Recall the Recall the real number line:real number line:

0 1 2 3 4 5 6-1-2-3-4-5-6

OriginOrigin

Pos. realPos. realnumbersnumbers

Neg. realNeg. realnumbersnumbers

Coordinate of a pointCoordinate of a point

13

3

We can use inequalities to describeWe can use inequalities to describeintervals intervals of real numbersof real numbers

(recall the (recall the symbolssymbols?)?)<< >><< >>

Ex: Describe and graph the interval of real numbers for the inequality given

1. x > –2 All real numbers greater thanAll real numbers greater thanor equal to negative twoor equal to negative two

10–1–2

10–1–2

Closed bracket – value included in solution.Closed bracket – value included in solution.

We can use inequalities to describeWe can use inequalities to describeintervals intervals of real numbersof real numbers

(recall the (recall the symbolssymbols?)?)<< >><< >>

Ex: Describe and graph the interval of real numbers for the inequality given

2. 0 < x < 3 All real numbers between zeroAll real numbers between zeroand three, including zeroand three, including zero

210–1 3

210–1 3

Interval NotationInterval NotationBounded Intervals of Real NumbersBounded Intervals of Real Numbers

(let a and b be real #s with a < b;(let a and b be real #s with a < b;a and b are the a and b are the endpoints endpoints of each interval)of each interval)

IntervalNotation

IntervalType

InequalityNotation 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

Interval NotationInterval NotationUnbounded Intervals of Real NumbersUnbounded Intervals of Real Numbers

(let a and b be real #s)(let a and b be real #s)

IntervalNotation

IntervalType

InequalityNotation Graph

[a, ) closed x > aa

(a, ) open x > aa

( , b] closed x < bb

( , b) open x < bb

88

88

More Examples…More Examples…

Convert interval notation to inequality notation or vice Convert interval notation to inequality notation or vice versa. Find the endpoints and state whether the versa. Find the endpoints and state whether the interval is bounded, its type, and graph the interval.interval is bounded, its type, and graph the interval.

3. –3 < x < 7 [–3, 7]

Endpoints: –3, 7

Bounded, closed interval

–3 0 7

More Examples…More Examples…

Convert interval notation to inequality notation or vice Convert interval notation to inequality notation or vice versa. Find the endpoints and state whether the versa. Find the endpoints and state whether the interval is bounded, its type, and graph the interval.interval is bounded, its type, and graph the interval.

4. (– , –9) x < –9

Endpoint: –9

Unbounded, open interval

0–9

8

Some new/old info…

Consider the magically appearing expression below:

32x p 3b

Constants

Variables

AlgebraicExpression

FactoredFactoredFormForm

ExpandedExpandedFormForm

ExpandedExpandedFormForm

FactoredFactoredFormForm

3y z c 3yc z c

3 3a z a w 3a z w

Additive inverses Additive inverses are two numbersare two numberswhose sum is zero (whose sum is zero (opposites?opposites?))

Example:Example:

Multiplicative inverses Multiplicative inverses are two numbersare two numberswhose product is one (whose product is one (reciprocals?reciprocals?))

Example:Example:

Other Properties from AlgebraLet 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)

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

Exponential Notation

Let a be a real number, variable, or algebraic expression and n is a positive integer. Then:

a = a a a … a,

n factors

n

n is the exponent, a is the base, and a is thenth power of a, read as “a to the nth power”

n

Properties of Exponents(All bases are assumed to be nonzero)

1. u u = um

n

m + n

2. =uu

m

um – n

3. u = 10

4. u =– n

u

1n

n

Properties of Exponents(All bases are assumed to be nonzero)

5. (uv) = u v

6. (u ) = u

7. =vu

m m m

m n mn

( )m

vum

m

Scientific Notation

c x 10m Where 1 < c < 10,

and m is any integer

Let’s do some practice problems…Let’s do some practice problems…

Guided Practice

1. Proctor’s brain has approximately 102,390,000,000 Neurons (at least before the rugby season). Write this number in scientific notation

2. Write the number 8.723 x 10 in decimal form– 9

1.0239 x 101.0239 x 101111

0.0000000087230.000000008723

Guided Practice

For #3 and 4, simplify the expression.

3. 4.(3x) y

12x y

2 3

5–1ab( )2

b3

2

a

b

2

2

3x

4y

3

2

Homework: p. 11-12 5-31 odd, 37-63 odd Note: Name and assignment should be written on the top line of you paper.

Use scientific notation to multiply:

5. (3.7 x 10 )(4.3 x 10 )

2.5 x 10

– 7 6

7

6.364 x 10 6

Guided Practice