Algebra and brackets opening

Question

I'm trying to clarify, how to open brackets. Simple example:

$$2a - {4 - [3b -(5a -7 + b)] + 2} = 2a - {4 - 3b +5a +7 - b + 2}$$ As you can see, need to change sign. Opened $()$ changed $-$ to $+$ and vice versa. Any good explanation? Proof, articles ?

Thanks.

Answer 1

You have a mistake in your calculations \begin{align}-(5a-7+b)&=-5a+7-b\\[0.2cm]-[3b-(5a-7+b)]&=-[3b-5a+7-b]=-3b+5a-7+b\end{align}

Answer 2

You only need to change the sign when you have two minuses, since:

$-(-a) = +a$

If you have a plus and a minus, then the minus overrules:

$-(+a) = -a$

$+(-a) = -a$

Answer 3

$$2a - 4 - [3b -(5a -7 + b)] + 2 =2a - 4 - 1\cdot[3b -1\cdot(5a -7 + b)] + 2.$$

Now just multiply out as usual. "Changing signs" isn't a calculation; just remember in such cases that every '$-$' is an abbreviation of '$-1\cdot$'.

If there was a "changing sign rule" we would obtain $$-(-5)^2=+5^2.$$