Simplifying Expressions with Negative Signs: Understanding Parentheses and Signs

[kenewbie]

Silly question but this kind of threw me:

a + 2b - (-a + b) =

So, when you remove the parantheses, you change the sign. So I figured it would go like this:

a + 2b - a - b = b

(by this logic: - (-a + b) = - +a -b = -a -b)

But my books says

a + 2b + a - b = 2a + b

I'm guessing the book is right, but where am I stepping wrong?

k

 

[cristo]

You are basically looking to expand -1\cdot(-a+b)=(-1)(-a)+(-1)(b)

Can you simplify this?

 

[kenewbie]

I thought I did, but it seems I don't :/

(-1)(-a) + (-1)(b) = a + (-b) = a + b

k

 

[GTrax]

You got real close. You correctly figured that (-1)(-a) amounts to the 'a'
So look close at (-1)(b), and note that you think it amounts to 'b', which is wrong!

(-1)(-a) + (-1)(b) = a + (-b) = a + b

is what you were thinking

(-1)(-a) + (-1)(b) = a + (-b) = a - b is what I am thinking.

OK - so you now know that a minus sign in front of anything, (even stuff in brackets) is like saying it is multiplied by -1. Now move on and get it all quicker like this... start again... Take it from the left.
a + 2b - (-a + b)

There's a+2b which we leave alone for now. Work on the -(-a+b)

Think "two minusus multiplied out make a plus" so the first term will be 'a'
Now make the minus sign outside the work on the next term inside.
Ahh, its a minus times a plus, which makes a minus. That results in -b
Put them together. You get a - b

Now finally bring back the a + 2b, and you get a + 2b + a - b

I leave it to you to finally push the a's and b's together, and come up with the final simplest expression.

 

[kenewbie]

(-1)(-a) + (-1)(b) = a + (-b) = a + b

Ugh, that was a nasty little typo, it is supposed to read = a - b.

Thanks for the responses, I appreciate it.

k