Target question: Is |x| = 1?

Statement 1: \(x = \frac{y}{|y|}\)
First recognize that |y| must positive.
With this in mind let's consider two possible cases: y is positive and y is negative.
Case a: If y is negative, then x = \(\frac{y}{|y|}\) = (negative y)/(positive y) = -1, in which case |x| = |-1| = 1. So, the answer to the target question is YES, |x| = 1
Case b: If y is positive, then x = \(\frac{y}{|y|}\) = (positive y)/(positive y) = 1, in which case |x| = |1| = 1. So, the answer to the target question is YES, |x| = 1
Since both possible cases yield the same answer to the target question, statement 1 is SUFFICIENT

Statement 2: |x| = -x
There are infinitely many values of x that satisfy the equation |x| = -x
Here are two:
Case a: x = -1 (notice that |x| = -x becomes |-1| = -(-1), which simplifies to be |-1| = 1 (which is true). In this case, the answer to the target question is YES, |x| = 1
Case b: x = -2 (notice that |x| = -x becomes |-2| = -(-2), which simplifies to be |-2| = 2 (which is true). In this case, the answer to the target question is NO, |x| does not equal 1
Since we can’t answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent
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Question stem, is \(|x| = 1\) ?

St1:- \(x = \frac{y}{|y|}\)
When y>0, \(x = \frac{y}{y}\) (where y is non-negative and non-zero)
Or, x=1; so |x|=1

when y<0, \(x = \frac{y}{-y}\)
Or, x=-1; so |x|=1
Sufficient.

St2:- \(|x| = -x\)
Or, \(|x|+x=0\)
Or, \(x\leq0\)
Insufficient.

Ans. (A)
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A.

(1) If \(y > 0\), then \(x = 1\) [ positive number divided by its absolute value is 1 ]
If \(y < 0\), then \(x = -1\) [ negative number divided by its absolute value is -1 ]
In both cases, value of \(x\) is either 1 or -1, so \(|x| = 1\). SUFFICIENT.

(2) \(|x| = -x\)
This is true for all x less than or equal to 0. NOT SUFFICIENT.

From statement 1) x |y| = y

the only possible way for both sides to be equal if x = 0 or 1 or -1, however y cannot be zero.

so x is 1 or -1, sufficient.

From statement 2) |x| = -x

x can be 0 or -1

Two different answers.

Insufficient.

Answer choice A

For statement 2, x could be negative anything or zero. x = -5, -x=5, |x| = 5

\(|x|\) means \(x\) will take positive value only.

(1) \(x=1, \ if \ y>0, \ x=-1 \ if \ y<0,\) but ultimately \(x=1 \ as \ x=|x|\); Sufficient.

(2) It could be numerous values less than 0. Insufficient.

The answer is \(A\)
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If y≠0, is |x|=1?


(1) x=y/|y|
y = 1 YES
y = -1 YES
y = 2 YES
y = -2 YES

ad infinitum for any real integer.

Sufficient.

(2) |x|=−x
x = -1 YES
x = 0 NO
Insufficient.

Answer is A.
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Hi,

Can someone please explain what does the sign |...| means? (e.g |x|)

Thank you !

|x| denotes the absolute value or modulus of a real number x.

10. Absolute Value

• Theory
  Math: Absolute value (Modulus)
  Absolute Value: Tips and hints
  Properties of Absolute Values on the GMAT
  Absolute modulus : A better understanding
  GMAT Question Patterns: Abolute Value
  The Reason Behind Absolute Value Questions on the GMAT
  
  • Questions
    Inequality and absolute value questions from my collection
    The E-GMAT Question Series on ABSOLUTE VALUE
    DS Questions
    PS Questions

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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BrentGMATPrepNow please help

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