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18 Jun 2012, 15:46
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
64% (01:54) correct
36%
(02:06)
wrong
based on 3668
sessions
History
Date
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Not Attempted Yet
18 Jun 2012, 16:15
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Good question. +1.
Is x> 1?
(1) (x+1) (|x| - 1) > 0. Consider two cases:
If \(x>0\) then \(|x|=x\) so \((x+1) (|x| - 1) > 0\) becomes \((x+1) (x - 1) > 0\) --> \(x^2-1>0\) --> \(x^2>1\) --> \(x<-1\) or \(x>1\). Since we consider range when \(x>0\) then we have \(x>1\) for this case;
If \(x\leq{0}\) then \(|x|=-x\) so \((x+1) (|x| - 1) > 0\) becomes \((x+1) (-x - 1) > 0\) --> \(-(x+1) (x+1) > 0\) --> \(-(x+1)^2>0\) --> \((x+1)^2<0\). Now, since the square of a number cannot be negative then for this range given equation has no solution.
So, we have that \((x+1) (|x| - 1) > 0\) holds true only when \(x>1\). Sufficient.
(2) |x| < 5 --> \(-5<x<5\). Not sufficient.
Answer: A.
Hope it's clear.
Is x> 1?
(1) (x+1) (|x| - 1) > 0. Consider two cases:
If \(x>0\) then \(|x|=x\) so \((x+1) (|x| - 1) > 0\) becomes \((x+1) (x - 1) > 0\) --> \(x^2-1>0\) --> \(x^2>1\) --> \(x<-1\) or \(x>1\). Since we consider range when \(x>0\) then we have \(x>1\) for this case;
If \(x\leq{0}\) then \(|x|=-x\) so \((x+1) (|x| - 1) > 0\) becomes \((x+1) (-x - 1) > 0\) --> \(-(x+1) (x+1) > 0\) --> \(-(x+1)^2>0\) --> \((x+1)^2<0\). Now, since the square of a number cannot be negative then for this range given equation has no solution.
So, we have that \((x+1) (|x| - 1) > 0\) holds true only when \(x>1\). Sufficient.
(2) |x| < 5 --> \(-5<x<5\). Not sufficient.
Answer: A.
Hope it's clear.
17 Oct 2021, 12:20
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enigma123
Is x > 1?
(1) \((x+1)(|x| - 1) > 0\)
(2) \(|x| < 5\)
Hello, everyone. I came across this question today in my studies and thought I would post my thoughts on Statement (1) in an effort to assist the community. (It should be quite clear that Statement (2) is NOT sufficient on its own if you have even scratched the surface of absolute value.)(1) \((x+1)(|x| - 1) > 0\)
(2) \(|x| < 5\)
To answer the question, we only need to keep in mind three questions:
1) Can x = 1?
2) Does the inequality in the statement work if x > 1?
3) Does the inequality in the statement work if x < 1?
That is it. We can use simple arithmetic to check for answers.
1) Let x = 1.
\(((1)+1)(|(1)| - 1) > 0\)
\((2)(0) > 0\) X
The answer to question 1 above is NO, x cannot equal 1 itself.
2) Let x = 1.1 (or something close to 1 from the greater end to check one extreme).
\(((1.1)+1)(|(1.1)| - 1) > 0\)
\((2.1)(0.1) > 0\) √
Even though the product would be close to 0, a positive times a positive will be positive, still greater than 0, no further calculation necessary. Moreover, we do not need to test larger positive numbers, since we can see that we would only end up with a larger positive product. The answer to question 2 above, then, is YES, the inequality works if x > 1.
3) Let x = 0.99 (or something close to 1 from the lesser end to check another extreme).
\(((0.99)+1)(|(0.99)| - 1) > 0\)
\((1.99)(-0.01) > 0\) X
We know that the left-hand side will be negative, so the inequality would fail. Now, there is no point in testing the unique integer 0 because we can appreciate that the positive * negative dynamic would persist. The only thing left to test is a negative value, but by this point, we do not actually need to work through anything. We can see that -1 would give us a product of 0, and that other negatives would yield the following:
a) x is between -1 and 0.
(positive) * (negative) = negative
b) x is less than -1.
(negative) * (positive) = negative
No matter what kind of number we would want to test, if x < 1, the product will always be negative, so the answer to question 3 above is NO. To recap:
1) Can x = 1? NO
2) Does the inequality in the statement work if x > 1? YES
3) Does the inequality in the statement work if x < 1? NO
We know beyond any doubt that x > 1, and the answer must be (A). Although explaining the chain of logic step by step took time, the mental process, as well as writing down numbers to ensure accuracy, took me little over a minute, and I answered confidently. If I can do it, so can you.
Good luck with your studies.
- Andrew
