Prove that PSK|a−b|\geq|a|−|b|ZSK.

expeditiupc 2021-12-10 Answered
Prove that \(\displaystyle{\left|{a}−{b}\right|}\geq{\left|{a}\right|}−{\left|{b}\right|}\).

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Expert Answer

Terry Ray
Answered 2021-12-11 Author has 5276 answers
It is called the reverse triangle inequality.
\(\displaystyle{\left|{a}−{b}\right|}\geq{\left|{\left|{a}\right|}−\right|}{b}{\mid}{\mid}\)
\(\displaystyle{\left|{a}\right|}={\left|{a}-{b}+{b}\right|}\le{\left|{a}-{b}\right|}+{\left|{b}\right|}\)
\(\displaystyle{\left|{b}\right|}={\left|{b}-{a}+{a}\right|}\le{\left|{a}-{b}\right|}+{\left|{a}\right|}\)
Thus, we have:
\(\displaystyle-{\left|{a}-{b}\right|}\le{\left|{a}\right|}-{\left|{b}\right|}\le{\left|{a}-{b}\right|}\)
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Tiefdruckot
Answered 2021-12-12 Author has 2128 answers
The length of any side of a triangle is greater than the absolute difference of the lengths of the other two sides:
\(\displaystyle{\left|{\left|{a}\right|}−\right|}{b}{\left|\le\right|}{a}−{b}{\mid}\)
So,
\(\displaystyle{\left|{a}−{b}\right|}\geq{\left|{a}\right|}−{\left|{b}\right|}\)
\(\displaystyle{\left|{a}−{b}\right|}\geq{\left|{b}\right|}−{\left|{a}\right|}\)
That's it.
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