Question

Найдите сумму целых решений неравенства

Answer 1

$\frac{-x^2+10x-9+1-2\sqrt{-x^2+10x-9}}{\sqrt{-x^2+10x-9}} \geq 0\\\\ t=\sqrt{-x^2+10x-9}\ \textgreater \ 0\\\\ \frac{t^2+1-2t}{t} \geq 0\\\\ \frac{(t-1)^2}{t} \geq 0\\\\ \frac{(t-1)^2}{t} =0\ \ or\ \ \frac{(t-1)^2}{t} \ \textgreater \ 0\\\\ \left \{ {{(t-1)^2=0} \atop {t\neq0}} \right. \ \ or\ \ \frac{1}{t}\ \textgreater \ 0\\\\ \left \{ {{t=1} \atop {t\neq0}} \right. \ \ or\ \ t\ \textgreater \ 0\\\\ t=1\ \ or\ \ t\ \textgreater \ 0\\\\ t\ \textgreater \ 0\\\\ \sqrt{-x^2+10x-9}\ \textgreater \ 0$

$\left \{ {{0 \geq 0} \atop {-x^2+10x-9\ \textgreater \ 0^2}} \right. \ \ or\ \ \left \{ {{0\ \textless \ 0} \atop {-x^2+10x-9 \geq 0}} \right. \\\\ -x^2+10x-9\ \textgreater \ 0\\\\ x^2-10x+9\ \textless \ 0\\\\ x^2-x-9x+9\ \textless \ 0\\\\ x(x-1)-9(x-1)\ \textless \ 0\\\\ (x-1)(x-9)\ \textless \ 0\\\\ x\in(1;\ 9)$

Целые сумма целых решений: $2+3+4+5+6+7+8=35$

Ответ: $35$

$\sqrt{T}\ \textgreater \ U\ \ \ \textless \ -\ \textgreater \ \\\\ \ \textless \ -\ \textgreater \ \ \ \left \{ {{T \geq 0} \atop {T\ \textgreater \ U^2}} \right. \ or\ \left \{ {{U\ \textless \ 0} \atop {T \geq 0}} \right.$