Ответ:
№3
x∈(0;1] ∪ (2;6]
№4
(2,5; 1,5)
Пошаговое объяснение:
№3
введем ограничения на подкоренное и логарифмируемое выражения:
0 \end{matrix}\right. \Leftrightarrow \ \left\{\begin{matrix}x^2\leq 36 \\ x>0 \end{matrix}\right. \Leftrightarrow \ \left\{\begin{matrix}|x|\leq6 \\ x>0 \end{matrix}\right. \Leftrightarrow \ x \in (0;6]" alt="\left\{\begin{matrix}36-x^2\geq 0 \\ x>0 \end{matrix}\right. \Leftrightarrow \ \left\{\begin{matrix}x^2\leq 36 \\ x>0 \end{matrix}\right. \Leftrightarrow \ \left\{\begin{matrix}|x|\leq6 \\ x>0 \end{matrix}\right. \Leftrightarrow \ x \in (0;6]" align="absmiddle" class="latex-formula">
Корни числителя:
![\sqrt{36-x^2} *\log_{0.5}x=0 \\ \\ \left[ \begin{gathered} 36-x^2=0\\ \log_{0.5}x=0\end{gathered} \right. \Leftrightarrow \left[ \begin{gathered} x^2=36\\x=0.5^0\end{gathered} \right. \Leftrightarrow \left[ \begin{gathered} x=\pm 6\\x=1\end{gathered} \right. \sqrt{36-x^2} *\log_{0.5}x=0 \\ \\ \left[ \begin{gathered} 36-x^2=0\\ \log_{0.5}x=0\end{gathered} \right. \Leftrightarrow \left[ \begin{gathered} x^2=36\\x=0.5^0\end{gathered} \right. \Leftrightarrow \left[ \begin{gathered} x=\pm 6\\x=1\end{gathered} \right.](https://tex.z-dn.net/?f=%5Csqrt%7B36-x%5E2%7D%20%2A%5Clog_%7B0.5%7Dx%3D0%20%5C%5C%20%5C%5C%20%5Cleft%5B%20%5Cbegin%7Bgathered%7D%2036-x%5E2%3D0%5C%5C%20%5Clog_%7B0.5%7Dx%3D0%5Cend%7Bgathered%7D%20%5Cright.%20%5CLeftrightarrow%20%5Cleft%5B%20%5Cbegin%7Bgathered%7D%20x%5E2%3D36%5C%5Cx%3D0.5%5E0%5Cend%7Bgathered%7D%20%5Cright.%20%5CLeftrightarrow%20%5Cleft%5B%20%5Cbegin%7Bgathered%7D%20x%3D%5Cpm%206%5C%5Cx%3D1%5Cend%7Bgathered%7D%20%5Cright.)
x=-6 - не подходит под наши ограничения
Корень знаменателя:
x-2=0
x=2
(0)-------[1]---------(2)-------[6]>ₓ
Возьмем пробную точку x=0.5 чтобы узнать знак интервала (0;1)
![\frac{\sqrt{36-0.5^2}*\log_{0.5}0.5}{0.5-2} =-\frac{\sqrt{35.75}}{1.5} <0 \frac{\sqrt{36-0.5^2}*\log_{0.5}0.5}{0.5-2} =-\frac{\sqrt{35.75}}{1.5} <0](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%7B36-0.5%5E2%7D%2A%5Clog_%7B0.5%7D0.5%7D%7B0.5-2%7D%20%3D-%5Cfrac%7B%5Csqrt%7B35.75%7D%7D%7B1.5%7D%20%3C0)
(0)----[1]++++(2)----[6]>ₓ
![x \in (0;1] \cup (2;6] x \in (0;1] \cup (2;6]](https://tex.z-dn.net/?f=x%20%5Cin%20%280%3B1%5D%20%20%5Ccup%20%282%3B6%5D)
№4
Замена:
![\sqrt{x+y}=a, \ \ a\geq 0 \\ \sqrt{x-y}=b, \ \ b\geq 0 \sqrt{x+y}=a, \ \ a\geq 0 \\ \sqrt{x-y}=b, \ \ b\geq 0](https://tex.z-dn.net/?f=%5Csqrt%7Bx%2By%7D%3Da%2C%20%5C%20%5C%20%20a%5Cgeq%200%20%5C%5C%20%5Csqrt%7Bx-y%7D%3Db%2C%20%5C%20%5C%20b%5Cgeq%200)
