2-8 Inverse of a sum and
simplifying
Inverse of a Sum
What happens when we multiply a rational number by -1?
The product is the additive inverse of the number.
The Property of -1
For any rational number a, 1  a   a
(negative one times a is the additive inverse of a)
the property of -1 enables us to find an equivalent expression for the additive inverse of a sum
Examples
-(3 + x) = -1(3 + x)
= -3 – x
Using the distributive property
-(3x + 2y + 4) = -1 (3x + 2y + 4)
= -3x – 2y – 4
Try this:
a)
-(x+2)
c)
-(a – 7)
b)
Using the distributive property
- (5x +2y +8)
d) -(3c – 4d + 1)
Inverse of a Sum
•
Inverse of a sum property
For any rational numbers a and b,
-(a + b) = -a + -b
(The additive inverse of a sum is the sum of the additive inverses)
The inverse of a sum property holds for differences as well as sums because
any difference can be expressed as a sum. When we apply the inverse of
a sum property we sometimes say that we “change the sign of every term”
Example: rename each additive inverse without using parentheses
-(5 – y) = -5 + y
-(2a – 7b – 6) = -2a + 7b +6
Try this
-(6 – t)
-(-4a + 3t – 10)
-(18 – m – 2n + 4t)
Simplifying Expressions Involving
Parentheses
•
When an expression inside parentheses is added to another expression as in
5x + (2x + 3), the associative property allows us to move the parentheses and
simplify the expression to 7x + 3. When an expression inside parentheses is
subtracted from another expression as in 3x – (4x + 2), We can subtract by adding
the inverse.
Example: Simplify
3x – (4x + 2) = 3x + ( - (4x + 2))
= 3x + (-4x – 2)
= -x – 2
Example
5y – (3y + 4) = 5y -3y – 4
= 2y – 4
3y – 2 – ( 2y – 4) = 3y – 2 – 2y + 4
= (3y – 2y) + ( -2 + 4)
=y+2
Example
x – 3(x + y) = x – 3x + (-3y)
= x – 3x – 3y
= -2x – 3y
3y – 2(4y – 5) = 3y – 8y + 10
= -5y + 10
h) 5x – (3x + 9)
i) 5x – 2y – (2y – 3x – 4)
j.) y – 9(x + y)
k) 5a – 3(7a – 6)
Grouping Symbols
• Some expressions contain more than one grouping symbol.
Parentheses (), Brackets [ ] , and Braces { } are all grouping symbols we use
in algebra. When an expression contains more than one grouping symbol,
we start with the inner more parentheses and work our way out.
[3 – (7+3) = [ 3 – 10]
= -7
[8 – [9 – (12 + 5)]]
= [ 8 – [ 9 – (17)]
= [ 8 – [9 – 17]
= [ 8 – [-8]
= [8 - -8]
= 16
Try this
[ 9 – (6+ 4)]
3(4 + 2) – [ 7 – [ 4 – (6 + 5)]]
[3(4 + 2) + 2x] – [4 (y + 2) – 3 (y – 2)]