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)