If x is a positive number, what is the value of x ? (1) |x – 2| = 1

Given: \(x>0\). Question: \(x=?\)

(1) \(|x-2|=1\) --> if \(0<x<2\) then \(|x-2|=-(x-2)=1\) and \(x=1\) BUT if \(x\geq{2}\) then \(|x-2|=x-2=1\) and \(x=3\). Two different answers, hence not sufficient.

(2) \(x^2=4x-3\) --> \((x-3)(x-1)=0\) --> \(x=1\) or \(x=3\). Not sufficient.

(1)+(2) \(x\) still can be either 3 or 1. Not sufficient.

Answer: E.
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I am not satisfied with your solution, Why X can not have two values both 3 and 1 . In that case both the statements are individually sufficient .

Not Sure what you mean, We need a value for x. Both statements provide two values for x = {1,3} Hence both are insufficient
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There are two kinds of data sufficient questions: YES/NO DS questions and DS questions which ask to find a value (our case).

In a Yes/No Data Sufficiency questions, statement is sufficient if the answer is “always yes” or “always no” while a statement is insufficient if the answer is "sometimes yes" and "sometimes no".

When a DS question asks about the value of some variable, then the statement is sufficient ONLY if you can get the single numerical value of this variable.

Thus for our question the statements are not sufficient since we have TWO values of x possible 1 and 3.

Hope it's clear.
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Lets evaluate it step by step
Statement A:-
|X-2|=1

From above statement, we will have two values of X, since X is positive we can have X=1 or X=3, both the positive values of X satisfies the statement A. Since we have two values coming out of statement A, Statment A stands insufficient for any certain answer. With this the option D also gets away.

Statement B:-
X^2=4X-3

If we solve this equation i.e. X^2-4X+3=0, then it can also be written as (X-3)(X-1)=0
Again from Statement B, we have two values of X, so Statement B is insufficient

Combining the Two Statments also, we do not get any one value of X. Hence Option C also gets away. Hence the only answer left is E i.e. both the statements are not sufficient.

from st(1), x=1 or 3
from st(2), x= 1 or 3.
but the question was, "what is the value of x?"
but we found two values from both statements. thats why "what is?" is not evaluated at all and its an obvious double case.
so Answer is (E)

St. 1. gives us
x-2 = 1 ==> x = 3
-(x-2) = 1 ==> x = 1
1<x<3
Not Suff

St. 2 gives us
\(x2\)-4x+3 = 0
(x-3)(x-1)=0
Not Suff

Ans:E

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