a, Ta có :
\(\left|x+3\right|+\left|x-7\right|=\left|x+3\right|+\left|7-x\right|\)
Áp dụng bất đẳng thức \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\) ta có :
\(\left|x+3\right|+\left|7-x\right|\ge\left|x+3+7-x\right|=\left|10\right|=10\)
Dấu "=" xảy ra khi :
\(\left(x+3\right)\left(7-x\right)\ge0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x+3\ge0\\7-x\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}x+3\le0\\7-x\le0\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge-3\\7\ge x\end{matrix}\right.\\\left\{{}\begin{matrix}x\le-3\\7\le x\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}-3\le x\le7\\x\in\varnothing\end{matrix}\right.\)
Vậy ..
Vì \(\left|x+3\right|\ge0\) và \(\left|x-7\right|\ge0\)
\(\Rightarrow\left|x+3\right|+\left|x-7\right|\ge0\)
Dấu = khi : \(\left\{{}\begin{matrix}\left|x+3\right|=0\\\left|x-7\right|=0\end{matrix}\right.\) \(\)
\(\Rightarrow\left\{{}\begin{matrix}x=0-3\\x=0+7\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=-3\\x=7\end{matrix}\right.\)
\(\Rightarrow\) GTNN của \(\left|x+3\right|+\left|x-7\right|\) = 0 khi \(x\in\left\{-3,7\right\}\)
Vì \(\left\{\left|x+1\right|,\left|x+5\right|,\left|x+8\right|,\left|x+10\right|\right\}\ge0\)
\(\Rightarrow\left|x+1\right|+\left|x+5\right|+\left|x+8\right|+\left|x+10\right|\ge0\)
Dấu = khi \(\left\{{}\begin{matrix}\left|x+1\right|=0\\\left|x+5\right|=0\\\left|x+8\right|=0\\\left|x+10\right|=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=-1\\x=-5\\x=-8\\x=-10\end{matrix}\right.\)
\(\Rightarrow\) GTNN của \(\left|x+1\right|+\left|x+5\right|+\left|x+8\right|+\left|x+10\right|=0\)
khi \(x\in\left\{-1,-5,-8,-10\right\}\)
2
bạn ghi sai đề nên mình sẽ chữa lại đề theo 2 cách để bạn lựa chọn 
C1 Tìm GTNN của : \(\left|x+5\right|+\left|x-3\right|\)
Vì \(\left|x+5\right|\ge0,\left|x-3\right|\ge0\)
\(\Rightarrow\) \(\left|x+5\right|-\left|x-3\right|\ge0\)
Dấu = khi \(\left\{{}\begin{matrix}\left|x+5\right|=0\\\left|x-3\right|=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=-5\\x=3\end{matrix}\right.\)
\(\Rightarrow\) GTNN của \(\left|x+5\right|+\left|x-3\right|\) = 0 khi \(x\in\left\{-5,3\right\}\)
C2 Tìm GTNN của \(\left|x+5\right|+\left|x-3\right|\)
Vì \(\left|x+5\right|\ge0,\left|x-3\right|\ge0\)
\(\Rightarrow\left|x+5\right|+\left|x-3\right|\ge0\)
Dấu = khi \(\left\{{}\begin{matrix}x+5=0\\x-3=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=-5\\x=3\end{matrix}\right.\)
\(\Rightarrow\) GTNN của \(\left|x+5\right|+\left|x-3\right|\) = 0 khi \(x\in\left\{-5,3\right\}\)
