Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0

Hi Bunuel and VeritasKarishma

A follow up to the discussion on roots. If all positive roots give only one answer that is a positive answer then why does \(\sqrt{x^2}\) (positive root) become |x| giving two answers (x could be +ve if x>=0 or -ve if x<0)? Should this also not give only one answer that is +ve?

Maybe i am confused as I have been reading too much on this topic but I am really lost in this regard...

Also, the article shared by ccooley (have attached the snapshot of it) for A) the simplification would be =>|x|/x => |x| = x if x>= 0 OR |x| = -x if x<0

Therefore it would give us two answers -1 and 1. But the article says it will only give us one answer ie -1

\(\sqrt{x^2}=|x|\).

If x > 0, then \(\sqrt{x^2}=|x|=x=positive\).
If x < 0, then \(\sqrt{x^2}=|x|=-x=-negative=positive\).

As you can see in any case the square root gives positive result.

Ok, thanks a lot Bunuel I understood now!! one last thing, is my simplification for option A in the attached image correct? will x take two values in A ie 1 and -1?

If x < 0, then \(\frac{\sqrt{x^2}}{x}=\frac{|x|}{x}=\frac{-x}{x}=-1\). You can plug any negative number to check it.

Hi VeritasKarishma - can you confirm per the yellow highlight in the above post,

can the question stem be re-phrased as

X <= 3 or x < 3 ?

Per my understanding, the question can be re-phrased as x < 3 only (and not x <= 3)

Reason
|x| = x , if and only if x >=0
and |x| = -x if and only if x < 0

In this question , it is the latter case [|x| = -x if and only if x < 0 ] that shows up

Hence x < 3 should be the re-phrased question and not x <= 3

|x| = x , if and only if x >=0
and |x| = -x if and only if x < 0
is correct

and so is

|x| = x , if and only if x >0
and |x| = -x if and only if x <= 0

Since |0| = +0 = -0 = 0

You have to take the x = 0 case in one definition, either in |x| = x or in |x| = -x. We usually take the first case but the second is correct too.
This understanding becomes pertinent in a case like this.

When we know that |x| = -x, we know that x is either 0 or negative. Note that x = 0 satisfies here so you cannot ignore it.

[(x-3)2]1/2 can be x-3 or 3-x..
The first data x not=3 doesn't help as the square root of (x-3)^2 may be x-3 or 3-x...as it depends on the value of x, whether it is smaller than x or greater
The second data says -x|x| is positive. This is possible only if x is negative... because, then -x is positive and |x| is positive and naturally the product is positive. So 3-x will be positive...the square root of the expression had to be positive...and x-3 is negative..so B is sufficient.
Hence B

Posted from my mobile device

We can re-write the question as: Is \(|x-3| = 3-x\)?

This equation only holds true if \(x ≤ 3\). Think about it. If \(x = 4\), then \(|4-3|\) is not equal to \(3-4\).

Therefore the question we have to answer is: Is \(x ≤ 3\)?

(1) Clearly not sufficient. X could be 4 or x could be 2. INSUFFICIENT.

(2) This statement tells us -x is positive. We can conclude x is negative. SUFFICIENT.

Answer is B.

A much easier way to approach this non-value DS question is to simply plug in integers and to seek both a Yes and No answer.

Also, even though you can simplify aka "translate" the stem, I don't recommend that you do so, as the original equation works fine, and you run the risk of human error. In the case of this question, it's actually condition #2 that benefits the most from a "translation."

1) Plug in x = 2 to get YES, and x = 4 to get NO. Not sufficient. Cross off A and D.

2) This condition is essentially telling us that x must be negative. Plug in any negative number (-1, -2, -3, etc.) and you will see that it works, and that the answer is always YES. Sufficient.

Hence, the correct answer is Choice B.

Solution:

We can rephrase or re represent the Question stem here as Is |x−3|=3−x?

