Here, we shall discuss something called the Triangle inequality which states that $|x+y| \le |x| + |y|$. I thought this is rather important as I see too many students writing things like $|z_1 + z_2| = |z_1| + |z_2|$\$ in complex numbers. This is really scary because it shows that they are not thinking when they are doing the question.

Students can argue that adding two number and applying modulus should give positive so we can split the positive numbers up. Now this is sufficiently true. But then what if we have negative numbers coming into play? Consider this

$|2+(-2)| = 0$

Then we see that $|2|+|-2| = 4$.

So this is a big problem as $4 \ne 0$ and thus we cannot assume that relation above! So students please be really careful!

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