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Inequality + Absolute value

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Source: — Data Sufficiency |
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by GMATGuruNY » Tue Mar 07, 2017 4:19 am
Is |x| = y - z?

(1) x + y = z
(2) x < 0
Statement 1: x = z-y
Case 1: x = z-y = 0, which case y-z = 0.
In this case, |x| = y-z.

Case 2: x = z-y = 1, in which case y-z = -1.
In this case, |x| ≠ y-z.
INSUFFICIENT.

Statement 2: x < 0
No information about y-z.
INSUFFICIENT.

Statements combined:
Case 3: x = z-y = -1, in which case y-z = 1.
In this case, |x| = y-z.

Case 4: x = z-y = -10, in which case y-z = 10.
In this case, |x| = y-z.

Case 5: x = z-y = -1/2, in which case y-z = 1/2.
In this case, |x|= y-z.

Since |x|= y-z in every case, SUFFICIENT.

The correct answer is C.
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by Jay@ManhattanReview » Tue Mar 07, 2017 4:33 am
Mo2men wrote:Is |x| = y - z ?

(1) x + y = z
(2) x < 0


OA: C
Source: Manhattan
Hi Mo2men,

This is a question based on testing the concept of absolute values.

You must know that |x| always yields a positive value.

We have to see whether |x| = y - z.

Let's see each statement one by one.

S1: x + y = z

=> x = z - y = -(y - z)

Thus, the prompt |x| = y - z = -x

Or, |x| = -x

Since LHS (|x|) is always positive, RHS (-x) must also be positive.

If x is negative, -x is positive, the answer is Yes; however, if x is positive, -x is negative, |x| > -x, the answer is No. Not sufficient.

S2: x < 0

Clearly, not sufficient as we do not have any information about y and z.

S1 and S2 together:

Since S2 states that x is negative, we have |x| = -x. Sufficient.

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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by [email protected] » Tue Mar 07, 2017 10:58 am
Hi Mo2men,

This question can be solved by TESTing VALUES. We're asked if |X| = Y - Z. This is a YES/NO question.

1) X + Y = Z

IF....
X=0
Y=0
Z=0
then the answer to the question is YES.

IF....
X=1
Y=1
Z=2
then the answer to the question is NO.
Fact 1 is INSUFFICIENT

2) X < 0

This Fact tells us nothing about Y and Z, so the answer to the question can vary.
Fact 2 is INSUFFICIENT

Combined, we know....
X + Y = Z
X < 0

From here, it helps to think about how the Facts "relate" to the question that is asked. We're asked if |X| = Y - Z.... Fact 1 tells us that Z = X+Y, so if we substitute that information into the question, we're asked....

Is |X| = Y - (X+Y)?
Is |X| = -X?

Since we're also told that X is NEGATIVE, we know that |X| will ALWAYS equal -X, so the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

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