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16 Sep 2014, 00:40
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
33% (01:39) correct
67%
(01:49)
wrong
based on 344
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Not Attempted Yet
If \(x \ne 0\) and \(\frac{x}{|x|} \lt x\), which of the following must be true?
A. \(x \gt 1\)
B. \(x \gt -1\)
C. \(|x| \lt 1\)
D. \(|x| > 1\)
E. \(-1 \lt x \lt 0\)
A. \(x \gt 1\)
B. \(x \gt -1\)
C. \(|x| \lt 1\)
D. \(|x| > 1\)
E. \(-1 \lt x \lt 0\)
16 Sep 2014, 00:40
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Official Solution:
If \(x \ne 0\) and \(\frac{x}{|x|} \lt x\), which of the following must be true?
A. \(x \gt 1\)
B. \(x \gt -1\)
C. \(|x| \lt 1\)
D. \(|x| > 1\)
E. \(-1 \lt x \lt 0\)
To begin with, note that the problem is asking to identify which option MUST be true, not just which one COULD be true. Next, let's determine the range(s) for \(x\), given that \(x\) is not equal to 0 and \(\frac{x}{|x|} \lt x\). Consider two cases:
Case 1: If \(x \lt 0\), then \(|x|=-x\). In this case, we have \(\frac{x}{-x} \lt x\), which simplifies to \(-1 \lt x\). Since we are considering the range where \(x \lt 0\), then for this case we have \(-1 \lt x \lt 0\).
Case 2: If \(x \gt 0\), then \(|x|=x\). In this case, we have \(\frac{x}{x} \lt x\), which simplifies to \(1 \lt x\).
Thus, \(\frac{x}{|x|} \lt x\) implies that \(-1 \lt x \lt 0\) or \(x \gt 1\). The only option that is always true is B, which states that \(x \gt -1\). This is because ANY \(x\) from the range \(-1 \lt x \lt 0\) or \(x \gt 1\) will definitely be greater than \(-1\). Thus, B, which states that \(x\) is greater than -1, must be true!
Option A, \(x \gt 1\), is not necessarily true since \(x\) could be -0.5.
Option C, \(|x| \lt 1\), is not necessarily true since \(x\) could be 2.
Option D, \(|x| \gt 1\), is not necessarily true since \(x\) could be -0.5.
Option E, \(-1 \lt x \lt 0\), is not necessarily true since \(x\) could be 2.
Answer: B
If \(x \ne 0\) and \(\frac{x}{|x|} \lt x\), which of the following must be true?
A. \(x \gt 1\)
B. \(x \gt -1\)
C. \(|x| \lt 1\)
D. \(|x| > 1\)
E. \(-1 \lt x \lt 0\)
To begin with, note that the problem is asking to identify which option MUST be true, not just which one COULD be true. Next, let's determine the range(s) for \(x\), given that \(x\) is not equal to 0 and \(\frac{x}{|x|} \lt x\). Consider two cases:
Case 1: If \(x \lt 0\), then \(|x|=-x\). In this case, we have \(\frac{x}{-x} \lt x\), which simplifies to \(-1 \lt x\). Since we are considering the range where \(x \lt 0\), then for this case we have \(-1 \lt x \lt 0\).
Case 2: If \(x \gt 0\), then \(|x|=x\). In this case, we have \(\frac{x}{x} \lt x\), which simplifies to \(1 \lt x\).
Thus, \(\frac{x}{|x|} \lt x\) implies that \(-1 \lt x \lt 0\) or \(x \gt 1\). The only option that is always true is B, which states that \(x \gt -1\). This is because ANY \(x\) from the range \(-1 \lt x \lt 0\) or \(x \gt 1\) will definitely be greater than \(-1\). Thus, B, which states that \(x\) is greater than -1, must be true!
Option A, \(x \gt 1\), is not necessarily true since \(x\) could be -0.5.
Option C, \(|x| \lt 1\), is not necessarily true since \(x\) could be 2.
Option D, \(|x| \gt 1\), is not necessarily true since \(x\) could be -0.5.
Option E, \(-1 \lt x \lt 0\), is not necessarily true since \(x\) could be 2.
Answer: B
21 Jul 2016, 08:24
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lshoshiashvili
This is a property of an absolute value:
When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|={-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);
When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|={some \ expression}\). For example: \(|5|=5\).
You should brush-up fundamentals on modulus:
Theory on Absolute Values: math-absolute-value-modulus-86462.html
The E-GMAT Question Series on ABSOLUTE VALUE: the-e-gmat-question-series-on-absolute-value-198503.html
Properties of Absolute Values on the GMAT: properties-of-absolute-values-on-the-gmat-191317.html
Absolute Value: Tips and hints: absolute-value-tips-and-hints-175002.html
DS Absolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Absolute Values Questions to practice: search.php?search_id=tag&tag_id=58
Hard set on Absolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html
The E-GMAT Question Series on ABSOLUTE VALUE: the-e-gmat-question-series-on-absolute-value-198503.html
Properties of Absolute Values on the GMAT: properties-of-absolute-values-on-the-gmat-191317.html
Absolute Value: Tips and hints: absolute-value-tips-and-hints-175002.html
DS Absolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Absolute Values Questions to practice: search.php?search_id=tag&tag_id=58
Hard set on Absolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html
