# Range Of $\frac{|a+b|}{|a|+|b|}+\frac{|b+c|}{|b|+|c|}+\frac{|c+a|}{|c|+|a|}$

Find the range of $\dfrac{|a+b|}{|a|+|b|}+\dfrac{|b+c|}{|b|+|c|}+\dfrac{|c+a|}{|c|+|a|}$

$a$,$b$,$c$ are real numbers, where $a\neq 0$ , $b\neq 0$ , $c\neq 0$

• – math110 Sep 2 '13 at 9:20

Hint: At least two of the numbers $a$, $b$, $c$ have the same sign.
Therefore $$1 \le \dfrac{|a+b|}{|a|+|b|}+\dfrac{|b+c|}{|b|+|c|}+\dfrac{|c+a|}{|c|+|a|} \le 3.$$
Without loss of generality we may assume $a, b > 0$. Then if $c>0$ then the sum equals 3. Otherwise the first term is 1 and each of the two remaining terms can be equal to any number between $0$ and $1$.