Recall the real number line: Origin Pos. real numbers Neg. real numbers Coordinate of a point 13 3

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 1. x > –2 All real numbers greater than or equal to negative two 10–1–2 10–1 –2 Closed bracket – value included in solution.

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 2. 0 < x < 3 All real numbers between zero and three, including zero 210–

Interval Notation 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]closeda < x < b ab (a, b)opena < x < b ab [a, b)half-opena < x < b ab (a, b]half-opena < x < b ab

Interval Notation Unbounded Intervals of Real Numbers (let a and b be real #s) Interval Notation Interval Type Inequality Notation Graph [a, )closedx > a a (a, )openx > a a (, b]closedx < b b (, b)openx < b b

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. –3 < x < 7 [–3, 7] Endpoints: –3, 7 Bounded, closed interval –307

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. 4. (–, –9) x < –9 Endpoint: –9 Unbounded, open interval 0–9 8

Some new/old info… Consider the magically appearing expression below: Constants Variables Algebraic Expression

Factored Form Expanded Form Expanded Form Factored Form

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

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)

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 the nth power of a, read as “a to the nth power” n

Properties of Exponents (All bases are assumed to be nonzero) 1. u u = u m n m + n 2. = u u m u m – n 3. u = u = – n u 1 n n

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

Scientific Notation c x 10 m Where 1 < c < 10, and m is any integer 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 x 10 in decimal form – x

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

Homework: p odd, 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 – x 10 6 Guided Practice