If $c$ is a positive number, $|a|<c \Leftrightarrow -c<a<c$ and $|a|>c \Leftrightarrow a<-c$ or $a>c$

Question

If $c$ is a positive number, $|a|<c \Leftrightarrow -c<a<c$ and $|a|>c \Leftrightarrow a<-c$ or $a>c$.

I am not sure I understand the statement. Can you explain to me why is it true?

Answer 1

Hint: The modulus of a real number $x$ is defined as $$\lvert x \rvert = \begin{cases} x, &\text{if } x\geq 0,\\ -x, &\text{if } x<0. \end{cases}$$ Now, you can show $\Longrightarrow$ and $\Longleftarrow$ by using a case differentiation.

Answer 2

Another way to see it: \begin{align*} |a| < c \Longleftrightarrow |a|^{2} < c^{2} \Longleftrightarrow a^{2} - c^{2} < 0 \Longleftrightarrow (a-c)(a+c) < 0 \Longleftrightarrow -c < a < c \end{align*}