Transcript a < b

P.1 Real Numbers
and Algebraic
Expressions
The Real Number Line
The real number line is a graph used to represent the set of real numbers. An
arbitrary point, called the origin, is labeled 0;
Negative numbers
-4
-3
-2
Units to the left of the
origin are negative.
-1
Positive numbers
0
the
Origin
1
2
3
4
Units to the right of
the origin are
positive.
Ordering the Real Numbers
On the real number line, the real numbers increase from left to right. The lesser
of two real numbers is the one farther to the left on a number line. The greater
of two real numbers is the one farther to the right on a number line.
-2
-1
0
1
2
3
4
5
6
Since 2 is to the left of 5 on the number line, 2 is less than 5. 2 < 5
Since 5 is to the right of 2 on the number line, 5 is greater than 2. 5 > 2
Inequality Symbols
Symbols Meaning
Example
Explanation
a<b
3<7
Because 3 < 7
7<7
Because 7 =7
b is greater than or equal to a. 7 > 3
Because 7 > 3
b>a
a is less than or equal to b.
-5 > -5
Because -5 = -5
Be especially careful when the numbers are negatively signed.
The relationship will be the opposite of what it would be if
positive.
Ex: 5<7, yet –5>-7.
Always “grab” the larger (further to the right) number.
Ex: -  ______ -3.14
(ans: < since 3.14 ends & neg.)
Absolute Value
Absolute value describes the distance from 0 on a real number line. If a
represents a real number, the symbol |a| represents its absolute value, read
“the absolute value of a.”
For example, the real number line below shows that
|-3| = 3 and |5| = 5.
|–3| = 3
-3
-2
-1
The absolute value of –3 is 3
because –3 is 3 units from 0
on the number line.
|5| = 5
0
1
2
3
4
5
The absolute value of 5 is 5
because 5 is 5 units from 0
on the number line.
Definition of Absolute Value
The absolute value of x is given as follows:
{
x if x > 0
|x| =
-x if x < 0
Note: the absolute value of any real number will always be
positive or zero. If it is originally negatively signed (less
than zero), we must take the opposite sign. (|-3| = - (-3)) If it
is originally positively signed (greater than zero) we want to
keep it positively signed.
Ex: Find | \/7 - |
(ans: pi-root7 since neg)
Ex: Evaluate the given expression for x = -5 and y = 7
|x-y|/(x-y)
(ans: -1)
Q: If |x| = 5, what possible values could x equal?
(ans5 or –5)
Distance Between Two Points on
the Real Number Line
You live 80 miles east off I-4, your partner lives 20
miles east off I-4.
What is the distance between you and your partner?
If your partner lived 20 miles WEST off I-4 what
would the distance be?
If a and b are any two points on a real number line, then the distance
between a and b is given by
|a – b| or |b – a|
Show how we could write the above using integers and absolute value
symbols.
(ans: |80-20| and if WEST: |80-(-20)|)
Ex:
Find the distance between –5 and 3 on the real number line.
Solution Because the distance between a and b is given by |a – b|, the
distance between –5 and 3 is | -5-3 |
= | -8 |
=
8.
?
-5
-4
-3
-2
-1
0
1
2
We obtain the same distance if we reverse the order of subtraction:
| 3-(-5) | = | 8 | = 8.
3
Algebraic Expressions
A combination of variables and numbers using the operations of addition,
subtraction, multiplication, or division, as well as powers or roots, is called an
algebraic expression. Note: there is NO EQUAL SIGN, and we can only
SIMPLIFY, not SOLVE!
Here are some examples of algebraic expressions:
x + 6, x – 6, 6x, x/6, 3x + 5.
Note: In none of these cases can you SOLVE for the
variable, the best you can do is SIMPLIFY. In what case
could you “solve”? What would that be called (instead of an
expression?)
The Order of Operations Agreement
1.
2.
3.
4.
•
Perform operations within the innermost parentheses (or any other
grouping symbol such as a radical sign, absolute value, or division
bar) and work outward. If the algebraic expression involves
division, treat the numerator and the denominator as if they were
each enclosed in parentheses.
Evaluate all exponential expressions.
Perform multiplication or division as they occur, working from left
to right.
Perform addition or subtraction as they occur, working from left to
right.
That is: remove parenthesis and use order of
operations to combine “like” terms.
Ex: Simplify 123•2
(ans: 8, NOT 2!)
Text Example
The algebraic expression 2.35x + 179.5 describes the population of the
United States, in millions, x years after 1980. Evaluate the expression
when x = 20. Describe what the answer means in practical terms.
Solution We begin by substituting 20 for x. Because x = 20, we will be
finding the U.S. population in 1980 + 20 or the year 2000.
2.35x + 179.5
Replace x with 20.
= 2.35(20) + 179.5
= 47 + 179.5
Perform the multiplication.
= 226.5
Perform the addition.
Thus, in the year 2000 the population of the United States was 226.5
million.
Note the FORM of the simplification. Your test papers should look like
this.
Example
Simplify: 6(2x – 4y) + 10(4x + 3y).
Solution
6(2x – 4y) + 10(4x + 3y)
=
Use the distributive property.
=
Multiply.
=
Combine like terms.
Note: when we add “like” terms, we add the coefficient parts,
but leave the variable part (the like part) the same.
(ans: 53x+6y)
Think: 2(ab) vs 2(a+b): In which case do we use the
distributive law and why?
(2nd only, 1st is all mult.)
From this point I have provided these sheets as
review only. Please review and answer the
questions. Answers will be discussed next class.
The Basics About Sets
The set {1, 3, 5, 7, 9} has five elements.
•
A set is a collection of objects whose contents can be clearly determined.
•
•
The objects in a set are called the elements of the set.
We use braces to indicate a set and commas to separate the elements of
that set.
For example,
The set of counting numbers can be represented by {1, 2, 3, … }.
The set of even counting numbers are {2, 4, 6, …}.
The set of even counting numbers is a
subset of the set of counting numbers,
since each element of the subset is
also contained in the set.
Important Subsets of the Real Numbers
Name
Description
Examples
Natural Numbers
N
{1, 2, 3, …}
These are the counting numbers
4, 7, 15
Whole Numbers
W
{0, 1, 2, 3, … }
Add 0 to the natural numbers
0, 4, 7, 15
Integers
Z
{…, -2, -1, 0, 1, 2, 3, …}
Add the negative natural
numbers to the whole numbers
-15, -7, -4, 0, 4, 7
Important Subsets of the Real Numbers
Name
Description
Examples
Rational
Numbers
Q
These numbers can be expressed as an
integer divided by a nonzero integer:
Rational numbers can be expressed as
terminating or repeating decimals.
17 
Irrational
Numbers
I
This is the set of numbers whose decimal
representations are neither terminating nor
repeating. Irrational numbers cannot be
expressed as a quotient of integers.
17
5
,5 
,3, 2
1
1
0,2,3,5,17
2
 0.4,
5
2
 0.666666...  0.6
3
2  1.414214
 3  1.73205
  3.142


