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If q, s, and t are all different numbers, is q < s < t ? [#permalink]
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If q, s, and t are all different numbers, is q < s < t ? (1) t  q = t  s + s  q (2) t > q
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Last edited by Bunuel on 19 Jun 2017, 04:59, edited 1 time in total.
Renamed the topic and edited the question.

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Re: If q, s, and t are all different numbers, is q < s < t ? [#permalink]
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19 Jun 2017, 05:28
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If q, s, and t are all different numbers, is q < s < t ?(1) t  q = t  s + s  q Notice that the right hand side is positive (it's the sum of two absolute values, so two nonnegative values, in fact, in our case two positive values, since we know that the variables are distinct). Thus the left hand side must also be positive, which means that t > q. So, we can have 3 cases for s: a. sqtIn this case \(s < q < t\): \(t  s > 0\) and \(s  q < 0\), which would mean that \(t  s = t s\) and \(s  q = (s  q)\) (recall that x = x when x > 0 and x = x when x <= 0). So, \(t  s + s  q = (t s)  (s  q) = t  2s + q\). So, in his case we'd have \(t  q = t  2s + q\) or \(q=s\). But we are told that q, s, and t are all different numbers, so this case is out. b. qstIn this case \(q < s < t\): \(t  s > 0\) and \(s  q > 0\), which would mean that \(t  s = t s\) and \(s  q = s  q\). So, \(t  s + s  q = (t s) + (s  q) = t  q\). This matches the info given in the statement. c. qtsIn this case \(q < t < s\): \(t  s < 0\) and \(s  q > 0\), which would mean that \(t  s = t s\) and \(s  q = (s  q)\). So, \(t  s + s  q = (t s) + (s  q) = t + 2s  q\). So, in his case we'd have \(t  q = t + 2s  q\) or \(t=s\). But we are told that q, s, and t are all different numbers, so this case is out. Only q < s < t case is possible. Sufficient. (2) t > q. Not sufficient. Answer: A.
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Re: If q, s, and t are all different numbers, is q < s < t ? [#permalink]
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19 Jun 2017, 13:27
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AbdurRakib wrote: If q, s, and t are all different numbers, is q < s < t ?
(1) t  q = t  s + s  q
(2) t > q With less math: Start with statement 2. This is insufficient, since s could be greater than t, or s could be between q and t. Eliminate answers B and D.Statement 1: When you see x  y, think 'distance between x and y on the number line'. That's all that means. So, this statement says that t  q is equal to the distance between t and s, plus the distance between s and q. In other words, s has to be between t and q. Jot down some diagrams on your paper to convince yourself of that: in order for the distances to make sense, s has to be in the middle. Also, tq has to be positive, since it's the sum of two absolute values. So, t is greater than q. If t is greater than q and s is in the middle, you know that q < s < t. Sufficient.
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