About a question with parenthesis

[d.mehdoi]

Hello everyone, first of all I am sorry for any mistakes in my message. My English is not perfect yet.

I have a question which exactly looks like this:

2(x-3(x-2)) so it is mainly about the use of parenthesis in Maths. Very basic question.

While solving this question, I start with parenthesis as it's how it should be according to order of operations and multiply (x-2) by 3, and have 3x-6. But when I look at the answer, I see they multiply -3 by (x-2).

The problem is, I see that - as minus, rather than negative sign. Because as far as I know if it were negative then it was supposed to be written as (-3) in order to avoid a confusion.

Now, how am I supposed to know if that (x-3(x-2)) is x minus 3 times (x-2), or x times negative 3 times (x-2)?

I know it is a very basic question but I recently started learning Maths and it probably explains my confusion. Am I supposed to take the operation sign, which is minus here, into consideration while multiplying 3 by (x-2), or is it already negative 3 there? How can I distinguish these two if they write them the same in the question?

Thank you for your help and have a nice day!

 

[Dr.Peterson]

Hello everyone, first of all I am sorry for any mistakes in my message. My English is not perfect yet.

I have a question which exactly looks like this:

2(x-3(x-2)) so it is mainly about the use of parenthesis in Maths. Very basic question.

While solving this question, I start with parenthesis as it's how it should be according to order of operations and multiply (x-2) by 3, and have 3x-6. But when I look at the answer, I see they multiply -3 by (x-2).

The problem is, I see that - as minus, rather than negative sign. Because as far as I know if it were negative then it was supposed to be written as (-3) in order to avoid a confusion.

Now, how am I supposed to know if that (x-3(x-2)) is x minus 3 times (x-2), or x times negative 3 times (x-2)?

I know it is a very basic question but I recently started learning Maths and it probably explains my confusion. Am I supposed to take the operation sign, which is minus here, into consideration while multiplying 3 by (x-2), or is it already negative 3 there? How can I distinguish these two if they write them the same in the question?

Thank you for your help and have a nice day!

The easiest way to think of this is to rewrite 2(x - 3(x - 2)) as 2(x + (-3)(x - 2)). That is, subtracting a number is the same as adding the opposite: a - b = a + (-b). So we do distribute a -3, but then we add the result to what comes before it.

It does not mean 2(x(-3)(x - 2)), that is, 2 times x times -3 times (x-2), because a "-" is always interpreted as a subtraction if possible; it only means negation when there is nothing on the left to subtract from. So here, the "-" does mean subtraction, as you took it; but that amounts to a negation and an addition.

You could also distribute only the 3, but then you still have to subtract: 2(x - 3(x - 2)) = 2(x - (3x - 6)). Then you still have to distribute the subtraction; that is, you subtract the 3x and add the 6 (subtracting a negative 6). Here again, it is easiest to think of it as 2(x - (3x + (-6))).

 

[lev888]

It doesn't matter, the result is the same. You can multiply by (-3) and add the product to x or multiply by 3 and subtract the product from x.
1. Minus: 2(x-3(x-2)) = 2(x-(3x-6)) = 2(x-3x+6) = 2(-2x+6) = -4x + 12
2. Negative: 2(x-3(x-2)) = 2(x-3x+6) = 2(-2x+6) = -4x + 12

 

[d.mehdoi]

I understand! Thank you very much for the answers Peterson and lev888!