This is because sqrt(x^2) =x if x>=0 or -x if x<0

St(1):- x≠3
We have no information on whether it is less than or greater than 3.(Insufficient)

St(2):-−x |x|>0

=> x<0,
=> x<3. Sufficient (option b)

Devmitra Sen
GMAT SME

But why is 0 counted as negative? I have not reflected on this.

|x-3| = |x-3| if x = 3.
|x-3| = |3-x| if x = 3.

If x = 3, then the expressions are both the same. Why do we include |0| as a -|x|?

chetan2u Bunuel IanStewart

Initial Approach: Square both sides

\(\sqrt{(x-3)^2} = 3-x\)?

Square both sides

(x-3)\(^{2}\) = (3-x)\(^{2}\)?
x^2 + 9 - 6x = 9 + x^2 - 6x?
0 = 0?
LHS = RHS?

I got stuck on this approach. I know the correct solution now, but why did this simple operation of squaring both side give me such erroneous results?

Is it a rule that you shouldn't square a square root on GMAT or it might screw up your analysis?

The fundamental issue here is that the "√" square root symbol does not mean exactly the same thing as the phrase "square root". The number 4, for example, has two square roots, 2 and -2, because if you square either of those numbers, you get 4. But the square root symbol "√" means only the positive square root (or, technically, the non-negative square root if you might be taking the root of zero). Since we must get something positive (or zero) when we apply the "√" to something, we sometimes have to account for that before doing anything else. So if you have this equation:

a = √1

there's obviously no need to do anything to solve, because √1 = 1, so here a = 1 is obviously the only solution. But if instead we decided to solve by squaring both sides, we'd get:

a^2 = 1

and from this new equation, you might think a = 1 and a = -1 are both valid solutions. And they would be if the symbol "√" meant "either the positive and the negative square root", but that's not what it means. It is only equal to one of those two roots. So when we see "a = √(something)", we first need to recognize that a must be at least zero before doing anything else, and then after we square and solve, we need to discard any solutions that are negative. To give a non-trivial example, if you see, say, z^3 = √(z^2), then you might indeed want to square both sides, but before you do that, you have to recognize that z^3, and therefore z, must be 0 or greater, or else you might solve and think that z can equal -1, and it can't. It can only equal 0 or 1 in this case.

Applying that to the GMAT question in this thread, we have this equation:

3 - x = √(x - 3)^2

and since 3 - x is equal to "√(something)", we first need to recognize that 3 - x must be at least zero. So 3 - x > 0, and x < 3. Now you can go ahead and square both sides and solve as you did, and when you arrive at "0 = 0", that just means the equation is always true, for any valid values. Since we know in advance x < 3 must be true, we then can correctly conclude that the equation is true for every value of x that is 3 or less.

That's not how I'd solve this question -- instead we can notice that √(x - 3)^2 = |x - 3| = |3 - x|. So the question is asking: is 3 - x = |3 - x|? If that equation is true, that means the absolute value isn't doing anything (it's not changing "3 - x" into something else; it's leaving it completely unchanged), so this equation is only going to be true if 3 - x is positive or zero, or in other words if x < 3. From that point on, analyzing the statements is straightforward.

VeritasKarishma Bunuel

How is B sufficient ? It does not cover the case when x = 0,1,2. All these values satisfy the stem equation.

What am I missing ?

Namangupta1997

You are not looking for the exact range of values that will satisfy the inequality given in the question stem. Simplify the question:

Stmnt 2: x is negative

Stmnt 2 tells us that x is negative.

Question stem: Can we say whether x is 3 or greater?
If using statement 2 alone, we can answer the question with a "Yes" or a "No", it will make statement 2 sufficient. If our answer is "May be, may be not", then statement 2 is not sufficient.

Stmnt 2 tells us that x is negative. So we know that for sure it is less than 3. So we can answer the question asked with a "NO, x is not 3 or greater. x is certainly less than 3"
So stmnt 2 is sufficient.

Check these two posts:
https://anaprep.com/algebra-game-must-b ... questions/
https://anaprep.com/algebra-must-be-tru ... questions/

Got it. Thanks VeritasKarishma. It really clears things up. And thanks for the blog links. They are always very insightful. Much appreciated.

Can someone post similar questions please? Love this!