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the number of solutions of [2x]-3{2x}=1

the number of solutions of [2x]-3{2x}=1? How to solve such problems

Note by Priyankar Kumar
4 years ago

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If \(\lfloor 2x \rfloor - 3\{2x\} = 1\) then \(3\{2x\} = \lfloor 2x\rfloor - 1\) is an integer, so \(\{2x\} = 0,1/3,2/3\).

  1. If \(\{2x\} = 0\) then \(\lfloor 2x\rfloor - 1 = 0\), so \(\lfloor 2x\rfloor = 1\), and hence \(2x=1\), so \(x=1/2\).

  2. If \(\{2x\} = 1/3\) then \(\lfloor 2x\rfloor - 1 = 1\), so \(\lfloor 2x \rfloor = 2\), and hence \(2x=7/3\), so \(x=7/6\).

  3. If \(\{2x\} = 2/3\) then \(\lfloor 2x\rfloor - 1 = 2\), so \(\lfloor 2x \rfloor = 3\), and hence \(2x=11/3\), so \(x=11/6\).

Mark Hennings · 4 years ago

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@Mark Hennings Thanks Priyankar Kumar · 4 years ago

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(I assume that the square and curly brackets represent floor and fractional part function respectively)

I am not the right person to help out in such problems but the following is worth a try. Add and subtract {2x} in the LHS. The equation now becomes 2x-4{2x}=1 or 2x=1+4{2x}. You can now plot a graph to find the number of solutions. Pranav Arora · 4 years ago

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@Pranav Arora Thanks. Priyankar Kumar · 4 years ago

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