If x ≠0, Then What Is The Value Of (|x|)/x? GMAT Data Sufficiency

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Question - If x ≠0, then what is the value of (|x|)/x?
(1) \(\sqrt{(x^2)=x}\)
(2) |x-4|=\(\frac{x}{3}\)

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.

‘If x ≠0, then what is the value of (|x|)/x?’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". 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 sufficiencycomprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

let us rephrase the question,
If x > 0 , (|x|)/x = 1
If x < 0 , (|x|)/x = -1
Thus, we must determine if x is positive or negative.
Statement 1 says \(\sqrt{(x^2)=x}\)
The key to properly analysing this claim is understanding that the literal translation of the symbol sqrt is "the positive square root of."
Statement 1 therefore informs us that x is positive.
Hence Statement 1 is sufficient
Now lets look at statement 2
Statement 2 says |x-4|=\(\frac{x}{3}\)

Left Hand Side is always a postive side
Therefore, the Right Hand Side must also be positive.
Consequently, x is positive.
Hence Statement 2 is Sufficient.
Therefore, statement 1 and statement 2 are sufficient
Answer is D, which is statement 1 and statement 2 are sufficient.

Correct Answer: D

Approach Solution 2:

There is another approach to this question which is fairly simple
If x > 0 , (|x|)/x = 1 since you would have a pospos
If x < 0 , (|x|)/x = -1 since you would have a posneg
Thus, we must determine if x is positive or negative.
Statement 1 says \(\sqrt{(x^2)=x}\)
The key to properly analysing this claim is understanding that the literal translation of the symbol sqrt is "the positive square root of."
Statement 1 therefore informs us that x is positive.
Hence Statement 1 is sufficient
Now lets look at statement 2
Statement 2 says|x-4|=\(\frac{x}{3}\)
When we remove absolute value we get 2 possibilities,
x-4=-\(\frac{x}{3}\)
And
x-4=\(\frac{x}{3}\)
Let us solve both
x-4=-\(\frac{x}{3}\)
3x - 12 = -x
4x = 12
x = 3
Positive
x-4=\(\frac{x}{3}\)
3x -12 = x
2x = 12
x = 6
Positive
Given that both possibilities give us the same answer to the question (a positive one), statement 2 is sufficient
Therefore, statement 1 and statement 2 are sufficient
Answer is D, which is statement 1 and statement 2 are sufficient.

Correct Answer: D

Approach Solution 3:

CASE (i) (x2)−−−−√=x(x2)=x
|x|=x|x|=x
Note:(x2)−−−−√=|x|
Note:(x2)=|x|

(|x|)x=1(|x|)x=1Sufficient

Alternate way
just to be 100% sure, we can try some numbers (use x= 2, x = -2)
use x = 2 --- > (2)2−−−−√=(2)(2)2=(2)
use x = -2 --- >(−2)2−−−−−√≠(−2)(−2)2≠(−2)
implies x is a positive number,
hence
(|x|)x=1(|x|)x=1 Sufficient

CASE (ii) |x−4|=x3|x−4|=x3

Solving both side (x−4)=x3−−>x=6(x−4)=x3−−>x=6 & (x−4)=−x3−−>x=3(x−4)=−x3−−>x=3
both cases
(|x|)x=1(|x|)x=1 Sufficient

Correct Answer: D

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If x ≠0, Then What Is The Value Of (|x|)/x? GMAT Data Sufficiency

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