What is restriction in x in f(x)=ln x-2(x-1)/x+1 if ln x> 2(x-1)/x+1?

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You need to notice that the problem provides the information that `ln x> 2(x-1)/(x+1), ` hence  `ln x -2(x-1)/(x+1) > 0` .

Notice that the expression `ln x - 2(x-1)/(x+1)`  represents the equation of the function f(x) such that:

`f(x) > 0`

The inequality `f(x) > 0 `  tells that the function has the minimum value equal to 0, hence, you need to solve for x the following equation, such that:

`ln x - 2(x-1)/(x+1) = 0`

Notice that for `x = 1` , the equation is checked, such that:

`ln 1 - 2(1-1)/(1+1) = 0 => 0 - 2*0/2 = 0 => 0 - 0 = 0 => 0 = 0`

Since the function reaches its minimum at `x = 1` , hence, the inequality holds for `x in (1,oo).`

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