college algebra sixth edition james stewart lothar redlin saleem watson
TRANSCRIPT
![Page 1: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/1.jpg)
College AlgebraSixth EditionJames Stewart Lothar Redlin Saleem Watson
![Page 2: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/2.jpg)
PrerequisitesP
![Page 3: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/3.jpg)
The Real NumbersP.2
![Page 4: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/4.jpg)
Types of Real Numbers
![Page 5: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/5.jpg)
Introduction
Let’s review the types of numbers
that make up the real number
system.
![Page 6: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/6.jpg)
Natural Numbers
We start with the natural
numbers:
1, 2, 3, 4, …
![Page 7: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/7.jpg)
The integers consist of the natural
numbers together with their negatives
and 0:
. . . , –3, –2, –1, 0, 1, 2, 3, 4, . . .
Integers
![Page 8: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/8.jpg)
Rational Numbers
We construct the rational numbers
by taking ratios of integers.
• Thus, any rational number r can be expressed as:
where m and n are integers and n ≠ 0.
m
rn
![Page 9: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/9.jpg)
Rational Numbers
Examples are:
• Recall that division by 0 is always ruled out.
• So, expressions like 3/0 and 0/0 are undefined.
3 461 172 7 1 10046 0.17
![Page 10: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/10.jpg)
Irrational Numbers
There are also real numbers, such as ,
that can’t be expressed as a ratio of integers.
Hence, they are called irrational numbers.
• It can be shown, with varying degrees of difficulty, that these numbers are also irrational:
2
32
33 5 2
![Page 11: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/11.jpg)
Set of All Real Numbers
The set of all real numbers is
usually denoted by:
• The symbol
![Page 12: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/12.jpg)
Real Numbers
When we use the word ‘number’
without qualification, we will mean:
• “Real number”
![Page 13: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/13.jpg)
Real Numbers
Figure 1 is a diagram of the types
of real numbers that we work with
in this book.
![Page 14: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/14.jpg)
Repeating Decimals
Every real number has a decimal
representation.
If the number is rational, then its
corresponding decimal is repeating.
![Page 15: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/15.jpg)
Repeating Decimals
For example,
• The bar indicates that the sequence of digits repeats forever.
12
23
157495
97
0.5000... 0.50
0.66666... 0.6
0.3171717... 0.317
1.285714285714... 1.285714
![Page 16: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/16.jpg)
Non-Repeating Decimals
If the number is irrational, the decimal
representation is non-repeating:
2 1.414213562373095...
3.141592653589793...
![Page 17: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/17.jpg)
Approximation
If we stop the decimal expansion of
any number at a certain place, we get
an approximation to the number.
• For instance, we can write π ≈ 3.14159265
where the symbol ≈ is read “is approximately equal to.”
• The more decimal places we retain, the better our approximation.
![Page 18: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/18.jpg)
Properties of Real Numbers
![Page 19: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/19.jpg)
Introduction
We all know that: 2 + 3 = 3 + 2
5 + 7 = 7 + 5
513 + 87 = 87 + 513
and so on.
• In algebra, we express all these (infinitely many) facts by writing:
a + b = b + a where a and b stand for any two numbers.
![Page 20: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/20.jpg)
Commutative Property
In other words, “a + b = b + a” is a concise
way of saying that:
“when we add two numbers, the order
of addition doesn’t matter.”
• This is called the Commutative Property for Addition.
![Page 21: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/21.jpg)
Properties of Real Numbers
From our experience with numbers, we
know that these properties are also valid.
![Page 22: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/22.jpg)
Distributive Property
The Distributive Property
applies:
• Whenever we multiply a number by a sum.
![Page 23: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/23.jpg)
Distributive Property
Figure 2 explains why this property works
for the case in which all the numbers are
positive integers.
• However, it is true for any real numbers a, b, and c.
