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Is |x|/x equal to -1?

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by Max@Math Revolution » Wed Nov 14, 2018 12:20 am

Timer

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Answers

A

B

C

D

E

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Difficulty—

[Math Revolution GMAT math practice question]

Is |x|/x equal to -1?

1) x>0
2) x<1
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Source: — Data Sufficiency |
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by fskilnik@GMATH » Wed Nov 14, 2018 3:38 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

Is |x|/x equal to -1?

1) x>0
2) x<1
Important: the stem implicitly assumes x is nonzero. (Reason: when x=0, |x|/x is not defined and the question itself would not make sense!)

\[\frac{{\left| x \right|}}{x}\,\, = \,\,\left\{ \begin{gathered}
\,\,\frac{x}{x} = 1\,\,\,{\text{if}}\,\,x > 0 \hfill \\
\,\frac{{ - x}}{x} = - 1\,\,\,{\text{if}}\,\,x < 0\, \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\boxed{\,\,x\,\,\mathop < \limits^? \,\,0\,\,}\]
\[\left( 1 \right)\,\,x > 0\,\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \]
\[\left( 2 \right)\,\,\,x < 1\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,x = 0.5\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,x = - 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\]

The correct answer is therefore (A).


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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Jay@ManhattanReview » Thu Nov 15, 2018 10:32 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

Is |x|/x equal to -1?

1) x > 0
2) x < 1
Looking at the fraction |x|/x, we see that the numerator |x| is a positive number, thus for |x|/x to be equal to -1, x must be negative.

Let's take each statement one by one.

1) x > 0

=> x is positive. Thus, |x|/x equal to 1 ≠ -1. The answer is no. Unique answer. Sufficient.

2) x < 1

Case 1: If 0 < x < 1, we have x a positive number. The answer is no.
Case 2: If x < 0, we have x a negative number. The answer is yes.

No unique answer. Insufficient.

The correct answer: A

Hope this helps!

-Jay
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by Max@Math Revolution » Fri Nov 16, 2018 2:04 am
=>

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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The question |x|/x = -1 is equivalent to x ≤ 0 as shown below:
|x|/x = -1
=> |x| = -x
=> x ≤ 0

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)
If x > 0, then "x ≤ 0" is always false, and the answer is 'no'.
Since 'no' is also a unique answer by CMT (Common Mistake Type) 1, condition 1) is sufficient.

Condition 2)
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
Condition 2) is not sufficient, since the solution set of the question does not include the solution set of condition 2).

Therefore, A is the answer.
Answer: A

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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