1\\|x-1|>-x-2\\\begin{cases}x-1>-x-2\\x-1-1\\0<1\end{cases}\Rightarrow x>-\frac12\\
2.\;\;\log(x^2-17)-\log(2x-2)=0\\O.O.\Phi.:\begin{cases}x^2-17>0\\2x-2>0\end{cases}\Rightarrow \begin{cases}x>\sqrt{17}\\x<-\sqrt{17}\\x>1\end{cases}\Rightarrow x>\sqrt{17}\approx4,12\\
\log(x^2-17)-\log(2x-2)=0\\\log\left(\frac{x^2-17}{2x-2}\right)=0\\\frac{x^2-17}{2x-2}=1\\x^2-17=2x-2\\x^2-2x-15=0\\x_1=-3,\;\;x_2=5" alt="1.\;\;|x-1|+x+3>1\\|x-1|>-x-2\\\begin{cases}x-1>-x-2\\x-1-1\\0<1\end{cases}\Rightarrow x>-\frac12\\
2.\;\;\log(x^2-17)-\log(2x-2)=0\\O.O.\Phi.:\begin{cases}x^2-17>0\\2x-2>0\end{cases}\Rightarrow \begin{cases}x>\sqrt{17}\\x<-\sqrt{17}\\x>1\end{cases}\Rightarrow x>\sqrt{17}\approx4,12\\
\log(x^2-17)-\log(2x-2)=0\\\log\left(\frac{x^2-17}{2x-2}\right)=0\\\frac{x^2-17}{2x-2}=1\\x^2-17=2x-2\\x^2-2x-15=0\\x_1=-3,\;\;x_2=5" align="absmiddle" class="latex-formula">
![4.\;\;\sin x\cdot\sin(3x)+\cos(4x)=0\\\frac{\cos2x-\cos4x}2+\cos4x=0\\\cos2x+\cos4x=0\\\cos2x+2\cos^22x-1=0\\\cos2x=t,\;\;\cos^22x=t^2,\;\;t\in[-1;\;1]\\2t^2+t-1=0\\D=9,\;t_1=-1,\;t_2=-\frac12\\\cos2x=-1\Rightarrow2x=\pi+2\pi n\Rightarrow x=\frac\pi2+\pin,\;n\in\mathbb{Z}\\\cos2x=\frac12\Rightarrow 2x=\frac\pi3+2\pi n\Rightarrow x=\frac\pi6+\pi n,\;n\in\mathbb{Z}](https://tex.z-dn.net/?f=4.%5C%3B%5C%3B%5Csin+x%5Ccdot%5Csin%283x%29%2B%5Ccos%284x%29%3D0%5C%5C%5Cfrac%7B%5Ccos2x-%5Ccos4x%7D2%2B%5Ccos4x%3D0%5C%5C%5Ccos2x%2B%5Ccos4x%3D0%5C%5C%5Ccos2x%2B2%5Ccos%5E22x-1%3D0%5C%5C%5Ccos2x%3Dt%2C%5C%3B%5C%3B%5Ccos%5E22x%3Dt%5E2%2C%5C%3B%5C%3Bt%5Cin%5B-1%3B%5C%3B1%5D%5C%5C2t%5E2%2Bt-1%3D0%5C%5CD%3D9%2C%5C%3Bt_1%3D-1%2C%5C%3Bt_2%3D-%5Cfrac12%5C%5C%5Ccos2x%3D-1%5CRightarrow2x%3D%5Cpi%2B2%5Cpi+n%5CRightarrow+x%3D%5Cfrac%5Cpi2%2B%5Cpin%2C%5C%3Bn%5Cin%5Cmathbb%7BZ%7D%5C%5C%5Ccos2x%3D%5Cfrac12%5CRightarrow+2x%3D%5Cfrac%5Cpi3%2B2%5Cpi+n%5CRightarrow+x%3D%5Cfrac%5Cpi6%2B%5Cpi+n%2C%5C%3Bn%5Cin%5Cmathbb%7BZ%7D "4.\;\;\sin x\cdot\sin(3x)+\cos(4x)=0\\\frac{\cos2x-\cos4x}2+\cos4x=0\\\cos2x+\cos4x=0\\\cos2x+2\cos^22x-1=0\\\cos2x=t,\;\;\cos^22x=t^2,\;\;t\in[-1;\;1]\\2t^2+t-1=0\\D=9,\;t_1=-1,\;t_2=-\frac12\\\cos2x=-1\Rightarrow2x=\pi+2\pi n\Rightarrow x=\frac\pi2+\pin,\;n\in\mathbb{Z}\\\cos2x=\frac12\Rightarrow 2x=\frac\pi3+2\pi n\Rightarrow x=\frac\pi6+\pi n,\;n\in\mathbb{Z}")
32^{\frac1x}\\2^{\frac1x}>2^{\frac5x}\\\frac1x>\frac5x\Rightarrow 1>5" alt="\sqrt{2^x}}>32^{\frac1x}\\2^{\frac1x}>2^{\frac5x}\\\frac1x>\frac5x\Rightarrow 1>5" align="absmiddle" class="latex-formula">
Последнее неравенство не выполняется ни при каких x, значит, решения нет.+2\end{cases}\rightarrow\begin{cases}2x>