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Absolute Value Inequality (I.S.I. Tomato subjective 78)

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)

August 3, 2015
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