![\left\{\begin{matrix} 3a-2b=4\\2a-b=3 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} 3a-2(2a-3)=4\\b=2a-3 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} 3a-4a+6=4\\b=2a-3 \end{matrix}\right. \Leftrightarrow \\ \\ \Leftrightarrow \left\{\begin{matrix} a=2\\b=2*2-3 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} a=2\\b=1 \end{matrix}\right. \left\{\begin{matrix} 3a-2b=4\\2a-b=3 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} 3a-2(2a-3)=4\\b=2a-3 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} 3a-4a+6=4\\b=2a-3 \end{matrix}\right. \Leftrightarrow \\ \\ \Leftrightarrow \left\{\begin{matrix} a=2\\b=2*2-3 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} a=2\\b=1 \end{matrix}\right.](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%203a-2b%3D4%5C%5C2a-b%3D3%20%5Cend%7Bmatrix%7D%5Cright.%20%5CLeftrightarrow%20%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%203a-2%282a-3%29%3D4%5C%5Cb%3D2a-3%20%5Cend%7Bmatrix%7D%5Cright.%20%5CLeftrightarrow%20%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%203a-4a%2B6%3D4%5C%5Cb%3D2a-3%20%5Cend%7Bmatrix%7D%5Cright.%20%5CLeftrightarrow%20%5C%5C%20%5C%5C%20%5CLeftrightarrow%20%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%20a%3D2%5C%5Cb%3D2%2A2-3%20%5Cend%7Bmatrix%7D%5Cright.%20%5CLeftrightarrow%20%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%20a%3D2%5C%5Cb%3D1%20%5Cend%7Bmatrix%7D%5Cright.)
Обратная замена:
![\left\{\begin{matrix} \sqrt{x+y}=2\\ \sqrt{x-y}=1 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} x+y=4\\ x-y=1 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} 1+y+y=4\\ x=1+y \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} 2y=3\\ x=1+y \end{matrix}\right. \Leftrightarrow \\ \\ \Leftrightarrow \left\{\begin{matrix} y=1.5\\ x=1+1.5 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} y=1.5\\ x=2.5 \end{matrix}\right. \left\{\begin{matrix} \sqrt{x+y}=2\\ \sqrt{x-y}=1 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} x+y=4\\ x-y=1 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} 1+y+y=4\\ x=1+y \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} 2y=3\\ x=1+y \end{matrix}\right. \Leftrightarrow \\ \\ \Leftrightarrow \left\{\begin{matrix} y=1.5\\ x=1+1.5 \end{matrix}\right. \Leftrightarrow \left\{\begin{matrix} y=1.5\\ x=2.5 \end{matrix}\right.](https://tex.z-dn.net/?f=%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%20%5Csqrt%7Bx%2By%7D%3D2%5C%5C%20%20%5Csqrt%7Bx-y%7D%3D1%20%5Cend%7Bmatrix%7D%5Cright.%20%5CLeftrightarrow%20%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%20x%2By%3D4%5C%5C%20%20x-y%3D1%20%5Cend%7Bmatrix%7D%5Cright.%20%5CLeftrightarrow%20%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%201%2By%2By%3D4%5C%5C%20%20x%3D1%2By%20%5Cend%7Bmatrix%7D%5Cright.%20%5CLeftrightarrow%20%20%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%202y%3D3%5C%5C%20%20x%3D1%2By%20%5Cend%7Bmatrix%7D%5Cright.%20%5CLeftrightarrow%20%5C%5C%20%5C%5C%20%20%5CLeftrightarrow%20%20%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%20y%3D1.5%5C%5C%20%20x%3D1%2B1.5%20%5Cend%7Bmatrix%7D%5Cright.%20%5CLeftrightarrow%20%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%20y%3D1.5%5C%5C%20%20x%3D2.5%20%5Cend%7Bmatrix%7D%5Cright.)