1. How do you feel about the first test?

2. How did you prepare for the first test?

3. Was it effective? Or how would you change your study habits?

4. What are you going to do to prepare for tests in the future?

Chapter 2: Properties of Real Numbers

Graphing Real Numbers

• All the numbers used in this course are real numbers

• You can represent real number visually by using a horizontal line called a number line

• The middle point on a number line is called the origin

• Points to the right of the origin are positive and points to the left are negative

-7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

Example 1: Graphing Real Numbers• Graph 4.7 and –3/4

-7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

Graph of 4.7Graph of –3/4

Example 2: Comparing Real Numbers• Graph 3 and –5• Write two inequalities that compare the

two numbers

-7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

Graph of 3

Inequality 1: 3 > -5 Inequality 2: -5 < 3

Graph of –5

Example 3: Ordering Real Numbers• Write the following numbers in decreasing

order:• 3½, 3.1, 4.8, -5, -5.8, -4½• It might help to graph these on a number

line:

-7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

Ready for the answer? 4.8, 3½, 3.1, -4½, -5, -5.8

Example 4: Comparing Real Real Numbers

• In golf, the total score is given as the number of strokes above or below par

• For a particular hole, if you sink the ball below par, your score is less than zero

• If you sink the ball above par, your score is above zero

• At the 2007 Arnold Palmer Invitational golf tournament, some of the world’s best golfers competed

Player ScoreTiger Woods +3

Vijay Singh -8

Phil Mickelson +6

Sergio Garcia -4

Boo Weekley +1

• Which player scored closest to par?• Which player scored farthest from par?• Which players scored above par?• Which players scored below par?• Who won the $5,500,000?

Example 5: Finding the opposite of a number

• Two numbers that are the same distance from the origin but on opposite sides of the origin are opposites

-7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

-4 can be stated as “negative 4” or the “opposite of 4”

negative a (-a) can also be called the opposite of a

If a = -13, then –a = -(-13) = 13 So, -a can be positive

Absolute Value

• The absolute value of a number is the distance between the origin and the point on the number line

• The symbol | a | represents the absolute value of the number a

• | 7 | = ?• | 0 | = ?• | -24 | = ?

Absolute Value

• The absolute value of a number is always positive (never negative)

• So, what do you get with the following?:• - | -83 | = ?• This is the same as – (83) or –83

Example 6: Absolute Value

• Evaluate the expressions:• | 5.3 | =• | - ¼ | =• - | - 32 | =

Example 7a: Solving Absolute Value Equations

• | -9 | = ?• | 9 | = ?• What about this?:• | x | = 9• What does x equal?• Since the absolute value of –9 and +9 both

equal 9, then x = +9, -9 or ( x = ± 9 )

Example 7b: Solving Absolute Value Equations

• Ok, so what about this?• | x | = -17

• What does x equal?• Remember, the result of an absolute value can

only be positive, so there is no solution to this equation

Example 8: Velocity and Speed

• Dictionary.com says that speed is the rate of motion

• Velocity indicates both speed and direction• From the web site, The Physics Factbook, “A

person has a terminal velocity of about 200 mph when balled up and about 125 mph with arms and feet fully extended to catch the wind.”

Example 8: Velocity and Speed

• If we assume that up is positive and down is negative, than how would you express the velocity of the balled up skydiver?

• Velocity = - 200 mph• What about the speed? • Speed = | -200 | or 200 mph