These double absoutes are always long and tricky.
|3x-2|-1=|5-x|
There are 4 possibilities that might lead to different answers.
\(1)\quad If\quad3x-2\ge0\quad and \quad 5-x\ge 0\\ \qquad x\ge 2/3\qquad and \qquad x\le 5\\ \qquad \qquad 2/3\le x\le5\\ then\\ 3x-2-1=5-x\\ \qquad\;\; \quad4x=8\\ \qquad\; \qquad x=2\\ So \;\;x=2\;\;is\;\;a\;\;solution\\~\\ 2)\quad If\quad3x-2\le0\quad and \quad 5-x\le 0\\ \qquad x\le 2/3\qquad and \qquad x\ge 5\\ \qquad No\;\;solution\\ \)
\(3)\quad If\quad3x-2\ge0\quad and \quad 5-x\le 0\\ \qquad x\ge 2/3\qquad and \qquad x\ge 5\\ \qquad\;\text{The union is }\;\; x\ge 5\\ \qquad3x-2-1=-(5-x)\\ \qquad3x-3=-5+x\\ \qquad2x=-2\\ \qquad x=-1\\ \qquad \text {This is not greater than 5 so there is no solution here} \)
\(4)\quad If\quad3x-2\le0\quad and \quad 5-x\ge 0\\ \qquad x\le 2/3\qquad and \qquad x\le 5\\ \qquad\;\text{The union is }\;\; x\le 2/3\\ \qquad -3x+2-1=5-x\\ \qquad \qquad -2x=4\\ \qquad \qquad x=-2\\ \qquad \text {This is less than 2/3 so x=-2 is a second solution}\\~\\ \text{So the 2 solutions are } \;\;x=2 \;\;and \;\;x=-2 \)
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