Ответы
Ответ дал:
1
ОДЗ:
![2x-2 \ \textgreater \ 0 \ \Rightarrow \ 2x\ \textgreater \ 2 \ \Rightarrow \ \boxed{x\ \textgreater \ 1} \\ \\ 8x^2 -23x+15 \neq 1 \ \Rightarrow \ 8x^2 -23+14 \neq 0; \\ x_{1,2} = \frac{23 \pm \sqrt{23^2 -4 \cdot 8 \cdot 14}}{2 \cdot 8}=\frac{23 \pm \sqrt{529 - 448}}{16}=\frac{23 \pm 9}{16}; \ \ \boxed{x \neq 2}, \ \ x \neq \frac{7}{8} \ \textless \ 1 \\ \\ 8x^2 -23x +15 \ \textgreater \ 0; \ \ 8x^2 -23x+15=0; \ \ x_{3,4}=\frac{23 \pm \sqrt{529-480}}{16}=\frac{23 \pm 7}{16}; \\ x_3=\frac{15}{8}, \ x_4=1 \ \Rightarrow \ x\ \textless \ 1 ; \ \ \ \boxed{x\ \textgreater \ \frac{15}{8}} 2x-2 \ \textgreater \ 0 \ \Rightarrow \ 2x\ \textgreater \ 2 \ \Rightarrow \ \boxed{x\ \textgreater \ 1} \\ \\ 8x^2 -23x+15 \neq 1 \ \Rightarrow \ 8x^2 -23+14 \neq 0; \\ x_{1,2} = \frac{23 \pm \sqrt{23^2 -4 \cdot 8 \cdot 14}}{2 \cdot 8}=\frac{23 \pm \sqrt{529 - 448}}{16}=\frac{23 \pm 9}{16}; \ \ \boxed{x \neq 2}, \ \ x \neq \frac{7}{8} \ \textless \ 1 \\ \\ 8x^2 -23x +15 \ \textgreater \ 0; \ \ 8x^2 -23x+15=0; \ \ x_{3,4}=\frac{23 \pm \sqrt{529-480}}{16}=\frac{23 \pm 7}{16}; \\ x_3=\frac{15}{8}, \ x_4=1 \ \Rightarrow \ x\ \textless \ 1 ; \ \ \ \boxed{x\ \textgreater \ \frac{15}{8}}](https://tex.z-dn.net/?f=2x-2+%5C+%5Ctextgreater+%5C+0+%5C+%5CRightarrow+%5C+2x%5C+%5Ctextgreater+%5C+2+%5C+%5CRightarrow+%5C+%5Cboxed%7Bx%5C+%5Ctextgreater+%5C+1%7D+%5C%5C+%5C%5C+8x%5E2+-23x%2B15+%5Cneq+1+%5C+%5CRightarrow+%5C+8x%5E2+-23%2B14+%5Cneq+0%3B+%5C%5C+x_%7B1%2C2%7D+%3D+%5Cfrac%7B23+%5Cpm+%5Csqrt%7B23%5E2+-4+%5Ccdot+8+%5Ccdot+14%7D%7D%7B2+%5Ccdot+8%7D%3D%5Cfrac%7B23+%5Cpm+%5Csqrt%7B529+-+448%7D%7D%7B16%7D%3D%5Cfrac%7B23+%5Cpm+9%7D%7B16%7D%3B+%5C+%5C+%5Cboxed%7Bx+%5Cneq+2%7D%2C+%5C+%5C+x++%5Cneq+%5Cfrac%7B7%7D%7B8%7D+%5C+%5Ctextless+%5C+1++%5C%5C+%5C%5C+8x%5E2+-23x+%2B15+%5C+%5Ctextgreater+%5C+0%3B+%5C+%5C+8x%5E2+-23x%2B15%3D0%3B+%5C+%5C+x_%7B3%2C4%7D%3D%5Cfrac%7B23+%5Cpm+%5Csqrt%7B529-480%7D%7D%7B16%7D%3D%5Cfrac%7B23+%5Cpm+7%7D%7B16%7D%3B+%5C%5C++x_3%3D%5Cfrac%7B15%7D%7B8%7D%2C+%5C+x_4%3D1+%5C+%5CRightarrow+%5C+x%5C+%5Ctextless+%5C+1+%3B+%5C+%5C+%5C+%5Cboxed%7Bx%5C+%5Ctextgreater+%5C++%5Cfrac%7B15%7D%7B8%7D%7D+)
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1 15/8
![\frac{15}{8} \ \textless \ x \ \textless \ 2, \ \ x\ \textgreater \ 2 \frac{15}{8} \ \textless \ x \ \textless \ 2, \ \ x\ \textgreater \ 2](https://tex.z-dn.net/?f=%5Cfrac%7B15%7D%7B8%7D+%5C+%5Ctextless+%5C++x+%5C+%5Ctextless+%5C++2%2C+%5C+%5C+x%5C+%5Ctextgreater+%5C+2)
