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1/[x]+1/[2x]={x}+1/3 ,Solve for x. where [•]is G.I.F. and {x} is Fraction part function.
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1/[x]+1/[2x]={x}+1/3 ,Solve for x. where [•]is G.I.F. and {x} is Fract...
Solving the Equation 1/[x] 1/[2x]={x} 1/3


Understanding the Notations


  • G.I.F. - Greatest Integer Function, also known as the floor function, represents the greatest integer less than or equal to a given number.

  • Fraction Part Function - Denoted by {x}, represents the decimal part of a given number.



Breaking Down the Equation

The given equation can be written as:

1/[x] + 1/[2x] = {x} + 1/3

where [x] represents the greatest integer less than or equal to x.


Solving the Equation


  1. Let's assume that [x] = n.

  2. Then, n ≤ x < />

  3. Similarly, 2n ≤ 2x < />

  4. Now, we can substitute these values in the given equation:

    5. 1/n + 1/2n + = x - n + 1/3

  5. where 2n + represents the smallest even integer greater than or equal to 2n.

  6. Since n ≤ x < n+1,="" the="" left-hand="" side="" of="" the="" equation="" is="" less="" than="" or="" equal="" to="" 1/n="" +="" />

  7. Therefore, we can write:

    8. 1/n + 1/2n + ≤ n - n + 1/3

    1/n + 1/2n + ≤ 1/3

  8. Now, we can solve for n using the above inequality:

    9. 1/n + 1/2n + ≤ 1/3

    3(2n + + n) ≤ 2n +n 2

    3n 2 + 9n ≤ 2n 2 + 2n +

    n 2 - 7n + 2n + ≥ 0

    (n - 7/2) 2 - 17/4 + 2n + ≥ 0

    (n - 7/2) 2 + 8n + - 17/4 ≥ 0

    Now, we can solve for n using the quadratic formula:

    n = (7 ± √(17 - 32n +)) / 2

  9. Since
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1/[x]+1/[2x]={x}+1/3 ,Solve for x. where [•]is G.I.F. and {x} is Fraction part function.
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