Confused about equations with absolute values

[Nat3]

My calc book rewrites this equation:

$$|y|=e^c|x|$$

As this:

$$y=\pm e^cx$$

But that doesn't really make any sense to me. I know I should understand why we're allowed to do that, but I don't. Could someone please try to explain it to me?

I really appreciate your help, thanks!

 

[Nat3]

Is it because:

$$|y| = e^c|x| = |e^cx|$$

And there are four cases:

$$y = e^cx$$

$$y = -e^cx$$

$$-y = e^cx$$

$$-y = -e^cx$$

With the inner two and outer two being equivalent, respectively, resulting in:

$$y = e^cx$$

$$y = -e^cx$$

Which can be written as (?):

$$y = \pm e^cx$$

 

[The_Duck]

First consider the simpler equation

$$|y|=|x|$$

Suppose you know the value of x. What values of y would make the equation true?

 

[HallsofIvy]

Nat3, in LaTex, use { } to group. That is, use e^{cx} to get $e^{cx}$. e^cx gives $e^cx$.