Absolute value of a negative number

I was reading 'The method of Coordinates - Gelfand' and in the section about the absolute value of a number, it is stated what follows :

if x > 0, then |x| = x,
if x < 0, then |x| = -x,
if x = 0, then |x| = 0

"Since the points a and -a are located at the same distance from the origin of coordinates, the numbers a and -a have the same absolute value: |x| = |-x|."

I don't understand the second statement : if x < 0, then |x| = -x. Why is the absolute value of a negative number negative if a distance between 2 points, I suppose, can't be negative ? Plus, the author contradicts himself by saying that |x| = |-x|, because if |x| = x and |x| = |-x|, then |-x| = x. Is it a typing error or am I missing something ?

• if x < 0, then |x| = -x Replace $x$ with $-1$ and see what you get for $|-1|$. Hint: it's not negative. – dxiv Sep 9 '18 at 19:19
• Because $-x> 0$ if $x < 0$ – ab123 Sep 9 '18 at 19:20
• $-x$ isn’t necessarily negative, it’s just $-1\cdot x$. – Michael Hoppe Sep 9 '18 at 19:26
• If x = -1, then |-1| = -(-1) = 1. So the author actually could not say that if x < 0, then |x| = x because that would mean, as x is a negative number, that it's modulus is a negative number (because x = -something), which is false. Now I understand, thanks for your help. – Cylian Barichard Sep 9 '18 at 19:54

$$|x|=\max(x,-x)$$
if $x<0$, $-x$ is bigger than $x$.