2
 1.571
The Real Numbers
Rational numbers
Irrational numbers
Integers
Whole numbers
Natural numbers
The set of real numbers is formed by combining the rational numbers and
the irrational numbers.
Graphing on the Number Line
Real numbers are graphed on the number line by placing a dot at the location
for each number. –3, 0, and 4 are graphed below.
-4
-3
-2
-1
0
1
2
3
4
Q: How could we graph –13/5 on the real number line?
(ans: -13/5 = -2and 3/5, so it would be a little over half way
from –2 to –3.)
Properties of Absolute Value
For all real number a and b,
1. |a| > 0
2. |-a| = |a|
3. a < |a|
4. |ab| = |a||b|
5.
6. |a + b| < |a| + |b| (the triangle inequality)
a |a|
= , b not equal to 0
b |b|
Example
• Find the following:
|-3|
and
|3|.
|-5•7|
and
|-5+7|
(Ans: 3 and 3, 35 and 2)
Properties of the Real Numbers
Name
Meaning
Examples
Commutative
Property of
Addition
Two real numbers can be added
in any order.
a+b=b+a
• 13 + 7 = 7 + 13
• 13x + 7 = 7 + 13x
Commutative Two real numbers can be
Property of
multiplied in any order.
Multiplication ab = ba
• x · 6 = 6x
Associative
Property of
Addition
• 3 + ( 8 + x)
= (3 + 8) + x
= 11 + x
If 3 real numbers are added, it
makes no difference which 2 are
added first.
(a + b) + c = a + (b + c)
Think: How is 3(8+x) different and why can’t we use the
associative property? (Ans: two operations: mult & addtn.)
Properties of the Real Numbers
Name
Meaning
Examples
Associative
If 3 real numbers are multiplied,
Property of
it makes no difference which 2
Multiplication are multiplied first.
(a · b) · c = a · (b · c)
• -2(3x) = (-2·3)x = -6x
Distributive
Multiplication distributes over
Property of
addition.
Multiplication a · (b + c) = a · b + a · c
over Addition
• 5 · (3x + 7)
= 5 · 3x + 5 · 7
= 15x + 35
Identity
Property of
Addition
• 0 + 6x = 6x
Zero can be deleted from a sum.
a+0=a
0+a=a
Properties of the Real Numbers
Name
Meaning
Examples
Identity
One can be deleted from a
Property of
product.
Multiplication a · 1 = a and 1 · a = a
• 1 · 2x = 2x
Inverse
Property of
Addition
• (-6x) + 6x = 0
The sum of a real number and its
additive inverse gives 0, the
additive identity.
a + (-a) = 0 and (-a) + a = 0
Inverse
The product of a nonzero real
• 2 · 1/2 = 1
Property of
number and its multiplicative
Multiplication inverse gives 1, the multiplicative
identity.
a · 1/a = 1 and 1/a · a = 1
Definitions of Subtraction and
Division
Let a and b represent real numbers.
Subtraction: a – b = a + (-b)
We call –b the additive inverse or opposite of b.
Division: a ÷ b = a · 1/b, where b = 0
We call 1/b the multiplicative inverse or reciprocal of b. The quotient of
a and b, a ÷ b, can be written in the form a/b, where a is the numerator
and b the denominator of the fraction.
Rewrite using multiplication:
10  2
10  (1/2)
(ans: 10*(1/2) and 10 * (2/1) or 10*2)
Properties of Negatives
•
1.
2.
3.
4.
5.
6.
Let a and b represent real numbers, variables, or
algebraic expressions.
(-1)a = -a
-(-a) = a
(-a)(b) = -ab
a(-b) = -ab
-(a + b) = -a - b
-(a - b) = -a + b = b - a
Algebra Translation Dictionary
You should add to this list as you discover new translations.
“Of” means use parenthesis (see p15 #111 ~118 to practice.)
Translation:
Example:
+ add, increased by, sum,
more than
- difference, decreased
by, less, minus, less than
Five times the sum of 4
and n: 5(4+n)
n decreased by 5: n-5
p less than 7: 7-p
x product, times, double
(etc.), half (etc.)
twice a number, decreased
by 5: 2n-5

quotient, divided by,
divided into
n divided by 5: n/5
p divided into 7: 7/p