Brackets in calculations: plus, times and minus brackets

Brackets are the top level of the order of operations, so they get priority when we work out calculations and solve equations. The sign in front of the bracket ($+$, $-$ or $\times$) decides what kind of bracket it is, and so also how we open it. Opening a bracket just means working it out. There are three cases, and only one of them has a trick in it.

When do I use this?

Every time a bracket turns up in a calculation or an equation. With only numbers inside, you can often just work the bracket out. But as soon as there are letters inside, you can't, and then you need to know what the sign in front does. The letters case with a times in front is covered in more depth in expanding a single bracket.

Plus brackets

$$3 + (3 + 2)$$

With plus brackets like this one, we can simply work out what's inside the bracket: $3 + 2 = 5$, and then $3 + 5 = 8$. We don't have to take anything special into account. It's the other types of brackets where we have to do things differently.

Times brackets

A bracket with a times sign in front, like this one:

$$3 \times (2x + 3y)$$

Here we have to multiply into the bracket, meaning we multiply 3 by both terms inside the bracket. So:

$$3 \times 2x \quad \text{and} \quad 3 \times 3y$$

$$6x + 9y$$

When we can't just work out what's inside the bracket, because there are letters in there, we have to multiply into it.

If there had only been numbers inside the bracket, like

$$3 \times (2 + 3)$$

we could simply work out the bracket first and then multiply by 3. In this example we'd start with the bracket, adding 2 and 3:

$$3 \times (5)$$

and then multiply:

$$3 \times 5 = 15$$

If we'd multiplied into the bracket instead, we'd have got

$$3 \times 2 + 3 \times 3 = 6 + 9 = 15$$

We multiply 3 by 2, and then 3 by 3. It gives us the same result. So with numbers only, either way works.

Minus brackets

$$3 - (2 + 3)$$

This is a minus bracket, since there's a minus in front. Here we could easily start by working out what's inside the bracket, $2 + 3 = 5$, and then subtract it from 3 to get $-2$. But when the bracket gets more complicated, we have to do it another way. We have to flip the signs, plus and minus, on all the terms inside the bracket:

$$3 - (2x + 3y)$$

$2x$ becomes $-2x$, and $+3y$ becomes $-3y$. Remember here that there's an invisible plus in front of $2x$. That's why it becomes $-2x$:

$$3 - 2x - 3y$$

It can seem strange to flip the signs, but there's a good reason for it. A bracket, or anything else, can always be written with a $\times 1$ in front of it, because the bracket is there 1 time. So we can also write this example as:

$$3 - 1 \times (2x + 3y)$$

Here we can see that we have a times bracket with $-1$. When we multiply $-1$ into the bracket, we see that the signs inside the bracket simply flip:

$$3 + (-2x) - 3y$$

So that's why we can just flip the signs when we open a minus bracket. It isn't a separate rule to learn. It's the times bracket rule with $-1$ as the number in front.

Worked examples

A plus bracket. $3 + (3 + 2) = 3 + 5 = 8$.

A times bracket with numbers. $3 \times (2 + 3)$. Bracket first: $3 \times 5 = 15$. Or multiply in: $3 \times 2 + 3 \times 3 = 6 + 9 = 15$. Same answer.

A times bracket with letters. $3 \times (2x + 3y)$. We can't work out the inside, so we multiply in: $6x + 9y$.

A minus bracket with numbers. $3 - (2 + 3)$. Bracket first: $3 - 5 = -2$. Or flip the signs: $3 - 2 - 3 = -2$. Same answer again.

A minus bracket with a minus inside. $10 - (4 - 1)$. Bracket first: $10 - 3 = 7$. Or flip every sign inside: the $4$ becomes $-4$ and the $-1$ becomes $+1$, so $10 - 4 + 1 = 7$. Every sign flips, including the ones that were already minus.

A minus bracket with letters. $3 - (2x + 3y) = 3 - 2x - 3y$. Here flipping the signs is the only way, because we can't work out the inside.

Common mistakes

• "A plus bracket needs special handling." It doesn't. You just work out what's inside.
• "I can work out the inside of $3 \times (2x + 3y)$ first." You can't, because $2x + 3y$ isn't a single number. With letters inside, you have to multiply into the bracket.
• "With a minus bracket I just work out the inside." With numbers only, yes. With letters inside you can't, so you flip the signs of every term instead.
• "In $3 - 2x$ the minus belongs to the 3." It belongs to the $2x$. There's an invisible plus in front of a term with no sign, and the sign in front of a term travels with it.
• "Flipping the signs is a magic rule." It's the $\times 1$ trick: $3 - (2x + 3y)$ is $3 - 1 \times (2x + 3y)$, a times bracket with $-1$, and $-1$ multiplied into the bracket flips every sign.

Related

The pyramid that puts brackets at the top is in order of operations (BIDMAS). The times bracket with letters, $3 \times (2x + 3y) = 6x + 9y$, is the start of algebra and gets its own guide on expanding a single bracket, which lives under simplifying algebraic expressions.