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Test of Mathematics Solution Subjective 78 -Absolute Value Inequality

Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution Subjective 78 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.


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Problem

For real numbers $ {x}$, $ {y}$ and $ {\displaystyle{z}}$, show that
$ {\displaystyle{|x| + |y| + |z| {\le} |x + y - z| + |y + z - x| + |z + x - y|}}$.


Solution

Applying Ravi transformation
$ {x = a + b}$, $ {y = b + c}$ and $ {z = c + a}$.
Our inequality reduces to $ {\displaystyle{|a + b| + |b + c| + |c + a| {\le} |2b| + |2c| + |2a|}}$.
$ {\Leftrightarrow}$ $ {\displaystyle{|a + b| + |b + c| + |c + a| {\le} 2(|a| + |b| + |c|)}}$.
Now we know, $ {\displaystyle{|m + n| {\le} |m| + |n|}}$.
Applying this we get
L.H.S = $ {\displaystyle{|a + b| + |b + c| + |c + a| {\le} |a| + |b| + |b| + |c| + |c| + |a|}}$
= $ {\displaystyle{2(|a| + |b| + |c|)}}$
= R.H.S (proved)

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