Is \(\sqrt{(x-3)^2}=3-x ?\) GMAT Data Sufficiency

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

X ≠ 3
-x|x|>0

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: (B)

Approach Solution - 1 :

The following are some important information regarding a square term's square root

The answer is only the positive root if the number is under the square root symbol, √[(-3)^2] = √9 = 3.
If there are variables under the square root symbol, you must take into account the possibility that the squared "thing" was initially either positive or negative, √[(-x)^2] = √[x^2] = |x|

Although the sign of x determines whether +x or -x is actually greater than zero, we are still searching for the positive root.

So the question asks whether √[(x-3)^2] = 3-x.
Let us rewrite that as |x-3| = 3-x.

When the component of the absolute value sign is either positive or zero, this becomes,

Is x-3 = 3-x?
Is 2x = 6?
Is x = 3?

When the component of the absolute value sign is negative, this becomes,

Is -(x-3) = 3-x?
Is -x+3 = 3-x?

Yes, for all x, but keep in mind that this was only "all x" because x-3 was negative, or x 3.
So, let us ask ourselves, Is x > 3?

For the final rephrase, combine the two cases: "Is x =<3 ? "

Statement - 1 : X ≠ 3

There is no information on whether x is less than or greater than 3, but it is not 3.
Therefore this statement is not sufficient

Statement - 2 : -x|x|>0

|x| is positive, then -x must also be positive since positive*positive > 0 but negative*positive < 0. X is thus negative. X must be less than 3 if it is negative. The answer is yes as a result.
Therefore this statement is sufficient.

Approach Solution - 2 :

Remember that x^2 = |x|.

This is because The key here is that the square root function cannot produce a negative result, meaning that,
√(some expression) ≥ 0.

So √(x^2) ≥ 0. But what will be x^2 ?

Let us consider two examples.
If x = 5 => √(x^2) = √ 25 = 5 = x = positive.
If x=−5 => √(x^2) = √ 25 = 5 = -x = positive.

So now we got that,
√(x^2) = x, if x ≥ 0;
√(x^2) =−x, if x < 0.

The absolute value function does exactly the same thing. That is why √(x^2) = |x|

So, now let us rewrite the original question as, “ Is |x−3| = 3−x? “
When x>3, the right hand side (RHS) is negative but the absolute value LHS is never negative, so the equations are invalid in this situation.
If x ≤ 3, then LHS = |x−3| = −x+3 = 3−x = RHS, hence in this case the equation becomes true.

Basically the question asks whether x ≤ 3.

Statement - 1 : X ≠ 3

This statement is clear not sufficient

Statement - 2 : -x|x| > 0

It is clear that this inequality implies that x < 0.
From this we can know for sure that x < 3.
Therefore this statement is sufficient.

Approach Solution - 3 :

It is given that |x-3| = 3-x if x < 3.

Statement - 1 : X ≠ 3

This implies that x can also be greater than 3.
Therefore this statement is not sufficient

Statement - 2 : -x|x|>0

This implies that x is less than 0. That is x < 3, for all values of x.
Therefore this statement is sufficient.

“Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0” - is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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Is \(\sqrt{(x-3)^2}=3-x ?\) GMAT Data Sufficiency

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