If |x 2| x, which of the following must be true?

Question

If |x−2|>x , which of the following must be true? A. x<−1 B. x<1 C. −11 E. There is no value of x which meets the condition. This is a PS Butler Question Check the links to other Butler Projects: Data Sufficiency Butler Critical Reasoning Butler Sentence Correction Butler Reading Comprehension Butler Integrated Reasoning Butler

learnwithrohan · Answer

We can square both sides to remove the modulus, 
so

(x-2)^2 > x^2 
x^2 - 4x + 4 > x^2
-4x +4 > 0
4 > 4x
1 >x

hence answer B

sanchitb23 · Answer

Although the answer is Correct IMO, but is the process right? If we square both sides, aren't we losing the nature of x? what if x was negative, value of x^2 will always be positive hence possibly changing the inequality? Correct me if I am wrong

According to me,
We can take it case by case

Case 1 : x−2>x
x-x>2 (Not Possible)

Case 2: 2-x>x
=2>2x
=1>x

Therefore Correct Answer is B

If you are opening the Modulus in this manner, shouldn't the sign of inequality be reversed?

szcz · Answer

You are absolutely correct on both the questions. The inequalities can be squared only when both the sides of the inequality are non negative. This usually is not the method to go with. Better to make cases as you mentioned. Also on opening the modulus, one case will involve reversing the inequality sign.

rxb266 · Answer

Opening the modulus of given equality:-

Case 1-

x-2>x
x-x>2 , the value of x can not be determined.

Case 2-

2-x > x
2 > 2x
1 > x

Hence, option b

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If |x 2| > x, which of the following must be true?

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