Ta có: \(\left|x+2\right|-\left|x+7\right|=0\Rightarrow\left|x+2\right|=\left|x+7\right|\)
\(\Rightarrow\left[{}\begin{matrix}\left|x+2\right|=x+7\\\left|x+2\right|=-x-7\end{matrix}\right.\)
+) Xét trường hợp |x+2|=x+7 . ĐK: \(x+7>0\Rightarrow x>-7\)\(\Rightarrow\left[{}\begin{matrix}x+2=x+7\\-x-2=x+7\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x-x=7-2\\-x-x=7+2\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}0x=5\\-2x=9\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}0x=5\left(ko-TM\right)\\x=-4,5\left(TM\right)\end{matrix}\right.\)
+) Xét trường hợp |x+2|=-x-7. ĐK: \(-x-7>0\Rightarrow-x>7\Rightarrow x< -7\)
\(\Rightarrow\left[{}\begin{matrix}x+2=-x-7\\-x-2=-x-7\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x+x=-7-2\\-x+x=-7+2\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=-4,5\left(ko-TM\right)\\0x=-5\left(ko-TM\right)\end{matrix}\right.\)
Vậy x=-4,5
cách gọn:
\(\left|x+2\right|-\left|x+7\right|=0\)
\(\Leftrightarrow\left|x+2\right|=\left|x+7\right|\)
\(\Leftrightarrow\left[{}\begin{matrix}x+2=x+7\\x+2=-\left(x+7\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}2=7\left(vôlý\right)\\x+2=-x-7\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x\in\varnothing\\x+x=-7-2\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x\in\varnothing\\2x=-9\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x\in\varnothing\\x=-\dfrac{9}{2}\end{matrix}\right.\)
Vậy \(x=-\dfrac{9}{2}\)