![1) 8x^2 -23x+15 \ \textgreater \ 1 \ \Rightarrow \ x\ \textgreater \ 2 \\ \log_{8x^2 -23x+15} (2x-2) \leq 0 \\\\ \log_{8x^2 -23x+15} (2x-2)\leq \log_{8x^2-23x+15}1 \\ \\ 2x-2 \leq 1 \ \Rightarrow \ 2x \leq 3 \Rightarrow \ x \leq \frac{3}{2} 1) 8x^2 -23x+15 \ \textgreater \ 1 \ \Rightarrow \ x\ \textgreater \ 2 \\ \log_{8x^2 -23x+15} (2x-2) \leq 0 \\\\ \log_{8x^2 -23x+15} (2x-2)\leq \log_{8x^2-23x+15}1 \\ \\ 2x-2 \leq 1 \ \Rightarrow \ 2x \leq 3 \Rightarrow \ x \leq \frac{3}{2}](https://tex.z-dn.net/?f=1%29+8x%5E2+-23x%2B15+%5C+%5Ctextgreater+%5C+1+%5C+%5CRightarrow+%5C+x%5C+%5Ctextgreater+%5C+2+%5C%5C+%5Clog_%7B8x%5E2+-23x%2B15%7D+%282x-2%29+%5Cleq+0+%5C%5C%5C%5C+%5Clog_%7B8x%5E2+-23x%2B15%7D+%282x-2%29%5Cleq+%5Clog_%7B8x%5E2-23x%2B15%7D1+%5C%5C+%5C%5C+2x-2+%5Cleq+1+%5C+%5CRightarrow+%5C+2x+%5Cleq+3+%5CRightarrow+%5C+x+%5Cleq+%5Cfrac%7B3%7D%7B2%7D+)
Не удовлетворяет ОДЗ
![2) \ 0\ \textless \ 8x^2 -23x+15\ \textless \ 1 \\ \\ \left \{ {{x\ \textgreater \ \frac{15}{8}} \atop {\frac{7}{8} \ \textless \ x\ \textless \ 2}} \right. \ \Rightarrow \ \frac{15}{8}\ \textless \ x\ \textless \ 2
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\log_{8x^2 -23x+15} (2x-2) \leq \log_{8x^2-23x+15}1 \\ \\ 2x-2 \geq 1 \\ \\ x \geq \frac{3}{2} 2) \ 0\ \textless \ 8x^2 -23x+15\ \textless \ 1 \\ \\ \left \{ {{x\ \textgreater \ \frac{15}{8}} \atop {\frac{7}{8} \ \textless \ x\ \textless \ 2}} \right. \ \Rightarrow \ \frac{15}{8}\ \textless \ x\ \textless \ 2
\\ \\
\log_{8x^2 -23x+15} (2x-2) \leq \log_{8x^2-23x+15}1 \\ \\ 2x-2 \geq 1 \\ \\ x \geq \frac{3}{2}](https://tex.z-dn.net/?f=2%29+%5C+0%5C+%5Ctextless+%5C+8x%5E2+-23x%2B15%5C+%5Ctextless+%5C+1+%5C%5C+%5C%5C++%5Cleft+%5C%7B+%7B%7Bx%5C+%5Ctextgreater+%5C++%5Cfrac%7B15%7D%7B8%7D%7D+%5Catop+%7B%5Cfrac%7B7%7D%7B8%7D+%5C+%5Ctextless+%5C+x%5C+%5Ctextless+%5C+2%7D%7D+%5Cright.+%5C+%5CRightarrow+%5C+%5Cfrac%7B15%7D%7B8%7D%5C+%5Ctextless+%5C+x%5C+%5Ctextless+%5C+2%0A%5C%5C+%5C%5C+%0A%5Clog_%7B8x%5E2+-23x%2B15%7D+%282x-2%29+%5Cleq+%5Clog_%7B8x%5E2-23x%2B15%7D1+%5C%5C+%5C%5C+2x-2+%5Cgeq+1+%5C%5C+%5C%5C+x+%5Cgeq+%5Cfrac%7B3%7D%7B2%7D)
Ответом будет пересечение![\frac{15}{8}\ \textless \ x\ \textless \ 2 \ \cup\ x \geq \frac{3}{2} \frac{15}{8}\ \textless \ x\ \textless \ 2 \ \cup\ x \geq \frac{3}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B15%7D%7B8%7D%5C+%5Ctextless+%5C+x%5C+%5Ctextless+%5C+2+%5C+%5Ccup%5C+x+%5Cgeq+%5Cfrac%7B3%7D%7B2%7D+)
Сравним![\frac{15}{8} \ \textgreater \ \frac{3}{2} \ \ \ (\frac{15}{8} \ \textgreater \ \frac{12}{8}) \frac{15}{8} \ \textgreater \ \frac{3}{2} \ \ \ (\frac{15}{8} \ \textgreater \ \frac{12}{8})](https://tex.z-dn.net/?f=%5Cfrac%7B15%7D%7B8%7D+%5C+%5Ctextgreater+%5C++%5Cfrac%7B3%7D%7B2%7D+%5C+%5C+%5C+%28%5Cfrac%7B15%7D%7B8%7D+%5C+%5Ctextgreater+%5C++%5Cfrac%7B12%7D%7B8%7D%29)
ОТВЕТ:![\frac{15}{8} \ \textless \ x \ \textless \ 2 \\ \\ (\frac{15}{8} ; 2) \frac{15}{8} \ \textless \ x \ \textless \ 2 \\ \\ (\frac{15}{8} ; 2)](https://tex.z-dn.net/?f=%5Cfrac%7B15%7D%7B8%7D+%5C+%5Ctextless+%5C++x+%5C+%5Ctextless+%5C++2+%5C%5C+%5C%5C+%28%5Cfrac%7B15%7D%7B8%7D+%3B+2%29)
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1 15/8
Не удовлетворяет ОДЗ
Ответом будет пересечение
Сравним
ОТВЕТ:
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