![Page 24: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/24.jpg)
E.g. 1—Using the Distributive Property
2(x + 3)
= 2 . x + 2 . 3 (Distributive Property)
= 2x + 6 (Simplify)
Example (a)
![Page 25: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/25.jpg)
(a + b)(x + y)
= (a + b)x + (a + b)y (Distributive Property)
= (ax + bx) + (ay + by) (Distributive Property)
= ax + bx + ay + by (Associative Property
of Addition)
• In the last step, we removed the parentheses.• According to the Associative Property, the order
of addition doesn’t matter.
E.g. 1—Using the Distributive Property Example (b)
![Page 26: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/26.jpg)
Addition and Subtraction
![Page 27: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/27.jpg)
Additive Identity
The number 0 is special for addition.
It is called the additive identity.
• This is because a + 0 = a for a real number a.
![Page 28: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/28.jpg)
Subtraction
Every real number a has a negative, –a,
that satisfies a + (–a) = 0.
Subtraction undoes addition.
• To subtract a number from another, we simply add the negative of that number.
• By definition, a – b = a + (–b)
![Page 29: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/29.jpg)
Properties of Negatives
To combine real numbers involving
negatives, we use these properties.
![Page 30: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/30.jpg)
Property 5 & 6 of Negatives
Property 6 states the intuitive fact
that: • a – b and b – a are negatives of each other.
Property 5 is often used with more than
two terms:• –(a + b + c) = –a – b – c
![Page 31: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/31.jpg)
E.g. 2—Using Properties of Negatives
Let x, y, and z be real numbers.
a) –(x + 2) = –x – 2 (Property 5: –(a + b) = –a – b)
b) –(x + y – z) = –x – y – (–z) (Property 5)
= –x – y + z (Property 2:
–(– a) = a)
![Page 32: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/32.jpg)
Multiplication and Division
![Page 33: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/33.jpg)
Multiplicative Identity
The number 1 is special for multiplication.
It is called the• multiplicative identity• This is because a 1 = a for any
real number a.
![Page 34: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/34.jpg)
Division
Every nonzero real number a has an inverse,
1/a, that satisfies a . (1/a).
Division undoes multiplication.• To divide by a number, we multiply by
the inverse of that number.
• If b ≠ 0, then, by definition, a ÷ b = a . 1/b
• We write a . (1/b) as simply a/b.
![Page 35: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/35.jpg)
Division
We refer to a/b as:
The quotient of a and b or as
the fraction a over b.
• a is the numerator. • b is the denominator (or divisor).
![Page 36: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/36.jpg)
Division
To combine real numbers using division,
we use these properties.
![Page 37: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/37.jpg)
Property 3 & 4
When adding fractions with different
denominators, we don’t usually use
Property 4.
• Instead, we rewrite the fractions so that they have the smallest common denominator (often smaller than the product of the denominators).
• Then, we use Property 3.
![Page 38: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/38.jpg)
LCD
This denominator is the Least
Common Denominator (LCD).
• It is described in the next example.
![Page 39: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/39.jpg)
E.g. 3—Using LCD to Add Fractions
Evaluate:
• Factoring each denominator into prime factors gives:
36 = 22 . 32
120 = 23 . 3 . 5
5 7
36 120
![Page 40: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/40.jpg)
E.g. 3—Using LCD to Add Fractions
We find the LCD by forming the product of all
the factors that occur in these factorizations,
using the highest power of each factor.
• Thus, the LCD is:
23 . 32 . 5 = 360
![Page 41: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/41.jpg)
E.g. 3—Using LCD to Add Fractions
So, we have:
(Use common denominator)
(Property 3: Adding fractions
with the same denominator)
5 7
36 1205 7
36 1
10
2050 21
360 3607
3
10 3
1
360
![Page 42: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/42.jpg)
The Real Line
![Page 43: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/43.jpg)
Introduction
The real numbers can be represented
by points on a line, as shown.
• The positive direction (toward the right) is indicated by an arrow.
![Page 44: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/44.jpg)
Origin
We choose an arbitrary reference point O,
called the origin, which corresponds to
the real number 0.
![Page 45: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/45.jpg)
Origin
Given any convenient unit of measurement,
• Each positive number x is represented by the point on the line a distance of x units to the right of the origin.
