Question

Find the values of 'x' satisfying this solution.
[x] + 1/[2x] ={x} +1/3
where [ ] represents greatest integer function and { } represents fractional part.
I want to know the procedure.

Answer 1

somebody help please.........

Answer 2

you can write \(x=n+\left\{ x \right\}\) where \(n=[x]\) and \(0\le\left\{ x \right\}<1\), put it in the equation:\[n+\frac{1}{[2(n+\left\{ x \right\})]}=\left\{ x \right\}+\frac{1}{3}\]you will get\[n+\frac{1}{2n+[2\left\{ x \right\}]}=\left\{ x \right\}+\frac{1}{3}\](why?)now note that \([2\left\{ x \right\}]\) can either be \(0\) or \(1\) (why?)

case 1: \([2\left\{ x \right\}]=0\) \[n+\frac{1}{2n}=\left\{ x \right\}+\frac{1}{3}\]\(n\) is a positive integer, hence no answer from here (why?)

case 2: \([2\left\{ x \right\}]=1\) \[n+\frac{1}{2n+1}=\left\{ x \right\}+\frac{1}{3}\]\(n\) is a positive integer, hence no answer from here (why?).