1.
a. \(\dfrac{3x-2}{4}+\dfrac{x+3}{2}=\dfrac{x-1}{3}-\dfrac{-x-1}{12}\)
\(\Leftrightarrow\) \(\dfrac{3\left(3x-2\right)}{12}\) + \(\dfrac{6\left(x+3\right)}{12}=\dfrac{4\left(x-1\right)}{12}-\dfrac{-x-1}{12}\)
\(\Leftrightarrow\) \(3\left(3x-2\right)+6\left(x+3\right)=4\left(x-1\right)-\left(-x-1\right)\)
\(\Leftrightarrow9x-6+6x+18=4x-4-x+1\)
\(\Leftrightarrow9x-6+6x+18-4x+4+x-1=0\)
\(\Leftrightarrow12x+15=0\)
\(\Leftrightarrow x=\dfrac{-15}{12}\)
\(S=\left\{\dfrac{-15}{12}\right\}\)
b. \(\left(3x+1\right)\left(x-2\right)=\left(x-2\right)\left(x+1\right)\)
\(\Leftrightarrow3x^2-6x+x-2=x^2+x-2x-2\)
\(\Leftrightarrow3x^2-6x+x-2-x^2-x+2x+2=0\)
\(\Leftrightarrow2x^2-4x=0\)
\(\Leftrightarrow2x\left(x-2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=0\\x-2=0\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=2\end{matrix}\right.\)
\(S=\left\{0,2\right\}\)
c. \(\dfrac{2}{x-1}+\dfrac{2}{x+1}-\dfrac{2x^2+2}{x^2-1}=0\)
\(\Leftrightarrow\dfrac{2}{x-1}+\dfrac{2}{x+1}-\dfrac{2x^2+2}{\left(x-1\right)\left(x+1\right)}=0\left(1\right)\)
\(ĐKXĐ\left\{{}\begin{matrix}x-1\ne0\\x+1\ne0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\ne1\\x\ne-1\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow\dfrac{2\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}+\dfrac{2\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}-\dfrac{2x^2+2}{\left(x-1\right)\left(x+1\right)}=0\)
\(\Leftrightarrow2\left(x+1\right)+2\left(x-1\right)-2x^2+2=0\)
\(\Leftrightarrow2x+2+2x-2-2x^2+2=0\)
\(\Leftrightarrow-2x^2+4x+2=0\)
Pt vô nghiệm
d. \(\left(x-2\right)^2-x+3>\left(x-1\right)\left(x+3\right)-2x+5\)
\(\Leftrightarrow x^2-4x+4-x+3>x^2+3x-x-3-2x+5\)
\(\Leftrightarrow x^2-4x+4-x+3-x^2-3x+x+3+2x-5>0\)
\(\Leftrightarrow-5x+5>0\)
\(\Leftrightarrow-5x>-5\)
\(\Leftrightarrow x< 1\)
\(S=\left\{x\uparrow x< 1\right\}\)