• Each negative number –x is represented by the point x units to the left of the origin.
![Page 46: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/46.jpg)
Coordinate
The number associated with
the point P is called:
• The coordinate of P
![Page 47: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/47.jpg)
Real Line
Then, the line is called any of
the following:
• Coordinate line
• Real number line
• Real line
![Page 48: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/48.jpg)
Real Line
Often, we identify the point with its
coordinate and think of a number as
being a point on the real line.
![Page 49: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/49.jpg)
Order of Numbers
The real numbers are ordered.
• We say that a is less than b, and write a < b if b – a is a positive number.
• Geometrically, this means that a lies to the left of b on the number line.
• Equivalently, we can say that b is greater than a, and write b > a.
![Page 50: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/50.jpg)
Symbol a ≤ b
The symbol a ≤ b (or b ≥ a):
• Means that either a < b or a = b.
• Is read “a is less than or equal to b.”
![Page 51: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/51.jpg)
Inequalities
For instance, these are true inequalities:
7 7.4 7.5
3
2 2
2 2
![Page 52: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/52.jpg)
Sets and Intervals
![Page 53: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/53.jpg)
Sets & Elements
A set is a collection of objects.
• These objects are called the elements of the set.
![Page 54: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/54.jpg)
Sets
If S is a set, the notation a S means that
a is an element of S.
b S means that b is not an element of S.
• For example, if Z represents the set of integers, then –3 Z but π Z.
![Page 55: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/55.jpg)
Braces
Some sets can be described by listing
their elements within braces.
• For instance, the set A that consists of all positive integers less than 7 can be written as:
A = {1, 2, 3, 4, 5, 6}
![Page 56: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/56.jpg)
Set-Builder Notation
We could also write A in set-builder
notation as:
A = {x | x is an integer and 0 < x < 7}
• This is read: “A is the set of all x such that x is an integer and 0 < x < 7.”
![Page 57: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/57.jpg)
Union of Sets
If S and T are sets, then their union
S T is:
• The set that consists of all elements that are in S or T (or both).
![Page 58: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/58.jpg)
Intersection of Sets
The intersection of S and T is the set
S T consisting of all elements that
are in both S and T.
• That is, S T is the common part of S and T.
![Page 59: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/59.jpg)
Empty Set
The empty set, denoted by Ø,
is:
• The set that contains no element.
![Page 60: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/60.jpg)
E.g. 4—Union & Intersection of Sets
If
S = {1, 2, 3, 4, 5}
T = {4, 5, 6, 7}
V = {6, 7, 8}
find the sets
S T, S T, S V
![Page 61: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/61.jpg)
E.g. 4—Union & Intersection of Sets
S T = {1, 2, 3, 4, 5, 6, 7} (All elements in S or T)
S T = {4, 5} (Elements common
to both S and T)
S V = Ø (S and V have no
elements in common)
![Page 62: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/62.jpg)
Intervals
Certain sets of real numbers occur
frequently in calculus and correspond
geometrically to line segments.
• These are called intervals.
![Page 63: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/63.jpg)
Open Interval
If a < b, the open interval from a to b
consists of all numbers between a
and b.
• It is denoted (a, b).
![Page 64: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/64.jpg)
Closed Interval
The closed interval from a to b
includes the endpoints.
• It is denoted [a, b].
![Page 65: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/65.jpg)
Open & Closed Intervals
Using set-builder notation,
we can write:
(a, b) = {x | a < x < b}
[a, b] = {x | a ≤ x ≤ b}
![Page 66: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/66.jpg)
Open Intervals
Note that parentheses ( ) in the interval
notation and open circles on the graph in
this figure indicate that:
• Endpoints are excluded from the interval.
![Page 67: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/67.jpg)
Closed Intervals
Note that square brackets and solid
circles in this figure indicate that:
• Endpoints are included.
![Page 68: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/68.jpg)
Intervals
Intervals may also include one endpoint
but not the other.
They may also extend infinitely far
in one direction or both.
![Page 69: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/69.jpg)
Types of Intervals
The following table lists the possible types of
intervals.
