What does it mean to have an absolute value equal an absolute value? I have no problem reading absolute value equations such as $|x -2| = 2$. 
I know this means that the distance of some real number is $2$ away from the origin. Because the origin splits the number line into a negative side and positive side then the numbers inside the absolute value symbol will be $2$ and $-2$, since those are the only two numbers $2$ units away from the origin. Then, it's just a matter of finding the values of $x$ which will give $2$ and $-2$ inside the absolute value.
Therefore, $|x - 2| = 2$ which is 
$x - 2 = 2$ 
or 
$x - 2 = -2$
And the solutions are $\{0, 4\}$
But when I see $|3x - 1| = |x + 5|$, I have no idea know what this means. I know how to solve it, but I don't know how this relates to the distance from the origin or how to interpret this on a number line. My initial interpretation is to say, "the absolute value of some unknown number is the absolute value of some unknown number," but that doesn't tell me the distance from $0$. 
My Algebra textbook gave the following definition: 
If $|u| = |v|$, then $u = v$ or $u = -v$. 
But I can't really tell why this is the case. 

Maybe trying to answer the following question can help you..
In real line, when do two points $u$ and $v$ have the same absolute value (the same distance away from the origin)?
