The Absolute Value of -4: Definition and Other Examples

We show how some of the properties work in the following examples.

• The symmetry property can be shown as |-7|=|7|=7.

• We show multiplicativity of the absolute value.
  \begin{align*}
  |2\times-4|=|-8|=8
  \end{align*}
  and
  \begin{align*}
  |2|\times|-4|=2\times4=8
  \end{align*}
  Thus, $|2\times-4|=|2|\times|-4|$.

• The preservation of division property is shown in this example.
  \begin{align*}
  \left|\dfrac{-4}{2}\right|=|-2|=2
  \end{align*}
  and
  \begin{align*}
  \dfrac{|-4|}{|2|} =\dfrac{4}{2}=2
  \end{align*}
  Then, $\left|\dfrac{-4}{2}\right|=\dfrac{|-4|}{|2|}$.

• Consider the following equations.
  \begin{align*}
  |2+(-4)|=|-2|=2
  \end{align*}
  and
  \begin{align*}
  |2|+|-4|=2+4=6.
  \end{align*}
  Note that $2<6$. Hence, $|2+(-4)|\leq|2|+|-4|$. Thus, the subadditivity property is satisfied.

• Note that $|7-10|=|-3|=3$. Say we take a real number $5$, then we have:
  \begin{align*}
  |7-5|+|5-10|&=|-2|+|-5|
  &=2+5=7.
  \end{align*}
  We know that $3<7$, which means that $|7-10|\leq|7-5|+|5-10|$. Hence, the triangle inequality is satisfied.

• Consider the expressions $|8-(-3)|$ and $\left||8|-|-3|\right|$, and let’s establish their relationship. Note that \begin{align*} |8-(-3)|=|8+3|=|11|=11 \end{align*} and \begin{align*} |(|8|-|-3|=|8-3|)|=|5|=5 \end{align*} Moreover, note that $11>5$. This implies that
  \begin{align*}
  |8-(-3)|\geq\left||8|-|-3|\right|.
  \end{align*}
  Therefore, the last property of absolute value is satisfied in this example.