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Question Number 114768 by ZiYangLee last updated on 21/Sep/20

$$\mathrm{Solve}\mathrm{for}\mid x-2\mid +\mid 1-x\mid =4$$

Answered by bobhans last updated on 21/Sep/20

$$\left(1\right)x<1\Rightarrow 2-x+1-x=4$$$$3-2x=4;x=-\frac{1}{2}$$$$\left(2\right)1⩽x<2\Rightarrow 2-x+x-1=4$$$$1=4\leftarrow nosolution$$$$\left(3\right)x⩾2\Rightarrow x-2+x-1=4$$$$2x=7;x=\frac{7}{2}$$$$\therefore x=-\frac{1}{2}orx=\frac{7}{2}$$

Answered by mathmax by abdo last updated on 22/Sep/20

$$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\mid \mathrm{x}-2\mid +\mid 1-\mathrm{x}\mid -4=0$$$$\mathrm{x}-\mathrm{\infty }12+\mathrm{\infty }$$$$\mid \mathrm{x}-2\mid -\mathrm{x}+2-\mathrm{x}+2\mathrm{x}-2$$$$\mid \mathrm{x}-1\mid -\mathrm{x}+1\mathrm{x}-1\mathrm{x}-1$$$$\mathrm{f}\left(\mathrm{x}\right)-2\mathrm{x}-1-32\mathrm{x}-7$$$$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{l}-2\mathrm{x}-1\mathrm{if}\mathrm{x}⩽1\\ -3\mathrm{if}1⩽\mathrm{x}⩽2\end{array}$$$$\{2\mathrm{x}-7\mathrm{if}\mathrm{x}⩾2$$$$\mathrm{so}\mathrm{x}⩽1\mathrm{we}\mathrm{get}\mathrm{f}\left(\mathrm{x}\right)=0\Rightarrow -2\mathrm{x}-1=0\Rightarrow \mathrm{x}=-\frac{1}{2}$$$$1⩽\mathrm{x}⩽2\mathrm{f}\left(\mathrm{x}\right)=0\Rightarrow -3=0\mathrm{imposdible}$$$$\mathrm{x}⩾2\mathrm{f}\left(\mathrm{x}\right)=0\Rightarrow 2\mathrm{x}-7=0\Rightarrow \mathrm{x}=\frac{7}{2}$$

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