Absolute value manipulation

Question

Is it always true to say that if $|a|=b$ then $a=\pm b$ or am I missing something?

Answer 1

Absolute value is defined as (if we talk about $a\in\mathbb{R}$, i.e real numbers):

$$|a| = \begin{cases}{a} & a\geq0 \\ {-a} & a<0 \end{cases}$$

Think of it as the distance of $a$ from $0$.

So for example, $|-5|=5, |6|=6$. So if we write $b=|a|$ we want to say "$b$ is in the same distance from $0$ as $a$". Note that absolute value is always a non negative number (because negative distance doesn't make sense).

For more information, take a look at Absolute value.

Answer 2

Yes, it's true.

$|a|= a$ or $-a$, so, if $|a|=b$, then $b=a$ or $-a$, which means $a=b$ or $-b$.