Confused about the $\pm$ sign? I have multiple questions about the $\pm$ sign, since it seems to confuse me in general...

Question 1:

Say I have $15=\pm(a+x)$, Can I use the distributive property so it becomes $15=\pm a \pm x$? Or does that mean I went from 2 to 4 solutions?

I get very confused when I encounter this sign in equations I need to simplify. Whenever I need to deal with these kind of situations I tend to just turn it into two equations, $15= a+x$ and $15=-a-x$. But if I were to do that with bigger equations that still need to be simplified it means I'm wasting alot of time since I'm doing twice the labor...

Question 2:

Does the $\pm$ sign only make sense in equations, or can they be used in normal expressions as well (e.g. $\pm y + 3$). But then I wonder, in what scenario would you do something like this? 

Question 3:

$\sqrt {x^2}+37=y+40$

Say I were to simplify $\sqrt{x^2}$ in that equation, I don't know where I should put the $\pm$ sign. Where would I put it? The extra terms 37 and 40 confuse me...
If the $\pm$ sign is confusing you, get rid of it.  If you have $$15 = \pm(a+x)$$ you can turn that into two equations:  $$15 = a+x\\15 = -(a+x)$$ and then deal with the two equations separately, one at a time.  That is exactly the meaning of the $\pm$ sign.



The reason you're confused is because the notation is confusing!  The expression $\pm a \pm b$ is actually ambiguous: in some contexts it means four values, and in other contexts it means two. Sometimes there is a convention that the two $\pm$ signs must represent the same sign; sometimes there isn't.

In your third example, I'd suggest that you write $z = \sqrt{x^2}$ and turn the equation into $z+37 = y + 40$.  Then solve as usual.  When you get to the end, you have $z$ and $y$.  Then you can conclude that $x$ could be either $z$ or $-z$.
what if variable x = +1 and y = -1, since it is a variable, maybe 4x could be 4+1= 5, not multiplication in this state? y= -1, 2y = 2-1 = 1.
4x -2y=3.
4x-2y=3
_____=3
           _

4-2y=4-2y
x = 3
    ______
     4-2y

3             = +1



4-2y

3
-   = +1
3

x=+1 , y=-1