I know I have worked problems before where we have been able to solve it by squaring both sides, but when done in this scenario the answer is entirely different than the OA.
Please help.
That is because if the question says:
Is 5 = -5?
And you do not know but you square both sides and get 25 = 25
Can you say then that 5 = -5? No!
I understand that you would have successfully used the technique of squaring both sides before but that would be in conditions like these:
Given equation: \(\sqrt{X} = 3\)
Squaring both sides: X = 9
Here you already know that the equation holds so you can square it. It will still hold. This is like saying:
It is given that 5 = 5.
Squaring both sides, 25 = 25 which is true.
This question is similar to the first case. It is asked whether \(\sqrt{((X-3)^2)} = (3 - X)\)?
LHS is positive because \(\sqrt{((X-3)^2)} = |X-3|\)
and by definition of mod, we know that
|X| = X if X is positive or zero and -X if X is negative (or zero).
Since |X-3| = - (X - 3), we can say that X - 3 <= 0 or that X <= 3
So the question is: Is X <= 3?Stmnt 1 not sufficient.
But stmnt 2 says -X|X| > 0
This means -X|X| is positive.
Since |X| will be positive, X must be negative to get rid of the extra negative sign in front. So this statement tells us that X < 0. Then X must be definitely less than 3. Sufficient.
Per my understanding, the question can be re-phrased as x < 3 only (and not x <= 3)
In this question , it is the latter case [|x| = -x if and only if x < 0 ] that shows up