Given that x is a real number, and:

{x}
|x-1| - 1 = ---------
|x-1|

where, [x] is the greatest integer less than or equal to x, and {x}= x-[x]

Find the largest possible value of x.

*** |x| denotes the

**absolute value** of x.

Let x = n + f, where n is an integer and 0 <= f <1

First, let's hope that the maximum x is >= 1

If so, then (n+f-2) = f/(n+f-1)

Multiplying by (n+f-1) and collecting terms gives:

n^2 + n(2f-3) + (f^2-4f+2) = 0

Solving for n has two roots, the larger of which is

n = (3 - 2f + sqrt(1-8f))/2

This is maximized when f = 0, and fortunately this makes n = 2, which is an integer. Since this is greater than 1, there are no other cases that need to be considered

Hope I didn't make any math mistakes.

**Final answer: x = 2**

Checking:

a) |2-1| - 1 = 0/|2-1|, so that works!

b) If x is between 2 and 3, say 2 + f, then by substitution we get

f = f/(1+f), which implies f is 0

c) If x is between 3 and 4, say 3 + f, then by substitution we get

1+f = f/(2+f)

But LHS >= 1 and RHS <1

And this only gets worse as X gets bigger.

So x = 2 checks out.

*Edited on ***February 20, 2014, 11:44 pm**