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.)
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.\(((1)+1)(|(1)| - 1) > 0\)
\((2)(0) > 0\)
XThe 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).
\(((0.99)+1)(|(0.99)| - 1) > 0\)
\((1.99)(-0.01) > 0\)
XWe 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? NOWe 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