# (1/x)>2 . why x<(1/2) is Not the answer

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Ines Ines 2019 年 9 月 21 日
Edited: James Tursa 2019 年 9 月 23 日

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madhan ravi 2019 年 9 月 21 日

Ines Ines 2019 年 9 月 21 日
My question is : why x<(1/2) is not the solution of (1/x)>2

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Matt J 2019 年 9 月 21 日

Because, for example, x=-1 satisfies x<1/2 but does not satisfy 1/x>2.

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James Tursa 2019 年 9 月 23 日

When you multiply (or divide) an inequality by a value, the direction of the inequality changes if the value is negative. So when you use a variable as the value you are multiplying by, you will need to split the inequality into two cases and work each case separately. E.g., start with this:
(1/x) > 2
Since x is in the denominator, you know right away that x can't be 0. So now multiply both sides by x. However, you don't yet know the sign of x, so you need to split this into two cases and work each one separately:
Case 1) If x > 0, then x*(1/x) > 2x => 1 > 2x => 1/2 > x
Case 2) If x < 0, then x*(1/x) < 2x => 1 < 2x => 1/2 < x
Case 1 yields 0 < x < 1/2
Case 2 turns out to be infeasible (x<0 and x>1/2 simultaneously can't happen)

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