Answer
See below.
Work Step by Step
I use the hint: $(a+b)^2=a^2+b^2+2ab\leq |a|^2+2|a||b|+|b|^2=(|a|+|b|)^2$
This implies that $(a+b)^2\leq(|a|+|b|)^2$, so after taking the square root of both sides: $|a+b|\leq|a|+|b|$, thus we proved what we had to.