1. Removing a bracket

Generally, there are two rules that have to be considered when removing a bracket:

Rule 1: Neglect a bracket if a plus sign is in front of this bracket.

Example:

$a + (b + c) = a + b + c$

Rule 2: If a minus sign is in front of a bracket then all terms in the bracket change their sign when the bracket is removed.

Examples:

$a - (b + c) = a - b - c$

$a - (b - c) = a -b + c$

$a - (-b + c) = a + b - c$

$a - (-b - c) = a + b + c$

$a - (b + c + d) = a - b - c - d$

etc.

Further examples:

$1. \quad 10 + x - (3 + x) =10+x-3-x=7$

$2. \quad (-x - y) -(4 - x) - (-20 - y) + 14 =-x-y-4+x+20+y+14=30$

$3. \quad x + 2ax + a -(x - 2ax + a) =x+2ax+a-x+2ax-a=4ax $

2. Removing more than one brackets

How can we remove brackets when there is more than one? The following rule is helpful.

Rule 3: For a multiple nesting of brackets, the brackets are removed step by step inside out.

Example:

$15 - [10 - (x - y)] + y=15-[10-x+y]+y=15-10+x-y+y=5+x$

Remark

In the previous example, the interior bracket is removed first according to rule 3. Then, the exterior bracket is removed. Please note that round and rectangular brackets have no different meaning. They are used to facilitate reading an equation.

Further examples

Example 1:

$ -13x - [-2y -(3x - 5y) - 2z - (-4z + 3x)] =-13x-[-2y-3x+5y-2z+4z-3x]$

$=-13x+2y+3x-5y+2z-4z+3x=-7x-3y-2z$

Example 2:

$5ab - (4a-2b-(3b - 10ab-(7a-b)) - 2ab ) = 5ab - (4a-2b-(3b - 10ab-7a+b) - 2ab)$

$ =5ab - (4a-2b-3b+ 10ab+7a-b - 2ab)=5ab - 4a+2b+3b- 10ab-7a+b +2ab =-5ab-11a+6b$

1)

$\left( +6 \right)+\left( -2 \right)=$

2)

$\left( +8 \right)-\left( +11 \right)-\left( -2 \right)=$

3)

$\left( 7x+4y \right)-\left( 7x+4y+z \right)=$

4)

$7-\left( -8+11-2 \right)=$

5)

$3{{x}^{2}}-\left( 6x+4y-7 \right)=$

6)

$-5ab-\left( -a+2-(2ab-2b-4a)+3a-7b \right)=$

7)

$-(1+(2-(3-(4-(5-6-(7+8)))+9)))=$

Solutions

Two additional websites dealing with the theory of removing brackets and discussing additional examples can be found on:

The Mathpage

Mathonweb