Transcript R.2 - Gordon State College

Sullivan Algebra and Trigonometry:
Section R.2
Algebra Review
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
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 a ,
denoted by the symbol a , is defined by
the rules
a a
if a  0
a   a if a  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.
d (- 3, 2) = - 3 - 2 = - 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 z in
13
the expression
z+ 3
Dom ain:
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

a



n
n facto rs
a 1
0
a
n
1
 n
a
if a  0
if a  0
Examples:
4  444
3
6 1
0
4
3
1
 3
4
Laws of Exponents
a a a
m n
mn
a 
m n
a
mn
ab  a b
m
a
1
mn
 a  nm if a  0
n
a
a
n
n
 a  a if b  0
 b  bn
 a 
 b
n
b

 
 a
n
if a  0, b  0
n
n n
Example:
3 2
x y
Write 1 4 so that all exponents are positive.
x y
3 2
3
2
x y
x y
 1  4
1 4
x y
x
y
x
3 ( 1)  2  4
y
4 6
x y
4
x
 6
y
Example:
Simplify the expression. Express the answer so
only positive exponents occur.
 3x y 
 3 
 x y 
2
3
2
4
2
 3x
x  y 
1  2
23
3 2
y

4 1  2
  3x y
1
2
2
3 x y
2
6

3 2
x
 6
9y
In general, if a is a nonnegative real number,
the nonnegative number b such that b2 = a is
the principal square root of a and is denoted
by b =
a.
a a
2
Absolute Value is needed here, since the
principal square root produces a positive value.
Example: (- 4) 2 =
16
= 4 = - 4
Using your calculator
For Scientific Calculators:
Evaluate: (3.4) 4
Keystrokes: 3.4
xy
3.4
=
133.6336
For Graphing Calculators:
Evaluate: (3.4) 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 x  10 N . If the original
number is between 0 and 1, the scientific notation is
x  10 N .
Examples: Scientific Notation
Write the number 5,100,000,000 in
scientific notation.
Solution: 5 .1  1 0 9
Write the number 0.00032 in scientific notation.
Solution: 3.2  1 0  4
5
4
.
3

10
Write
as a decimal.
Solution: 0.000043