Question

The number of solutions of the equation x = 2{x}, where {x} is the fractional part of x, is

Answer 1

we have to find the number of solutions of the equation, x = 2{x}

where {x} is the fractional part of x.

solution : x = 2{x}

⇒[x] + {x} = 2{x} , where [x] is greatest integers function. [ we used here , x = [x] + {x}

⇒[x] = 2{x} - {x} = {x}

⇒[x] = {x} .....(1)

we know, 0 ≤ {x} < 1

⇒0 ≤ [x] < 1

so, [x] = 0 ⇒{x} = 0 ⇒x = 0

Therefore only one solution is possible i.e., x = 0

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Answer 2

Given :  x = 2{x}, where {x} is the fractional part of x

To Find : number of solutions of the equation x = 2{x}

Solution:

{x} is the fractional part of x

=>  {x}  = x - [ x ]

[ x ] represent greatest integer function

x = 2{x}

=> x  = 2 ( x - [ x ] )

=> x = 2x   - 2  [ x ]

=> x  = 2 [ x ]

[ x ]  is integer => 2 [ x ] must be an integer

Hence  x must be integer

Integer x = 2x

=> x = 0

Hence x = 0 is the only solution

x = 2{x}

number of solutions of the equation x = 2{x}   is 1

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