a)

\(P = \left( {\frac{1}{{x - \sqrt x }} + \frac{1}{{\sqrt x - 1}}} \right):\frac{{\sqrt x }}{{x - 2\sqrt x - 1}}\) (với x > 0, x ≠ 1)

\(P = \left( {\frac{1}{{\sqrt x \left( {\sqrt x - 1} \right)}} + \frac{1}{{\sqrt x - 1}}} \right):\frac{{\sqrt x }}{{x - 2\sqrt x - 1}}\)

\(P = \left( {\frac{1}{{\sqrt x \left( {\sqrt x - 1} \right)}} + \frac{{\sqrt x }}{{\sqrt x \left( {\sqrt x - 1} \right)}}} \right):\frac{{\sqrt x }}{{x - 2\sqrt x - 1}}\)

\(P = \frac{{1 + \sqrt x }}{{\sqrt x \left( {\sqrt x - 1} \right)}}:\frac{{\sqrt x }}{{x - 2\sqrt x + 1}}\)

\(P = \frac{{1 + \sqrt x }}{{\sqrt x \left( {\sqrt x - 1} \right)}}.\frac{{x - 2\sqrt x + 1}}{{\sqrt x }}\)

\(P = \frac{{1 + \sqrt x }}{{\sqrt x \left( {\sqrt x - 1} \right)}}.\frac{{x - 2\sqrt x + 1}}{{\sqrt x }}\)

\(P = \frac{{1 + \sqrt x }}{{\sqrt x \left( {\sqrt x - 1} \right)}}.\frac{{{{\left( {\sqrt x - 1} \right)}^2}}}{{\sqrt x }} = \frac{{\left( {1 + \sqrt x } \right)\left( {\sqrt x - 1} \right)}}{x} = \frac{{x - 1}}{x}\).

b)

\(P > \frac{1}{2} \Rightarrow \frac{{x - 1}}{x} > \frac{1}{2}\) (1)

Với x > 0, x ≠ 1 ta có:

(1)

\(\begin{array}{l} \Leftrightarrow x > 2x - 2\\ \Leftrightarrow x - 2 < 0\\ \Leftrightarrow x < 2\end{array}\)