Question

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с решением, пожалуйста
 sqrt{x-2 sqrt{x-1}} - sqrt{x+2 sqrt{x-1}}

Answer 1

$displaystyle sqrt{x-2sqrt{x-1}}-sqrt{x+2sqrt{x-1}}=sqrt{(x-1)-2sqrt{x-1}+1}-\\\-sqrt{(x-1)+2sqrt{x-1}+1}=sqrt{(sqrt{x-1}-1)^2}-sqrt{(sqrt{x-1}+1)^2}\\\=|sqrt{x-1}-1|-|sqrt{x-1}+1|\\\1),,,sqrt{x-1}-1 textgreater 0\\sqrt{x-1} textgreater 1\\x-1 textgreater 1\\x textgreater 2\\sqrt{x-1}-1-(sqrt{x-1}-1)=-1-1=-2\\\2)sqrt{x-1}-1 leq 0\\sqrt{x-1} leq 1\\x-1 leq 1\\x leq 2$

$text{ODZ}:x geq 1quad rightarrow quad xin[1;2]\\-sqrt{x-1}+1-(sqrt{x-1}-1)=-2sqrt{x-1}\\\|sqrt{x-1}-1|-|sqrt{x-1}+1|=boxed{ left { {{-2sqrt{x-1},,,,xin[1;2]} atop {-2,,,,,,,,,,,,,,,x textgreater 2}} right. }$