is SQRT[(x-3)^2] = 3-x

[peter456]

Is SQRT[(x-3)^2] = 3-x
[/u]1st step
I rephrased as: is |x-3|=x-3
Solution a: x-3=3-x => x=3
Solution b: x-3 = -(3-x)... 0=0...not valid
So, only solution a is valid i.e is x=3?
Implying A is sufficient

2nd step
However, testing #s shows that all values less than 2 are valid.
In this case, B is sufficient.

Someone pls help identify my error in the 1st step as OA is not A.

Thanks.

[Brent@GMATPrepNow]

Here's the full question and one possible solution.

Is √[(x - 3)²] = 3 - x ?

1. x is not equal to 3
2. -x|x| > 0

Target question: Is √[(x - 3)²] = 3 - x ?

This question is a great candidate for rephrasing the target question.
Aside: We have a free video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

To begin, notice that we have two nice rules:
- If k > 0, then √(k²) = k
- If k < 0, then √(k²) = -k

Now observe that (3-x) = -(x-3)

Given the above information, under what conditions will √[(x-3)²] = 3-x?
In other words, under what conditions will √[(x-3)²] = -(x-3)?
This will occur IF x-3 is NEGATIVE.

So, we can now rephrase the target question as: Is (x-3) negative?
Or we can write: Is x-3 < 0?

. . . or better yet: Is x < 3?

Now that we've REPHRASED the target question in much simpler terms, we can check the statements.

Statement 1: x not equal to 3
This does not give us a definitive answer to the REPHRASED target question (Is x < 3? )
As such, statement 1 is NOT SUFFICIENT

Statement 2: -x|x| > 0
First notice that this implies that x does not equal zero.
Next, notice that, if x does not equal zero, then |x| will always be positive.
So, -x|x| > 0 is the same as saying (-x)(positive) > 0
In other words, the product (-x)(positive) results in a positive number.
This tells us that (-x) must be positive, which means x must be negative.
If x is negative, then x is definitely less than 3.
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent

[ceilidh.erickson]

Here's a general note about GMAT coded language:

Any time you see √n², it's a coded way of saying |n|. Because for any non-zero number n the square is positive, and the square root sign √ will always yield a positive result, then √n² will always be positive, even if n itself is negative.

It's a good idea to translate √n² into |n| whenever you see it.

[GMATGuruNY]

Is √(x-3)² = 3-x?

1) x is not equal to 3
2) -x*|x| > 0

Be definition:
√(x²) = |x|.
|x-y| is the DISTANCE between x and y.

Question rephrased: Is |x-3| = 3-x?
In other words:
Is the DISTANCE between x and 3 equal to the DIFFERENCE between 3 and x?

A DIFFERENCE can be negative, 0, or positive.
A DISTANCE must be greater than or equal to 0.
For the DIFFERENCE between two values to be equal to the DISTANCE between the two values, the DIFFERENCE -- like the DISTANCE -- must be greater than or equal to 0:
3-x≥0
x≤3.

Question rephrased: Is x≤3?

Statement 1: x is not equal to 3.
It is possible that x<3 or that x>3.
INSUFFICIENT.

Statement 2: -x*|x| > 0 .
Thus, the left-hand side must be (+)(+) or (-)(-).
Since |x| cannot be negative, both factors on the left-hand side must be positive.
Thus:
-x>0
x<0.
Since x<0, we know that x≤3.
SUFFICIENT.

The correct answer is B.