Objectives of this Section

• Graph Inequalities

• Find Distance on the Real Number Line

• Evaluate Algebraic Expressions

• Determine the Domain of a Variable

• Use the Laws of Exponents

• Evaluate Square Roots

• Use a Calculator to Evaluate Exponents

• Use Scientific Notation

Sullivan Algebra and Trigonometry: Section R.2

Algebra Review

The Real Number Line

The negative real numbers are the coordinates of points to the left of the origin 0.

The real number zero is the coordinate of the origin O.

The positive real numbers are the coordinates of points to the right of the origin O.

Comparing the position of two numbers on the number line is done using inequalities.

a < b means a is to the left of b

a = b means a and b are at the same location

a > b means a is to the right of b

Inequalities can also be used to describe the sign of a real number.

a > 0 is equivalent to a is positive.

a < 0 is equivalent to a is negative.

The absolute value of a real number ,

denoted by the symbol , is defined by

the rules

a

a

a a a

a a a

if

if

0

0

5 5 3 3 3

If P and Q are two points on a real number line with coordinates a and b, respectively, the distance between P and Q, denoted by

d (P, Q), is

d P Q b a, Example: Find the distance between –3 and 2 on

the number line.

( 3, 2) 3 2d - = - - 5= - 5=

Recall, letters such as x, y, z, a, b, and c are used to represent numbers. If the letter is used to represent any number from a given set of numbers, it is called a variable.

The set of values that a variable may assume is called the domain of the variable.

Example of Domain

Find the domain of the variable in

13the expression

3

z

z +

Domain: z z 3

The result is read “The set of all real numbers z such that z is not equal to –3”

Exponents: Basic Definitions

If a is a real number and n is a positive integer,

a a a an

n

factors

a a0 1 0 if

aa

ann

10 if

Examples:

4 4 4 43

6 10

41

43

3

Laws of Exponents

a a a a a ab a b

a

aa

aa

ab

a

bb

m n m n m n mn n n n

m

nm n

n m

n n

n

if

if

10

0

ab

ba

a bn n

if 0 0,

Example:

Write so that all exponents are positive.x y

x y

3 2

1 4

x y

x y

x

x

y

y

3 2

1 4

3

1

2

4

x y3 1 2 4( )

x y4 6 x

y

4

6

Example:

Simplify the expression. Express the answer so only positive exponents occur.

3 2 4

3

2x y

x y

3 2 3 4 1 2

x y 3 1 3 2

x y

3 2 1 2 3 2

x y 3 2 2 6x y x

y

2

69

2

In general, if is a nonnegative real number,

the nonnegative number such that is

the of and is denoted

by .

a

b b a

a

b a

=

=

principal square root

a a2

Absolute Value is needed here, since the principal square root produces a positive value.

Example: 2( 4)- 16= 4= 4= -

Using your calculator

For Scientific Calculators:

Evaluate: 4(3.4)

Keystrokes: 3.4 yx 3.4 =

133.6336

For Graphing Calculators:

Evaluate: 4(3.4)

Keystrokes: 3.4 3.4 =133.6336

^

Converting a Decimal to Scientific Notation

1. Count the number N of places that the decimal point must be moved in order to arrive at a number x, where 1 < x < 10.

2. If the original number is greater than or equal to 1, the scientific notation is If the original number is between 0 and 1, the scientific notation is

x N10 .

x N 10 .

Examples: Scientific Notation

Write the number 5,100,000,000 in scientific notation.

Solution: 51 109.

Write the number 0.00032 in scientific notation.

Solution: 3 2 10 4.

Write as a decimal.5103.4

Solution: 0.000043