> removing nested grouping signs
Removing nested grouping signs
Sometimes we are required to simplify an expression containing nested grouping signs. For example, to simply the expression 5x - [ 3x - 2(x - 1)] the nested grouping signs have to be removed first. It can not be simplified without removing the nested grouping signs first.
There are two ways to remove the nested grouping signs. One way is to start with removing the innermost parentheses. Another way is to remove the outermost parentheses first.
Example 1
simplify 5x - [ 3x - 2(x - 1)]
Solution
5x - [ 3x - 2(x - 1)]
= 5x - [3x -2x + 2]
= 5x - [x + 2]
= 5x - x - 2
= 4x - 2
By removing the outermost brackets first, we can get the same result:
5x - [ 3x - 2(x - 1)]
= 5x - 3x + 2(x - 1)
= 5x - 3x + 2x -2
= 4x - 2
Example 2
simplify -[3(x + 1) - 4] + [ 4x - 2(x - 1)]
Solution
-[3(x + 1) - 4] + [ 4x - 2(x - 1)]
= -[3x + 3 - 4] + [4x - 2x + 2]
= -[3x - 1] + [2x + 2]
= -3x + 1 + 2x + 2
= -3x + 2x + 1 + 2
= -x + 3
By removing the outermost brackets first, we can get the same result:
-[3(x + 1) - 4] + [ 4x - 2(x - 1)]
= -3(x + 1) +4 + 4x - 2(x - 1)
= -3x -3 + 4 + 4x -2x + 2
= -3x + 4x - 2x - 3 + 4 + 2
= -x + 3
Example 3
simplify x - {x - [x - (x - 1)]}
Solution
x - {x - [x - (x - 1)]}
= x - {x - [x -x + 1]}
= x - {x -[1]}
= x - {x - 1}
= x - x + 1
= 1
The same result can be achieved by removing the outermost braces first:
x - {x - [x - (x - 1)]}
= x - x + [x - (x - 1)]
= x - x + x - (x - 1)
= x - x + x - x + 1
= 1