Transcript a, b

Review Session #1
Outline
• Inequalities
• Absolute Value
• Exponents
Properties of Inequalities
If a, b, and c are any real numbers, then the following
properties are applicable.
Property 1 If a < b, then a + c < b + c.
Example: 5 < 12, so 5 – 7 < 12 – 7; that is, –2 < 5.
Interpretation: For any inequality, we can add or
subtract any number we like across the entire inequality
and the resulting inequality will still be true.
Exercise 1: Find the set of real numbers x that satisfy
each of the following:
a)
b)
x  3 1
0  x 1  4
Properties of Inequalities
Property 2 If a < b and c > 0, then ac < bc.
Example: –3 < 9, so –3(2) < 9(2); that is, –6 < 18.
Interpretation: For any inequality, we can multiply
across the entire inequality by any positive number we
like and the resulting inequality will still be true.
Exercise 2: Find the set of real numbers x that satisfy
each of the following:
a)
b)
4x  20
2x  3  1
and
x  2 1
Properties of Inequalities
Property 3 If a < b and c < 0, then ac > bc.
Example: –3 < 9, so –3(– 2) < 9(– 2); that is, 6 < –18.
Interpretation: For any inequality, we can multiply
across the entire inequality by any negative number we
like and the resulting inequality will still be true as long as
we switch the direction of the inequality.
Exercise 3: Find the set of real numbers x that satisfy
 3x  2  20 .
Absolute Value
The absolute value of a number a is denoted by |a| and is
defined as follows:
 a if a  0
| a | 
 a if a  0
Interpretation: If a is non-negative, then the absolute
value does nothing to change the value of a. However, if
a is negative, the absolute value makes it positive.
Example: | 5 | = 5 but | –3 | = 3.
Exercise 4: Evaluate the following:
a)
 2  5  2 | 4 |
b)
|   5 | 2
Absolute Value Properties
If a and b are any real numbers, then the following
properties are applicable.
Property 1 |ab| = |a| |b|
a a
 , b0
Property 2
b b
Exercise 5: Evaluate the following:
a)
b)
  4 5  1
0 .2  
2 .4  1 .6
Absolute Value and Inequalities
If a is any real number and f (x) is any function of x, then
the following statements are true.
1.
f ( x)  a
is equivalent to
 a  f ( x)  a
Example: Saying x  5 is equivalent to saying
 5  x  5 ; that is, x lies in the interval [5,5] .
2.
f ( x)  a
is equivalent to
Example: Saying
f ( x)  a or f ( x)  a
x  5 is equivalent to saying
x  5 or x  5 ; that is, x lies in the interval
(,5)  (5, ) .
Absolute Value and Inequalities
Exercise 6: Find the set of real numbers x that satisfy
each of the following:
a)
b)
2x  6
| 6 x  1 | 17
Exercise 7: Find the set of real numbers x that satisfy
each of the following:
a)
b)
8x  20
| 3x  1 | 19
Exponents
If b is any real number and n is any positive
integer, then the expression bn (read “b to the
power n”) is defined to be
b  b  b  b    b.
n
Example:
4  4  4  4  64
3
4
Example:
 2   2   2   2   2  16
        
 3   3   3   3   3  81
NOTE: b0 = 1 for any b.
Exponents
If is any positive integer, then the expression b1/n is
defined to be the number that, when raised to the
nth power, is equal to b. That is,
(b )  b.
1n n
If such a number exists, it is called the nth root of
b, and is sometimes written as
1n
b
 b
n
NOTE: The only time when the number b1/n does
not exist is when b < 0 and n is even.
Properties of Exponents
If a, b, m, and n are any real numbers, then the following
properties are applicable.
Property 1
Example:
a a  a
m
n
m n
x 2 x 5  x 25  x 7
Exercise 8: Simplify/evaluate each of the following:
a)
b)
2/3
4/5
x x
1/ 3 5 / 3
7 7
Properties of Exponents
m
Property 2
a
mn
a , a0
n
a
10
Example:
z
10 2
8
z
z
2
z
Exercise 9: Simplify/evaluate each of the following:
13
a)
4
10
4
3
b)
4/5
x x
2
x
Properties of Exponents
(a )  a
m n
Property 3
Example:
(y )  y
3 2
32
y
mn
6
Exercise 10: Simplify/evaluate each of the following:
a)
15 53
(8 )
b)
2 13 6
(x x )
Properties of Exponents
Property 4
Example:
(ab)  a  b
n
n
( xy )  x ( y )  x y
3 2
2
3 2
2
n
6
Exercise 11: Simplify/evaluate each of the following:
a)
13 2
(4 x )
b)
(3  7 )
12 6
Properties of Exponents
n
Property 5
x
 2
 y
13
Example:
4
a
a
   n , b0
b
b
n

(x )
x
  2 4  6
(y )
y

13 4
43
Exercise 12: Simplify/evaluate each of the following:
 2x
 4
 y
14
a)



3
 3x y
b) 
 xy

3
32



3