Предмет: Алгебра,
автор: Koshechka565656
Решите неравенство:
Log x+1 (x-1)*log x+1 (x+2)=<0
Ответы
Автор ответа:
0
Произведение будет меньше равно нуля, если либо первый множитель ≥ 0 и второй множитель ≤0, либо наоборот.
![displaystyle left[begin{array}{ccc}begin{cases}
& text{ } log_{x+1}(x-1) geq 0 \
& text{ }log_{x+1}(x+2) leq 0
end{cases}\ begin{cases}
& text{ } log_{x+1}(x-1) leq 0 \
& text{ } log_{x+1}(x+2) geq 0
end{cases}end{array}right~~~Rightarrow~~~](https://tex.z-dn.net/?f=displaystyle+++left%5Bbegin%7Barray%7D%7Bccc%7Dbegin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+log_%7Bx%2B1%7D%28x-1%29+geq+0+%5C+%0A+%26amp%3B+text%7B++%7Dlog_%7Bx%2B1%7D%28x%2B2%29+leq+0%0Aend%7Bcases%7D%5C+begin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+log_%7Bx%2B1%7D%28x-1%29+leq+0+%5C+%0A+%26amp%3B+text%7B++%7D+log_%7Bx%2B1%7D%28x%2B2%29+geq+0%0Aend%7Bcases%7Dend%7Barray%7Dright%7E%7E%7ERightarrow%7E%7E%7E++ "displaystyle left[begin{array}{ccc}begin{cases}
& text{ } log_{x+1}(x-1) geq 0 \
& text{ }log_{x+1}(x+2) leq 0
end{cases}\ begin{cases}
& text{ } log_{x+1}(x-1) leq 0 \
& text{ } log_{x+1}(x+2) geq 0
end{cases}end{array}right~~~Rightarrow~~~")
![displaystyle left[begin{array}{ccc}begin{cases}
& text{ } begin{cases}
& text{ } x-1 textgreater 0 \
& text{ } x+2 geq 1
end{cases}\
& text{ } begin{cases}
& text{ } x+2 textgreater 0 \
& text{ } x+2 leq 1
end{cases}
end{cases}
\ begin{cases}
& text{ } begin{cases}
& text{ } x-1 textgreater 0 \
& text{ } x-1 leq 1
end{cases} \
& text{ } begin{cases}
& text{ } x+2 textgreater 0 \
& text{ } x+2 geq 1
end{cases}
end{cases}end{cases}end{array}right~~~Rightarrow](https://tex.z-dn.net/?f=displaystyle+++left%5Bbegin%7Barray%7D%7Bccc%7Dbegin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+begin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+x-1+textgreater++0+%5C+%0A+%26amp%3B+text%7B++%7D+x%2B2+geq+1+%0Aend%7Bcases%7D%5C+%0A+%26amp%3B+text%7B++%7D+begin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+x%2B2+textgreater++0+%5C+%0A+%26amp%3B+text%7B++%7D+x%2B2+leq+1+%0Aend%7Bcases%7D+%0Aend%7Bcases%7D%0A%5C+begin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+begin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+x-1+textgreater++0+%5C+%0A+%26amp%3B+text%7B++%7D+x-1+leq+1+%0Aend%7Bcases%7D+%5C+%0A+%26amp%3B+text%7B++%7D+begin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+x%2B2+textgreater++0+%5C+%0A+%26amp%3B+text%7B++%7D+x%2B2+geq+1+%0Aend%7Bcases%7D+%0Aend%7Bcases%7Dend%7Bcases%7Dend%7Barray%7Dright%7E%7E%7ERightarrow "displaystyle left[begin{array}{ccc}begin{cases}
& text{ } begin{cases}
& text{ } x-1 textgreater 0 \
& text{ } x+2 geq 1
end{cases}\
& text{ } begin{cases}
& text{ } x+2 textgreater 0 \
& text{ } x+2 leq 1
end{cases}
end{cases}
\ begin{cases}
& text{ } begin{cases}
& text{ } x-1 textgreater 0 \
