# 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)