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\(\frac{x^2+3x-1}{2-x}+x>0\Leftrightarrow\frac{5x-1}{2-x}>0\Rightarrow\frac{1}{5}< x< 2\)
\(\frac{\left(x-1\right)^3\left(x+2\right)^2\left(x+6\right)}{\left(x-7\right)^3\left(x-2\right)^2}\le0\Leftrightarrow\left[{}\begin{matrix}x\le-6\\x=-2\\1\le x< 2\\2< x< 7\end{matrix}\right.\)
Kết hợp lại ta có: \(1\le x< 2\)
a/ ĐKXĐ: \(x\ne\left\{1;3\right\}\)
\(\Leftrightarrow\frac{x+5}{x-1}=\frac{x+1}{x-3}-\frac{8}{\left(x-1\right)\left(x-3\right)}\)
\(\Leftrightarrow\left(x+5\right)\left(x-3\right)=\left(x+1\right)\left(x-1\right)-8\)
\(\Leftrightarrow x^2+2x-15=x^2-9\)
\(\Leftrightarrow2x=6\Rightarrow x=3\) (ktm)
Vậy pt vô nghiệm
b/ ĐKXĐ: \(x\ne1\)
\(\Leftrightarrow\frac{x+1}{\left(x-1\right)\left(x+1\right)}+\frac{2}{x^2+x+1}=\frac{3x^2}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(\Leftrightarrow x^2+x+1+2\left(x-1\right)=3x^2\)
\(\Leftrightarrow2x^2-3x+1=0\Rightarrow\left[{}\begin{matrix}x=1\left(ktm\right)\\x=\frac{1}{2}\end{matrix}\right.\)
c/ ĐKXĐ: \(x\ne\pm4\)
\(\Leftrightarrow\frac{5\left(x^2-16\right)}{\left(x-4\right)\left(x+4\right)}+\frac{96}{\left(x-4\right)\left(x+4\right)}=\frac{2x-1}{x+4}+\frac{3x-1}{x-4}\)
\(\Leftrightarrow5x^2-80+96=\left(2x-1\right)\left(x-4\right)+\left(3x-1\right)\left(x+4\right)\)
\(\Leftrightarrow5x^2+16=5x^2+2x\)
\(\Rightarrow x=8\)
a/ \(\Leftrightarrow\left(x+2\right)^2-3\left|x+2\right|=0\)
\(\Leftrightarrow\left|x+2\right|^2-3\left|x+2\right|=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left|x+2\right|=0\\\left|x+2\right|=3\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=-2\\x+2=3\\x+2=-3\end{matrix}\right.\)
b/
\(\Leftrightarrow\left|x+2\right|^2-3\left|x+2\right|-4=0\)
\(\Leftrightarrow\left(\left|x+2\right|+1\right)\left(\left|x+2\right|-4\right)=0\)
\(\Leftrightarrow\left|x+2\right|-4=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x+2=4\\x+2=-4\end{matrix}\right.\)
c/
\(\Leftrightarrow\left|x^2-3\right|^2-6\left|x^2-3\right|+5=0\)
\(\Leftrightarrow\left(\left|x^2-3\right|-1\right)\left(\left|x^2-3\right|-5\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left|x^2-3\right|=1\\\left|x^2-3\right|=5\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2-3=1\\x^2-3=-1\\x^2-3=5\\x^2-3=-5\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x^2=4\\x^2=2\\x^2=8\\x^2=-2\left(l\right)\end{matrix}\right.\)
d/ ĐKXĐ: ...
\(\Leftrightarrow\frac{\left|x-2\right|^2}{\left(x-1\right)^2}+\frac{2\left|x-4\right|}{x-1}=3\)
Đặt \(\frac{\left|x-2\right|}{x-1}=a\)
\(a^2+2a-3=0\Rightarrow\left[{}\begin{matrix}a=1\\a=-3\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}\left|x-2\right|=x-1\\\left|x-2\right|=-3\left(x-1\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left|x-2\right|=x-1\left(x\ge1\right)\\\left|x-2\right|=3-3x\left(x\le1\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x-2=x-1\left(vn\right)\\x-2=1-x\\x-2=3-3x\\x-2=3x-3\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=\frac{3}{2}\\x=\frac{4}{5}\\x=\frac{1}{2}\end{matrix}\right.\)
e/ ĐKXĐ: ...
Đặt \(\left|\frac{2x-1}{x+2}\right|=a>0\)
\(a-\frac{2}{a}=1\Leftrightarrow a^2-a-2=0\)
\(\Rightarrow\left[{}\begin{matrix}a=-1\left(l\right)\\a=2\end{matrix}\right.\) \(\Rightarrow\left|\frac{2x-1}{x+2}\right|=2\)
\(\Rightarrow\left[{}\begin{matrix}2x-1=2\left(x+2\right)\\2x-1=-2\left(x+2\right)\end{matrix}\right.\)