\(2009-|x-2009|=x\)
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8 tháng 9 2023
\(\dfrac{\left(2009-x\right)^2+\left(2009-x\right)\left(x-2010\right)+\left(x-2010\right)^2}{\left(2009-x\right)^2-\left(2009-x\right)\left(x-2010\right)+\left(x-2010\right)^2}=\dfrac{19}{49}\left(1\right)\)
\(Đkxđ:x\ne2009;x\ne2010\)
Đặt \(t=x-2010\left(t\ne0\right)\)
\(\Rightarrow2009-x=-\left(t+1\right)\)
\(\left(1\right)\Leftrightarrow\dfrac{\left(t+1\right)^2-\left(t+1\right)t+t^2}{\left(t+1\right)^2+\left(t+1\right)t+t^2}=\dfrac{19}{49}\)
\(\Leftrightarrow\dfrac{t^2+2t+1-t^2-t+t^2}{t^2+2t+1+t^2+t+t^2}=\dfrac{19}{49}\)
\(\Leftrightarrow\dfrac{t^2+t+1}{3t^2+3t+1}=\dfrac{19}{49}\)
\(\Leftrightarrow49t^2+49t+49=57t^2+57t+19\)
\(\Leftrightarrow8t^2+8t-30=0\)
\(\Leftrightarrow4t^2+4t-15=0\)
\(\Leftrightarrow\left(4t^2+4t+1\right)-16=0\)
\(\Leftrightarrow\left(2t+1\right)^2=16=4^2\)
\(\Leftrightarrow\left[{}\begin{matrix}2t+1=4\\2t+1=-4\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}t=\dfrac{3}{2}\\t=-\dfrac{5}{2}\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x-2010=\dfrac{3}{2}\\x-2010=-\dfrac{5}{2}\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{4023}{2}\\x=\dfrac{4015}{2}\end{matrix}\right.\)
NT
0
\(2009-|x-2009|=x\)
\(\orbr{\begin{cases}2009-\left(x-2009\right)\left(x\ge2009\right)\\2009-\left(2009-x\right)=x\left(x\le2009\right)\end{cases}}\)
Vậy \(x\le2009\)
TH1: \(\left|x-2009\right|=x-2009\)\(\Leftrightarrow x\ge2009\)
\(\Rightarrow2009-\left(x-2009\right)=x\)\(\Leftrightarrow2009-x+2009=x\)
\(\Leftrightarrow2x=4018\)\(\Leftrightarrow2x=2009\)( thoả mãn \(x\ge2009\))
TH2: \(\left|x-2009\right|=-\left(x-2009\right)\)\(\Leftrightarrow x< 2009\)
\(\Rightarrow2009+\left(x-2009\right)=x\)\(\Leftrightarrow2009+x-2009=x\)\(\Leftrightarrow x=x\)
Vậy \(x\le2009\)