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16 Sep 2014, 01:29
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If \(x\) and \(y\) are integers and \(x + y = -12\), which of the following must be true?
A. Both \(x\) and \(y\) are negative
B. \(xy > 0\)
C. If \(y < 0\), then \(x > 0\)
D. If \(y > 0\), then \(x < 0\)
E. \(x-y > 0\)
A. Both \(x\) and \(y\) are negative
B. \(xy > 0\)
C. If \(y < 0\), then \(x > 0\)
D. If \(y > 0\), then \(x < 0\)
E. \(x-y > 0\)
02 Mar 2016, 21:35
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msk0657
As per the question, we know:
i. \(x\) and \(y\) are integers, and
ii. \(x + y = -12\)
From this, we get following five situations/ scenarios:
1) \(x < 0\), \(y < 0\) (eg. \(x = -5\), \(y = -7\) or \(x = -7\), \(y = -5\))
2) \(x < 0\), \(y > 0\) (eg. \(x = -13\), \(y = 1\))
3) \(x > 0\), \(y < 0\) (eg. \(x = 1\), \(y = -13\))
4) \(x = 0\), \(y < 0\) (\(x = 0\), \(y = -12\))
5) \(x < 0\), \(y = 0\) (\(x = -12\), \(y = 0\))
Let us now consider each of the options to determine whether they must be true for each of the above scenarios:
Option A:
Both \(x\) and \(y\) are negative
in other words: \(x <0\), \(y < 0\). This is true only for scenario 1) above. But this is not true for other 4 scenarios. Hence, this could be true, but not definitely true.
Option B:
\(xy>0\), this is only possible when \(x>0\) and \(y>0\) or when \(x<0 and y<0\). Again, this is true only for scenario 1) above. But this is not true for other 4 scenarios. Hence, this could be true, but not definitely true.
Option C:
If \(y<0\), then \(x>0\).
If we look at all the 5 scenarios, \(y<0\) in scenario 3 and 4. However, this statement is true for scenario 3 but not true for scenario 4. This is because, if \(y = -12\) (i.e. \(y<0\)), then \(x = 0\) (neither negative nor positive). Hence, this could be true, but not definitely true.
Option D:
If \(y>0\), then \(x<0\).
If we look at all the 5 scenarios, \(y<0\) only in scenario 2 and as per that scenario \(x>0\). So this must be true because whenever \(y\) is a negative integer, \(x\) has to be a positive integer. This is the correct answer.
Option E:
\(x-y>0\). This is true for scenario 4 but not true for scenario 5. Therefore, this could be true, but not definitely true. Remember, it is sufficient to consider 2 out of 5 scenarios, if we get different answers from them. We have to consider all the 5 scenarios only in cases where the statement is definitely true or must be true.
Hope this helps!!
Regards,
Nalin
General Discussion
16 Sep 2014, 01:29
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Official Solution:
If \(x\) and \(y\) are integers and \(x + y = -12\), which of the following must be true?
A. Both \(x\) and \(y\) are negative
B. \(xy > 0\)
C. If \(y < 0\), then \(x > 0\)
D. If \(y > 0\), then \(x < 0\)
E. \(x-y > 0\)
Note that the question asks which of the following statements must be true, rather than could be true.
Let's evaluate the options:
A. Both \(x\) and \(y\) are negative. This option is not always true, consider \(x = -20\) and \(y = 8\).
B. \(xy > 0\). This option is not always true either, consider \(x = -20\) and \(y = 8\).
C. If \(y < 0\), then \(x > 0\). This option is not always true, consider \(y = -2\) and \(x = -10\).
D. If \(y > 0\), then \(x < 0\). If \(y\) is positive, then \(x\) must be negative in order for the sum of \(x\) and \(y\) to be negative.
E. \(x - y > 0\). This option is not always true, consider \(x = -20\) and \(y = 8\).
Therefore, only option D is ALWAYS true: if \(y\) is positive, then \(x\) must be negative in order for the sum of \(x\) and \(y\) to be negative.
Answer: D
If \(x\) and \(y\) are integers and \(x + y = -12\), which of the following must be true?
A. Both \(x\) and \(y\) are negative
B. \(xy > 0\)
C. If \(y < 0\), then \(x > 0\)
D. If \(y > 0\), then \(x < 0\)
E. \(x-y > 0\)
Note that the question asks which of the following statements must be true, rather than could be true.
Let's evaluate the options:
A. Both \(x\) and \(y\) are negative. This option is not always true, consider \(x = -20\) and \(y = 8\).
B. \(xy > 0\). This option is not always true either, consider \(x = -20\) and \(y = 8\).
C. If \(y < 0\), then \(x > 0\). This option is not always true, consider \(y = -2\) and \(x = -10\).
D. If \(y > 0\), then \(x < 0\). If \(y\) is positive, then \(x\) must be negative in order for the sum of \(x\) and \(y\) to be negative.
E. \(x - y > 0\). This option is not always true, consider \(x = -20\) and \(y = 8\).
Therefore, only option D is ALWAYS true: if \(y\) is positive, then \(x\) must be negative in order for the sum of \(x\) and \(y\) to be negative.
Answer: D
