Answer
See the steps.
Work Step by Step
(a)
For $x=2$ and $y=3$
$$|x+y| = |2+3| = 5 \\ |x|+|y| = |2|+|3| = 5$$
$\therefore$ The inequality holds true.
For $x=-2$ and $y=-3$
$$|x+y| = |-2-3| = |-5| = 5 \\ |x|+|y| = |-2|+|-3| = 5$$
$\therefore$ The inequality holds true.
For $x=-2$ and $y=3$
$$|x+y| = |-2+3| =1 \\ |x|+|y| = |-2|+|3| = 5$$
$\therefore$ The inequality holds true.
(b)
\[|x+y] = \left\{
\begin{array}{lr}
|x|+|y| & : \text{x and y of the same sign} \\
||x|-|y|| & : \text{x and y of different signs} \\
\end{array}
\right.
\]
For $x$ and $y$ of the same sign:
$$|x+y] =|x|+|y|$$
For $x$ and $y$ of different signs:
$$|x+y] >||x|-|y||$$
$\therefore $ The Triangle Inequality is true for all real numbers $x$ and $y$.
