Tìm x: |x - 2011| = x - 2012
|x - 2010| + |x - 2011| = 2012
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suy ra hai truong hop
1 : x-2011=x-2012
suy ra x-x=2011-2012(loai)
2 : x-2011=-(x-2012)
suy ra : x-2011=-x+2012
2x=2011+2012
2x=4023
x=2011.5
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a) \(\frac{x+4}{2009}+1+\frac{x+3}{2010}+1=\frac{x+2}{2011}+1+\frac{x+1}{2012}\)
\(\frac{x+4+2009}{2009}+\frac{x+3+2010}{2010}=\frac{x+2+2011}{2011}+\frac{x+2+2012}{2012}\)
\(\frac{x+2013}{2009}+\frac{x+2013}{2010}-\frac{x+2013}{2011}-\frac{x+2013}{2012}=0\)
\(\left(x+2013\right).\left(\frac{1}{2009}+\frac{1}{2010}-\frac{1}{2011}-\frac{1}{2012}\right)=0\) (1)
Vì \(\left(\frac{1}{2009}+\frac{1}{2010}-\frac{1}{2011}-\frac{1}{2012}\right)\ne0\)
Nên biểu thức (1) xảy ra khi \(x+2013=0\)
\(x=-2013\)
b) \(\left(x-2011\right)\left(\frac{1}{2010}+\frac{1}{2011}+\frac{1}{2012}-\frac{1}{2013}-\frac{1}{2014}\right)=0\) (2)
Vì \(\left(\frac{1}{2010}+\frac{1}{2011}+\frac{1}{2012}-\frac{1}{2013}-\frac{1}{2014}\right)\ne0\)
Nên biểu thức (2) xảy ra khi \(x-2011=0\)
\(x=2011\)
Ta có: x=2011 \(\Rightarrow\)x+1=2012
\(\Rightarrow A=x^{2011}-\left(x+1\right).x^{2010}\)\(+\left(x+1\right)x^{2009}\)\(-\left(x+1\right)x^{2008}+...\)\(-\left(x+1\right)x^2+\left(x+1\right)x-1\)
=\(x^{2011}\)\(-x^{2011}-x^{2010}+x^{2010}+x^{2009}-x^{2009}-\)...\(-x^2+x^2+x-1\)
= \(x-1=2011-1=2010\)
=
a: \(\Leftrightarrow\left\{{}\begin{matrix}x>=2012\\\left(x-2012-x+2011\right)\left(x-2012+x-2011\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x>=2012\\2x=2023\end{matrix}\right.\Leftrightarrow x\in\varnothing\)
b: Trường hợp 1: x<2010
Pt sẽ là 2010-x+2011-x=2012
=>4021-2x=2012
=>2x=2009
hay x=2009/2(nhận)
TRường hợp 2: 2010<=x<2011
=>x-2010+2011-x=2012
=>1=2012(vô lý)
Trường hợp 3: x>=2011
=>x-2010+x-2011=2012
=>2x=2012+4021=6033
hay x=6033/2(nhận)