Is the Minus Sign Separating Terms or Part of the Term in Algebra?

[Holocene]

I’m currently on working on solving linear equations, and I’m having some trouble understanding whether a minus sign is actually separating two terms, or if it is actually part of the term.

Here’s something that snagged me, it’s the left member of the equation:

$$\displaystyle{7(2x - 3) - 4(x + 5)}$$

Regarding the minus sign just before the “4”, should I think of that as “separating” the two terms, or should I think of it as actually being a part of the term, and hence the value is -4?

I keep wanting to write: $$\displaystyle{14x - 21 - 4x + 20}$$, but apparently this is false.

It should be $$\displaystyle{14x - 21 - 4x - 20}$$

I’m just confused, because I clearly remember a rule stating that when subtracting a term, every sign has to change. For example:

$$\displaystyle{(7 - 3) - (2 - 1) = 7 - 3 - 2 + 1}$$

Any help would be greatly appreciated!

 

[NerfMonkey]

$$x+5$$ can be thought of as one term, so it's the same as if you were multiplying 4 by any regular number or variable. Because it's one term you multiply the whole thing by 4, which gives you $$-4x-20$$.

That's how I like to think of it, hope that helped.

 

[arildno]

I’m currently on working on solving linear equations, and I’m having some trouble understanding whether a minus sign is actually separating two terms, or if it is actually part of the term.

Here’s something that snagged me, it’s the left member of the equation:

$$\displaystyle{7(2x - 3) - 4(x + 5)}$$

Remember now the basic substitution rule:
Whenever substituting something by means of an identity, you will always get the correct result by setting a parenthesis around the new substitute.

Thus, since 4(x+5)=4x+20 and 7(2x-3)=14x-21, we may write:

7(2x-3)-4(x+5)=(14x-21)-(4x+20)=14x-21-4x-20, by removing the parentheses in the correct manner.

Since substitution with parantheses guarantees you the correct result (assuming you know your identities and how to remove parentheses), a good rule of thumb is always to substitute WITH parenthesis into a "complicated" expression.

 

[Holocene]

Thanks everyone.

 

[robert Ihnot]

You could think of it as an operation in subtraction of the qualtity 4(x+5), but the statement can also be written as 7(2x-3) + (-4)(x+5) and consider it addition of signed numbers.