a) \(\left|x-2001\right|=x-2012\)
\(\Leftrightarrow\left|x-2001\right|-x=-2012\)
\(\Leftrightarrow\left[{}\begin{matrix}x-2011-x=-2012\left(đk:x-2011\ge0\right)\\-\left(x-2001\right)-x=-2012\left(đk:x-2011< 0\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x\in\varnothing\left(đk:x\ge2011\right)\\x=\dfrac{4023}{2}\left(đk:x< 2011\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x\in\varnothing\\x\in\varnothing\end{matrix}\right.\)
\(\Leftrightarrow x\in\varnothing\)
b) \(\left|x-1\right|+\left|x+3\right|=4\)
\(\Leftrightarrow\left[{}\begin{matrix}x-1+x+3=4\left(đk:x-1\ge0;x+3\ge0\right)\\-\left(x-1\right)+x+3=4\left(đk:x-1< 0;x+3\ge0\right)\\x-1-\left(x+3\right)=4\left(đk:x-1\ge0;x+3< 0\right)\\-\left(x-1\right)-\left(x+3\right)=4\left(đk:x-1< 0;x+3< 0\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\left(đk:x\ge1;x\ge3\right)\\x\in R\left(đk:x< 1;x\ge3\right)\\x\in\varnothing\left(đk:x\ge1;x< 3\right)\\x=-3\left(đk:x< 1;x< -3\right)\end{matrix}\right.\)
..........
\(\left|x-1\right|+\left|x+3\right|=4\)
\(\left|x-1\right|\ge0\)
\(\left|x+3\right|\ge0\)
\(\Leftrightarrow\left|x-1\right|+\left|x+3\right|\ge0\)
\(\Leftrightarrow x-1+x+3=4\)
\(2x+2=4\)
\(2x=2\Leftrightarrow x=1\)
