1. Recall the real number line:
Coordinate of a point
Origin 13 3 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
Neg. real
numbers
Pos. real
numbers

2. < > < > We can use inequalities to describe
intervals of real numbers
<
(recall the symbols?)
>
<
>
Ex: Describe and graph the interval of real
numbers for the inequality given
x > –2
All real numbers greater than
or equal to negative two –2 –1 1 –1 1 –2
Closed bracket – value included in solution.

3. Ex: Describe and graph the interval of real
numbers for the inequality given
0 < x < 3
All real numbers between zero
and three, including zero –1 1 2 3 –1 1 2 3

4. Interval Notation [a, b] closed a < x < b (a, b) open
Bounded Intervals of Real Numbers
(let a and b be real #s with a < b;
a and b are the endpoints of each interval)
Interval
Notation
Interval
Type
Inequality
Notation
Graph
[a, b]
closed
a < x < b a b
(a, b)
open
a < x < b a b
[a, b)
half-open
a < x < b a b
(a, b]
half-open
a < x < b a b

5. 8 8 8 8 Interval Notation [a, ) closed x > a (a, ) open x > a
Unbounded Intervals of Real Numbers
(let a and b be real #s)
Interval
Notation
Interval
Type
Inequality
Notation
Graph
[a, ) 8
closed
x > a a
(a, ) 8
open
x > a a
( , b] 8
closed
x < b b
( , b) 8
open
x < b b

6. Bounded, closed interval
More Examples…
Convert interval notation to inequality notation or vice versa. Find the endpoints and state whether the interval is bounded, its type, and graph the interval.
[–3, 7]
–3 < x < 7
Endpoints: –3, 7
Bounded, closed interval –3 7

7. 8 x < –9 Endpoint: –9 Unbounded, open interval More Examples…
Convert interval notation to inequality notation or vice versa. Find the endpoints and state whether the interval is bounded, its type, and graph the interval.
x < –9
4. (– , –9) 8
Endpoint: –9
Unbounded, open interval –9

8. Additive inverses are two numbers
whose sum is zero (opposites?)
Example:
Multiplicative inverses are two numbers
whose product is one (reciprocals?)
Example:

9. Other Properties from Algebra
Let u, v, and w be real numbers, variables,
or algebraic expressions.
Commutative Property
Addition: u + v = v + u
Multiplication: uv = vu
Associative Property
Addition: (u + v) + w = u + (v + w)
Multiplication: (uv)w = u(vw)

10. Inverse Property
Addition: u + (– u) = 0
Multiplication:
Identity Property
Addition: u + 0 = u
Multiplication: (u)(1) = u
Distributive Property
u(v + w) = uv + uw
(u + v)w = uw + vw

11. Exponential Notation a = a a a … n factors
Let a be a real number, variable, or algebraic
expression and n is a positive integer. Then:
a = a a a … n
n factors n
n is the exponent, a is the base, and a is the
nth power of a, read as “a to the nth power”

12. Properties of Exponents
(All bases are assumed to be nonzero) m n
m + n
1. u u = u u m =
m – n u u n
3. u = 1 1
– n
4. u = u n

13. ( ) Properties of Exponents 5. (uv) = u v 6. (u ) = u u u 7. = v v
(All bases are assumed to be nonzero) m m m
5. (uv) = u v n m mn
6. (u ) = u
( ) m u m u = v m v

14. c x 10 Scientific Notation Where 1 < c < 10,m
Where 1 < c < 10,
and m is any integer
c x 10
Let’s do some practice problems…

15. Guided Practice
Proctor’s brain has approximately 102,390,000,000
Neurons (at least before the rugby season). Write this
number in scientific notation 11
x 10
– 9
2. Write the number x in decimal form

16. ( ) Guided Practice (3x) y ab b 12x y a 3x b 4y
For #3 and 4, simplify the expression.
( ) 2 2 3 2
(3x) y ab 3. 4. 3 –1 5 b
12x y 2 3 a 3x 2 2 b 4y

17. Guided Practice 6.364 x 10 (3.7 x 10 )(4.3 x 10 ) 2.5 x 10
Use scientific notation to multiply:
– 7 6
(3.7 x 10 )(4.3 x 10 ) 5. 7
2.5 x 10 6
6.364 x 10
Homework: odds p. 11 #5-21, 29, 31, 37-51, odd Note: Name and assignment should be written on the top line of your paper.