If\({\frac{x}{|x|}}\)< x Which of the Following Must be True About x? GMAT Problem Solving

Question: If \(\frac{x}{|x|}\)< x which of the following must be true about x?

x > 1
x > −1
|x| < 1
\(|x|^2\)= 1
\(|x|^2\) > 1

Correct Answer: B

Solution and Explanation
Approach Solution 1:

To find the value of x let us follow the steps-
We can multiply both sides by abs(x) without changing the sign because we know the absolute value is positive.
This will give us-
x This implies that x must be greater than one or between -1 and 0. It can be determined through intuition or by testing a number. 0 and -1 are ineffective because they make both sides equal.
Since We're looking for something that has to be true, so if we find a scenario for x that works outside the given parameters, we can rule it out right away.

does not have to be true because x could be true because -1/2 is valid for x.
has to be true because there is no value for x that works and is less than -1.
is not required to be true because -1/2 works.
does not have to be true because x=1 does not exist.
does not have to be true because -1/2 is sufficient.

Therefore the answer is x > −1
The answer is B which is x > −1

Approach Solution 2:

We can also use another approach to solve this question lets see that below
The primary idea of this question is to comprehend what is being asked.
Which of the following statements about x must be true?
It means that the question is asking about a Set of Values, which includes all of the x values that satisfy the given inequality.
However, the Big Idea would be that the set may also contain other values that do not fulfil the inequality.
It simply means that the set must contain all of the values of x that satisfy the inequality, but not vice versa.
Only after resolving the above-mentioned inequality,
-1 or
x>1 when x is positive
As it can be seen option B which is x>-1 is the only set of values that contains all of the above-mentioned x values.
The answer is B which is x > −1

Approach Solution 3:
First of all let's solve this inequality step by step and see what is the solution for it, or in other words let's see in which ranges this inequality holds true.

Two cases for x/|x|
A. x<0 --> |x|= −x --> x/−x −1 < x −1 < x < 0;

B. x>0 --> |x|= x--> x/x < x --> 1< x.

So given inequality holds true in the ranges: −11. Which means that x can take values only from these ranges.

------{-1}xxxx{0}----{1}xxxxxx

Now, we are asked which of the following must be true about x. Option A can not be ALWAYS true because x can be from the range −1
Only option which is ALWAYS true is B. ANY x from the ranges −11 will definitely be more the −1, all "red", possible x-es are to the right of -1, which means that all possible x-es are more than -1.

“If \(\frac{x}{|x|}\)< x which of the following must be true about x?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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