Is x < 0?

Target question: Is x < 0?

Statement 1: |x| = -x
Let's TEST some values.
There are several values of x that satisfy statement 1. Here are two:
Case a: x = -1. Notice that |-1| = -(-1). In this case, the answer to the target question is YES, x IS less than 0
Case b: x = 0. Notice that |0| = -0. In this case, the answer to the target question is NO, x is NOT less than 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: |x| > x
If |x| > x, then we can be certain that x does not equal 0.
So, let's see what happens if x is POSITIVE, and what happens if x is NEGATIVE

If x is positive, then |x| > x becomes |POSITIVE| > POSITIVE
This doesn't work, because |some POSITIVE number| is always equal to that same number.
For example, |3| = 3 and |12.9| = 12.9
So, if x is POSITIVE, it cannot be the case that |x| > x
In other words, x CANNOT by positive

So, if x does not equal 0 and x CANNOT by positive, then we can be certain that x is negative
In other words, the answer to the target question is YES, x IS less than 0
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

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\[x\,\,\mathop < \limits^? \,\,0\]

\[\left( 1 \right)\,\,\,\left| x \right| = - x\,\,\,\,\, \Leftrightarrow \,\,\,\,x \leqslant 0\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,x = 0\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\,{\text{Take}}\,\,x = - 1\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,\,x < \left| x \right|\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,x < 0\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{SUFF}}{\text{.}}\,\,\,\,\]
\[\left( * \right)\,\,x \geqslant 0\,\,\,\, \Rightarrow \,\,\,x = \left| x \right|\]


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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Fabio Skilnik :: www.GMATH.net (Math for the GMAT)
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=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
|x| = -x
=> x ≤ 0
In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient.
Since the solution set “x<0” of the question doesn’t include the solution set “x ≤ 0” of the condition 1), condition 1) is not sufficient.

Condition 2)
|x|>x
=> x < 0
Since condition 2) is equivalent to the question, condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

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