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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is root{(x-3)^2}=3-x?

(1) x≠3 (2) −x|x|>0

When you modify the original condition and the question, it becomes n-th power root (A^n)=|A| when n=even, and |A|=A when A>=0, |A|=-A when A<0. So, |x-3|=3-x=-(x-3)? becomes x-3<0?, x<3?. There is 1 variable(x), which should match with the number of equations. So you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer. For 1), x=/3-> x=2 yes, x=4 no, which is not sufficient. For 2), -x|x|>0 -> x<0<3, which is yes and sufficient. Therefore, the answer is B.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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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.

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.

How did you get this?

No, we are not establishing/using this. We are just simplifying the question. Since the question is:

Is \(|X-3| = - (X - 3)\) ? Is \((X - 3) <= 0\)? Is \(X <= 3\)?

And then we go on to evaluate each statement.
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quite simple , as it is known that root could not have -ve values so we are sure about the LHS

now go through the statements- statement 1. x is not equal to 3 so x can be anything except 3 , try plugging in say x=5 then LHS(2) not equal to RHS(-2) so we get A No

say x=-5 then LHS(8) = RHS(8) so we get A yes . so not sufficient

statement 2. it says that x must be -ve so we get A yes so sufficient

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