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.