& text{ } x-1 leq 1
end{cases} \
& text{ } begin{cases}
& text{ } x+2 textgreater 0 \
& text{ } x+2 geq 1
end{cases}
end{cases}end{cases}end{array}right~~~Rightarrow")
![Rightarrow~~ left[begin{array}{ccc}begin{cases}
& text{ } begin{cases}
& text{ } x textgreater 1 \
& text{ } x geq -1
end{cases} \
& text{ } begin{cases}
& text{ } x textgreater -2 \
& text{ } x leq -1
end{cases}
end{cases}\begin{cases}
& text{ } begin{cases}
& text{ } x textgreater 1 \
& text{ } x leq 2
end{cases} \
& text{ } begin{cases}
& text{ } x textgreater -2 \
& text{ } x geq -1
end{cases}
end{cases}end{array}right~~~Rightarrow~~](https://tex.z-dn.net/?f=Rightarrow%7E%7E++left%5Bbegin%7Barray%7D%7Bccc%7Dbegin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+begin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+x+textgreater++1+%5C+%0A+%26amp%3B+text%7B++%7D+x+geq+-1+%0Aend%7Bcases%7D+%5C+%0A+%26amp%3B+text%7B++%7D+begin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+x+textgreater++-2+%5C+%0A+%26amp%3B+text%7B++%7D+x+leq+-1+%0Aend%7Bcases%7D+%0Aend%7Bcases%7D%5Cbegin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+begin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+x+textgreater++1+%5C+%0A+%26amp%3B+text%7B++%7D+x+leq+2+%0Aend%7Bcases%7D+%5C+%0A+%26amp%3B+text%7B++%7D+begin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+x+textgreater++-2+%5C+%0A+%26amp%3B+text%7B++%7D+x+geq+-1%0Aend%7Bcases%7D+%0Aend%7Bcases%7Dend%7Barray%7Dright%7E%7E%7ERightarrow%7E%7E++ "Rightarrow~~ left[begin{array}{ccc}begin{cases}
& text{ } begin{cases}
& text{ } x textgreater 1 \
& text{ } x geq -1
end{cases} \
& text{ } begin{cases}
& text{ } x textgreater -2 \
& text{ } x leq -1
end{cases}
end{cases}\begin{cases}
& text{ } begin{cases}
& text{ } x textgreater 1 \
& text{ } x leq 2
end{cases} \
& text{ } begin{cases}
& text{ } x textgreater -2 \
& text{ } x geq -1
end{cases}
end{cases}end{array}right~~~Rightarrow~~")
![left[begin{array}{ccc}begin{cases}
& text{ } x textgreater 1 \
& text{ } -2 textless x leq -1
end{cases}\begin{cases}
& text{ } 1 textless x leq 2 \
& text{ } x geq -1
end{cases}end{array}right~~~Rightarrow~~~ 1 textless x leq 2](https://tex.z-dn.net/?f=++left%5Bbegin%7Barray%7D%7Bccc%7Dbegin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+x+textgreater++1+%5C+%0A+%26amp%3B+text%7B++%7D+-2+textless++x+leq+-1%0Aend%7Bcases%7D%5Cbegin%7Bcases%7D%0A+%26amp%3B+text%7B++%7D+1+textless++x+leq+2+%5C+%0A+%26amp%3B+text%7B++%7D+x+geq+-1%0Aend%7Bcases%7Dend%7Barray%7Dright%7E%7E%7ERightarrow%7E%7E%7E+1+textless++x+leq+2 "left[begin{array}{ccc}begin{cases}
& text{ } x textgreater 1 \
& text{ } -2 textless x leq -1
end{cases}\begin{cases}
& text{ } 1 textless x leq 2 \
& text{ } x geq -1
end{cases}end{array}right~~~Rightarrow~~~ 1 textless x leq 2")
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