![Page 70: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/70.jpg)
E.g. 5—Graphing Intervals
Express each interval in terms of
inequalities, and then graph the interval.
a) [–1, 2)
b) [1.5, 4]
c) (–3, ∞)
![Page 71: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/71.jpg)
E.g. 5—Graphing Intervals
[–1, 2)
= {x | –1 ≤ x < 2}
Example (a)
![Page 72: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/72.jpg)
E.g. 5—Graphing Intervals
[1.5, 4]
= {x | 1.5 ≤ x ≤ 4}
Example (b)
![Page 73: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/73.jpg)
E.g. 5—Graphing Intervals
(–3, ∞)
= {x | –3 < x}
Example (c)
![Page 74: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/74.jpg)
E.g. 6—Finding Unions & Intersections of Intervals
Graph each set.
(a) (1, 3) [2, 7]
(b) (1, 3) [2, 7]
![Page 75: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/75.jpg)
E.g. 6—Intersection of Intervals
The intersection of two intervals
consists of the numbers that are
in both intervals.
• Therefore, (1, 3) [2, 7] = {x | 1 < x < 3 and 2 ≤ x ≤ 7} = {x | 2 ≤ x < 3} = [2, 3)
Example (a)
![Page 76: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/76.jpg)
E.g. 6—Intersection of Intervals
This set is illustrated here.
Example (a)
![Page 77: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/77.jpg)
E.g. 6—Union of Intervals
The union of two intervals consists
of the numbers that are in either one
interval or the other (or both).
• Therefore, (1, 3) [2, 7] = {x | 1 < x < 3 or 2 ≤ x ≤ 7} = {x | 1 < x ≤ 7} =(1, 7]
Example (b)
![Page 78: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/78.jpg)
E.g. 6—Union of Intervals
This set is illustrated here.
Example (b)
![Page 79: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/79.jpg)
Absolute Value and Distance
![Page 80: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/80.jpg)
Absolute Value
The absolute value of a number a,
denoted by |a|, is:
• The distance from a to 0 on the real number line.
![Page 81: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/81.jpg)
Distance
Distance is always positive or zero.
So, we have: • |a| ≥ 0 for every number a
Remembering that –a is positive when a
is negative, we have the following definition.
![Page 82: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/82.jpg)
Absolute Value—Definition
If a is a real number, the absolute
value of a is:
if 0
if 0
a aa
a a
![Page 83: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/83.jpg)
E.g. 7—Evaluating Absolute Values of Numbers
a) |3| = 3
b) |–3| = –(–3) = 3
c) |0| = 0
d) |3 – π| = –(3 – π) = π – 3 (since 3 < π ⇒ 3 – π < 0)
![Page 84: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/84.jpg)
Properties of Absolute Value
When working with absolute values, we use
these properties.
![Page 85: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/85.jpg)
Absolute Value & Distance
What is the distance on the
real line between the numbers
–2 and 11?
![Page 86: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/86.jpg)
Absolute Value & Distance
From the figure,
we see the
distance is 13.
• We arrive at this by finding either:
| 11 – (– 2)| = 13 or |(–2) – 11| = 13
• From this observation, we make the following definition.
![Page 87: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/87.jpg)
Distance Between Points on the Real Line
If a and b are real numbers, then
the distance between the points a and b
on the real line is:
d(a, b) = |b – a|
![Page 88: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/88.jpg)
Distance Between Points on the Real Line
From Property 6 of negatives,
it follows that:
|b – a| = |a – b|
• This confirms, as we would expect, that the distance from a to b is the same as the distance from b to a.
![Page 89: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/89.jpg)
E.g. 8—Distance Between Points on the Real Line
The distance between the numbers
–8 and 2 is:
d(a, b) = | –8 – 2|
= |–10|
= 10
![Page 90: College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson](https://reader035.vdocuments.site/reader035/viewer/2022062221/56649f3e5503460f94c5ec7f/html5/thumbnails/90.jpg)
E.g. 6—Distance Between Points on the Real Line
We can check that calculation
geometrically—as shown here.