Brackets and removing them

Brackets are the master switches of an expression. They overrule BODMAS. Put a bracket around a piece, and that piece must be evaluated first.

Idea

The simplest use of a bracket: $5 + 3 \times 2 = 11$ but $(5 + 3) \times 2 = 16$. The bracket forces the addition before the multiplication.

Removing a $+$ bracket. If a bracket is preceded by $+$ (or nothing), you may remove the bracket without changing any signs inside: $a + (b + c) = a + b + c, \qquad a + (b - c) = a + b - c.$

Removing a $-$ bracket. If a bracket is preceded by $-$, every sign inside flips: $a - (b + c) = a - b - c, \qquad a - (b - c) = a - b + c.$

This is the rule that catches students most often. Imagine $a - (b + c) = a + (-(b+c)) = a - b - c$ , the minus distributes onto every term inside.

Multiplying a bracket. If a number multiplies a bracket, you use the distributive property: $k(b + c) = kb + kc, \qquad k(b - c) = kb - kc.$

Combining these rules lets you simplify even very complicated expressions. Always read the sign in front of the bracket carefully before removing it.

Worked examples

Example 1. Remove brackets and simplify: $20 + (5 - 3)$.

$+$ before the bracket, so signs stay: $20 + 5 - 3 = 22$.

Example 2. Remove brackets: $20 - (5 - 3)$.

$-$ before bracket, so signs flip: $20 - 5 + 3 = 18$. (Notice $20 - 2 = 18$ , same answer.)

Example 3. Simplify $10 - (4 + 2) + (3 - 1)$.

First bracket has $-$: $10 - 4 - 2 + (3 - 1) = 4 + 3 - 1 = 6$. Or just: $10 - 6 + 2 = 6$.

Example 4. Simplify $4(3 + 5) - 2(6 - 1)$ using the distributive property.

$4(3+5) = 12 + 20 = 32$ and $2(6-1) = 12 - 2 = 10$. So $32 - 10 = 22$.

Try it yourself

1. Remove the brackets in $15 + (7 - 4)$ and simplify.
2. Remove the brackets in $15 - (7 - 4)$ and simplify.
3. Simplify $8 - (3 + 2) + (4 - 1)$.
4. Use the distributive property to expand $5(8 + 2)$.
5. Simplify $6(4 - 1) - 2(3 + 2)$.
6. Put brackets in $7 - 4 + 2$ so that the value becomes $1$.
7. Simplify $20 - \{6 - (3 + 1)\}$.
8. Without computing each side, decide which is larger: $40 - (5 + 6)$ or $40 - 5 - 6$.

Activity

Take any value of $a, b, c$ between $1$ and $10$. Verify $a - (b - c) = a - b + c$ for three different choices.

Make a "sign-flip" poster: on one half write expressions before bracket removal, on the other half write after. Highlight which signs flipped.