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\begin{enumerate}[(a)]
    \item State the triangle inequality for two numbers $a$ and $b$.
    \item Find an example of two numbers $a$ and $b$ so that $|a+b| = |a| + |b|$, and a different pair of numbers $c$ and $d$ so that $|c + d| \neq |c| + |d|$.
    \item Among the conditions $a>0$, $a<0$, $b>0$, $b<0$, find two of them that will guarantee that $|a+b| \neq |a| + |b|$. Also find a second pair of these conditions with the same result.
    \item A student was faced with the problem ``Solve $|x-5|<8$." The student wrote
    \[ |x-5| = |x + (-5)| = |x| + |-5| = |x| + 5 \] and then solved $|x| + 5 < 8$, which gave $|x| < 3$, leading to $-3 < x < 3$. How would you explain to this student that this is incorrect? How would you correct this?
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