Hi..
x>0... Just important to know that x is NOT EQUAL to 0 or may be not required at all as this is also conveyed by second INEQUALITY
\(|x|=\frac{1}{x}\)...
This tells us that x is POSITIVE.
Which Number has its reciprocal EQUAL to itself .. 1 or -1..
But x is positive so answer is 1

C
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I'm trying to understand, unsuccessfully so far, that which you stress.

If x > 0, x is not 0.

Given the fraction involved, where by definition we cannot divide by 0, x is not 0.

Or are you saying that there is no such thing as |0| = \(\frac{0}{1}\)?(That is, that the variable x inside the absolute value brackets cannot be 0 irrespective of what is on the other side of the inequality?)

I'm trying to figure out what mistake you're trying to prevent from being made. I'm sure you make an important point. Please explain?
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Here is how I solved the problem

|x| = \(\frac{1}{x}\) => |x|*x=1

since x>0

\(x^2=1\)

=> x=1 (or) -1

since x>0, we have the answer as x=1. Ans C