Giải:
\(A=\left|x-5\right|+\left|x+\dfrac{3}{4}\right|\)
\(\Leftrightarrow A=\left|x-5\right|+\left|-x-\dfrac{3}{4}\right|\)
Ta có:
\(\left|x-5\right|+\left|-x-\dfrac{3}{4}\right|\ge\left|x-5+\left(-x-\dfrac{3}{4}\right)\right|\)
\(\Leftrightarrow\left|x-5\right|+\left|-x-\dfrac{3}{4}\right|\ge\left|x-5-x-\dfrac{3}{4}\right|\)
\(\Leftrightarrow\left|x-5\right|+\left|-x-\dfrac{3}{4}\right|\ge\left|-5-\dfrac{3}{4}\right|\)
\(\Leftrightarrow\left|x-5\right|+\left|-x-\dfrac{3}{4}\right|\ge\left|-\dfrac{23}{4}\right|\)
\(\Leftrightarrow\left|x-5\right|+\left|-x-\dfrac{3}{4}\right|\ge\dfrac{23}{4}\)
\(\Leftrightarrow A\ge\dfrac{23}{4}\)
\(\Leftrightarrow A_{Min}\ge\dfrac{23}{4}\)
Dấu "=" xảy ra:
\(\Leftrightarrow\left[{}\begin{matrix}x-5=0\\x+\dfrac{3}{4}=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=5\\x=-\dfrac{3}{4}\end{matrix}\right.\)
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