Свойство 1. $$ \sqrt {f(x)} > \sqrt {g(x)} \; \Leftrightarrow \left\{ \begin{array}{l} f(x) > g(x), \\ f(x) > 0,\;g(x) > 0. \\ \end{array} \right.\; $$

Свойство 2. $$f(x)\sqrt {g(x)} \le 0 \Leftrightarrow \left\{ \begin{array}{l} f(x)g^2 (x) \le 0, \\ g(x) \ge 0. \\ \end{array} \right.$$

Свойство 3. $$f(x)\sqrt {g(x)} \ge 0 \Leftrightarrow \left\{ \begin{array}{l} f(x)g^2 (x) \ge 0, \\ g(x) \ge 0. \\ \end{array} \right.$$

Свойство 4. $$\sqrt {f(x)} + \sqrt {g(x)} > \sqrt {h(x)} \Leftrightarrow \left\{ \begin{array}{l} f(x) + 2\sqrt {f(x) \cdot g(x)} + g(x) > h(x), \\ f(x) \ge 0,\quad g(x) \ge 0,\quad h(x) \ge 0. \\ \end{array} \right.$$

Свойство 5. $$\sqrt {f(x)} + \sqrt {g(x)} < \sqrt {h(x)} \Leftrightarrow \left\{ \begin{array}{l} f(x) + 2\sqrt {f(x) \cdot g(x)} + g(x) < h(x), \\ f(x) \ge 0,\quad g(x) \ge 0,\quad h(x) \ge 0. \\ \end{array} \right.$$

Свойство 6. $$\frac{{\sqrt {f(x)} - \sqrt {g(x)} }}{{h(x)}} \ge 0\;( \le 0) \Leftrightarrow \frac{{f(x) - g(x)}}{{h(x)}} \ge 0\;( \le 0)$$ в ОДЗ $$\left\{ \begin{array}{l} f(x) \ge 0, \\ g(x) \ge 0. \\ \end{array} \right.$$

Свойство 7. $$\frac{{\sqrt {f(x)} - g(x)}}{{h(x)}} \ge 0\;( \le 0) \Leftrightarrow \left[ \begin{array}{l} \left\{ \begin{array}{l} g(x) < 0, \\ h(x) > 0\;( < 0); \\ \end{array} \right. \\ \left\{ \begin{array}{l} g(x) \ge 0, \\ \frac{{f(x) - g^2 (x)}}{{h(x)}} \ge 0\;( \le 0). \\ \end{array} \right. \\ \end{array} \right.$$

Свойство 8. $$\sqrt {f^2 (x)} \ge g(x) \Leftrightarrow \left| {f(x)} \right| \ge g(x) \Leftrightarrow \left[ \begin{array}{l} g(x) < 0, \\ \left\{ \begin{array}{l} \left[ \begin{array}{l} f(x) \ge g(x), \\ f(x) \le - g(x), \\ \end{array} \right. \\ g(x) \ge 0. \\ \end{array} \right. \\ \end{array} \right.$$