Warm-up: m m m = - 1 m > 6 m = 6 m + 3 < 2 Evaluate  for   m HW: pg.10-12 (6, 22, 26 write in interval notation also, 33, 34, 36, 41, 42, 50, 52, 59, 62, 66, 76)

P.1: Real Numbers Objective: Classify real numbers Order real numbers and use inequalities Interpret and represent bounded and unbounded inequalities in words and in symbols. Evaluate absolute values Know the basic rules of algebra

The Real Number Line We can represent real numbers geometrically by points on a real number, or coordinate, line: This line includes all real numbers. Exactly one point on the line is associated with each real number, and vice-versa. Origin Negative Direction Positive Direction – 4 – 3 – 2 – 1 0 1 2 3 4 p

Real Numbers: numbers used in every day life. Ex) -5, 9, 0, , , , Subsets of real numbers: Natural {1, 2, 3, 4…} Whole numbers {0, 1, 2, 3, 4, …} Integers {…, -3, -2, -1, 0, 1, 2, 3, …}

Rational Numbers a number is rational if it can be written as a ratio of two integers such that the denominator is not zero. The decimal portion of a rational number either… repeats terminates

Irrational Number a real number that cannot be written as a ratio of two integers. The decimal portion of an irrational number is …. infinite and nonrepeating 

Ex) Determine which numbers in the set are (a)natural, (b)whole numbers, (c)integers, (d) rational, (e) irrational. Then plot them on the real number line. -1 3.6 125/111 2 4.132132…

Finite Intervals Open Intervals The set of real numbers that lie strictly between two fixed numbers a and b is called an open interval (a, b). It consists of all the real numbers that satisfy the inequalities a < x < b. It is called “open” because neither of its endpoints are included in the interval.

Finite Intervals Closed Intervals The set of real numbers that lie between two fixed numbers a and b, that includes a and b, is called a closed interval [a, b]. It consists of all the real numbers that satisfy the inequalities a  x  b. It is called “closed” because both of its endpoints are included in the interval.

Finite Intervals Half-Open Intervals The set of real numbers between two fixed numbers a and b, that contains only one of its endpoints a or b, is called a half-open interval (a, b] or [a, b). It consists of all the real numbers that satisfy the inequalities a < x  b or a  x < b.

Infinite Intervals Examples of infinite intervals include: (a, ), [a, ), (–, a), and (–, a]. The above are defined, respectively, by the set of real numbers that satisfy x > a, x  a, x < a, x  a.

Interpreting Inequalities Bounded and Unbounded Intervals on the Real Number Line: Notation Interval Type Inequality Graph [a, b] (a, b) [a, b) (-, b)

Ex) Use inequality notation to describe each of the following: C is at most -3 [-1, 4)

Ex) Give a verbal description of each interval (-2, 3) (-, 5] Ex) Write in interval notation. m > 6 3 < k ≤ 8

Absolute Value The absolute value of a number a is denoted |a| and is defined by

Ex) Evaluate for x > 0 x < 0

Ex) Evaluate Ex) Evaluate Ex) Evaluate

Ex) Find the distance between a and b if a = x – 3 and b = 2x – 3 Distance Between Two Points on the Real Number Line: Let a and b be real numbers. The distance between a and b is Ex) Find the distance between a and b if a = x – 3 and b = 2x – 3

Sneedlegrit: Find the d(a,b) if a = -x + 4 and b = 3x HW: pg.10 (6) pg.10 (22, 26) write in interval notation also pg.10 (33, 34, 36, 41, 42) pg.11 (50, 52, 59, 62, 66) pg.12 (76)