Put \(A=\left|m-2\right|+\left|m+3\right|\)

We have: \(A=\left|m-2\right|+\left|m+3\right|=\left|2-m\right|+\left|m+3\right|\)

Apply the inequality \(\left|a\right|+\left|b\right|\ge\left|a\right|+\left|b\right|\), we have:

\(A\ge\left|2-m+m+3\right|=\left|5\right|=5\)

The "=" sign occurs when \(\left\{{}\begin{matrix}2-m\ge0\\m+3\ge0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}m\le2\\m\ge-3\end{matrix}\right.\Rightarrow-3\le m\le2\)

So \(MIN_A=5\) when \(-3\le m\le2\)

Put A=|m−2|+|m+3|

We have: A=|m−2|+|m+3|=|2−m|+|m+3|

Apply the inequality |a|+|b|≥|a|+|b|

, we have:

A≥|2−m+m+3|=|5|=5

The "=" sign occurs when {2−m≥0m+3≥0⇒{m≤2m≥−3⇒−3≤m≤2

So MINA=5

 when −3≤m≤2

Find minimize |m−2|+|m+3|

 ?

By inequality |a|+|b|≥|a+b|

 we have:

|m−2|+|m+3|=|2−m|+|m+3|

≥|2−m+m+3|=5

Done !

Find minimize \(|m-2|+|m+3|\) ?

By inequality \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\) we have:

\(\left|m-2\right|+\left|m+3\right|=\left|2-m\right|+\left|m+3\right|\)

\(\ge\left|2-m+m+3\right|=5\)

Done !