Download Rules for Addition

Rules for Addition
Notes
If x, y , z ∈ R, then:
I (x + y ) + z = x + (y + z) (Law of associativity)
I x + y = y + x (Law of commutativity)
I x + 0 = x (Law of identity)
I −x + x = 0 (Law of inverses)
(University of Utah)
Math 1050
1/9
Rules Multiplication and the Distributive Law
Notes
If x, y , z ∈ R, then:
I (xy )z = x(yz) (Law of associativity)
I xy = yx (Law of commutativity)
I x · 1 = x (Law of idendity)
1
I If x 6= 0 then
x · x = 1 (Law of inverses)
The Distributive Law synthesises addition and
multiplication:
I x(y + z) = xy + xz (Distributive Law)
Take notes on the other forms of the distributive law.
(University of Utah)
Math 1050
2/9
Rules of Inequalities
Notes
If x, y ∈ R, then:
I If x > 0 and y > 0 then x + y > 0.
I If x > 0 and y > 0 then xy > 0.
I If x ∈ R, then either x > 0 or x < 0 or x = 0.
Summarize the first rule.
Summarize the second rule.
(University of Utah)
Math 1050
3/9
Intervals
Notes
Types of Intervals
Name of set Those x ∈ R contained in the set
[a, b]
a≤x ≤b
(a, b)
a<x <b
[a, b)
a≤x <b
(a, b]
a<x ≤b
[a, ∞)
a≤x
(−∞, b]
x ≤b
(a, ∞)
a<x
(−∞, b)
x <b
(University of Utah)
Math 1050
4/9
Examples - Distributive Law
x
Notes
x
x(y + z)
y +z
xy
xz
y
z
Example 1 ”Forwards” Distributive Law:
3(y + z) = 3y + 3z
(−2)(4y − 5z) = (−2)(4y ) − (−2)(5z) = −8y + 10z
True or false: 19(x + y ) = x + 19y
(University of Utah)
Math 1050
5/9
Examples - Distributive Law
Notes
Example 2 ”Backwards” Distributive Law:
3x + 6y = 3(x + 2y )
10x − 8y + 4z = 2(5x − 4y + 2z)
True or false: −54x + 18y + z = 9(−6x + 2y + z)
The ”backwards” use of the distributive law is also known as
factoring out a common factor.
(University of Utah)
Math 1050
6/9
Examples - Intervals
Notes
Example 3 - Intervals: Intervals must be written with the least of
the two numbers always on the left and the greater of the two
numbers always on the right.
Proper Example: [3, 7]
Improper Example: [7, 3]
True or false: −3 ∈ (−3, 4]
True or false: 2 ∈ (−∞, −1]
True or false: (5, 9] ⊆ [5, 9]
(University of Utah)
Math 1050
Homework Exercises:
7/9
Notes
This space is for in-class homework exercises.
Homework Exercises:
This space is for in-class homework exercises.
Notes