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I found the wrong answer when letting HP prime solving inequality:

The calculator gave result where x<=0 which should not be the case as x is under square root, and it cannot be 0 as the denominator will be 0 The rest interval are correct. So the calculator needs to exclude x<=0

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I see Wolfram alpha gives similar answer!
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Even if x is negative, the expression 1/(x+sqrt(x))+1/(x-sqrt(x)) is real valued :
normal(1/(x+sqrt(x))+1/(x-sqrt(x))) returns 2/(x-1)
Therefore the Prime answer is correct, and x=0 is also valid (since the limit at x=0 is -2 and -2<=1).

Thank you, parisse.

For x is negative, it means that each of 1/(x+sqrt(x)), 1/(x-sqrt(x)) to be complex number. If so, that is correct.

For x is zero,

(1/(x+sqrt(x))+1/(x-sqrt(x)))
= (x-sqrt(x) + x + sqrt(x))/(x^2-x)
= (2x)/(x*(x-1))
and this will be equal to 2/(x-1) if x<> 0 (otherwise, it will be 0/0)

So (1/(x+sqrt(x))+1/(x-sqrt(x))) when x=0 should be undefined. However, lim x->0 of (1/(x+sqrt(x))+1/(x-sqrt(x))) is -2.