Question

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$\sqrt{x} \leqslant x$

Answer 1

$\sqrt{x} \leq x\; \; \Leftrightarrow \; \; \left \{ {{x\geq 0} \atop {x\leq x^2}} \right. \; \left \{ {{x\geq 0} \atop {x^2-x\geq 0}} \right. \; \left \{ {{x\geq 0} \atop {x(x-1)\geq 0}} \right. \\\\x\, (x-1)\geq 0\; ;\; \; znaki:\; \; +++[\, 0\, ]---[\, 1\,]+++\\\\x\in (-\infty ,0\, ]\cup [1,+\infty )\\\\\left \{ {{x\geq 0} \atop {x\in (-\infty ,0\, ]\cup [\, 1,+\infty )}} \right. \; \; \Rightarrow \; \; \underline {\; x\in \{0\}\cup [\, 1,+\infty )\; }$

Answer 2

$\sqrt{x} \leqslant x \\ odz \\ x \geqslant 0 \\ \sqrt{x} = x \\ x = {x}^{2} \\ {x}^{2} - x = 0 \\ x(x - 1) = 0 \\ x = 0 \\ x = 1 \\ (0) + + (1) - - \\ x\in \: \{0\}\cup[1;\infty)$