# reverse triangle inequality

Given the reverse triangle inequality, mod(a+b+c) is greater than or equal to mod(a)- mod(b) - mod(c).

Suppose a, b, c are complex numbers such that mod(a)=4, mod(b)=10 and mod(c)=1. What is the smallest possible value that mod(a+b+c) can attain? Justify your answer.

I know the answer is 5, and that it relates to b's dominance over the over numbers but am struggling to explain how this is the case?

Thank you

• What is mod(...)? – hmakholm left over Monica Oct 16 '15 at 15:57
• The absolute value of... Apologies for my notation, I'm completely new to this! – B.Marsden Oct 16 '15 at 15:58
• Actually, mod is the complex modulus... – Michael Burr Oct 16 '15 at 15:58

Step 1: $|a+b+c|\geq|b|-|a+c|$ by the reverse triangle inequality.
Step 2: $|a+c|\leq |a|+|c|$ by the triangle inequality.
Step 3: $|a+b+c|\geq|b|-|a+c|\geq|b|-|a|-|c|=10-4-1=5$.