The absolute value [tex]|x|[/tex] returns the "positive version" of a number.
In other words, if the number is positive, it remains positive; if the number is negative, it changes sign.
So, if we want [tex]|x|=9[/tex], we want the "positive version" of x to be 9.
This can happen in two ways: if x is already 9, then its absolute value is still nine. If instead x=-9, its positive value will be 9 again.
In formula, we have
[tex]|x|=9 \iff x=\pm 9[/tex]
because
[tex]|9|=9,\quad |-9|=9[/tex]
