Download TGEA5 Chap 01

7/12/12
Objectives
Section 1.2
Fractions
Objective 1: Factor and Prime Factor
Natural Numbers
n 
To compute with fractions, we need to know how to
factor natural numbers. To factor a number means
to express it as a product of two or more numbers.
¨  For
example, some ways to factor 8
are 1  8, 4  2, and 2  2  2.
Factor and prime factor natural numbers
Recognize special fraction forms
n  Multiply and divide fractions
n  Build equivalent fractions
n  Simplify fractions
n  Add and subtract fractions
n  Simplify answers
n  Compute with mixed numbers
n 
n 
Objective 1: Factor and Prime Factor
Natural Numbers
n 
A prime number is a natural number greater than 1 that
has only itself and 1 as factors.
¨ 
n 
A composite number is a natural number, greater than 1,
that is not prime.
¨ 
n 
The numbers 1, 2, 4, and 8 that were used to write
the products are called factors of 8. In general, a
factor is a number being multiplied.
¨  Sometimes
a number has only two factors, itself and
1. We call such numbers prime numbers.
n 
The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and
29.
The first ten composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16,
and 18.
Every composite number can be factored into the product
of two or more prime numbers.
¨ 
This product of these prime numbers is called its prime
factorization.
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EXAMPLE 1
Find the prime factorization of 210.
Strategy We will use a series of steps to
express 210 as a product of only prime
numbers.
Why To prime factor a number means to
write it as a product of prime numbers.
EXAMPLE 1
Find the prime factorization of 210.
Solution
Writing the factors in order, from least to greatest, the prime-factored
form of 210 is 2  3  5  7. Two other methods for prime factoring 210
are shown below.
EXAMPLE 1
Find the prime factorization of 210.
Solution
First, write 210 as the product of two natural numbers other than 1.
Neither 10 nor 21 are prime numbers, so we factor each of them.
Objective 2: Recognize Special
Fraction Forms
n 
Fractions can describe the number of equal parts of
a whole. Consider the circle with 5 of 6 equal parts
colored red. We say that 5/6 (five-sixths) of the circle
is shaded.
n 
In a fraction, the number above the fraction bar is
called the numerator, and the number below is
called the denominator.
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Objective 2: Recognize Special
Fraction Forms
n 
Fractions are also used to indicate division. For example,
8/2 indicates that the numerator, 8, is to be divided by the
denominator, 2:
Objective 3: Multiply and Divide
Fractions
n 
Multiplying fractions: To multiply two fractions,
multiply the numerators and multiply the
denominators.
¨  For
¨ 
n 
n 
If the numerator and denominator of a fraction are the
same nonzero number, the fraction indicates division of a
number by itself, and the result is 1.
If a denominator is 1, the fraction indicates division by 1,
and the result is simply the numerator.
¨ 
n 
Multiply: 7/8 × 3/5.
Strategy To find the product, we will multiply the numerators,
7 and 3, and multiply the denominators, 8 and 5.
Dividing fractions: To divide two fractions, multiply
the first fraction by the reciprocal of the second.
¨  For
any two fractions a/b and c/d,
where c ≠ 0, a/b ÷ c/d = (a/b)(d/c).
Special fraction forms: For any nonzero number a, a/a = 1 and a/
1 = a.
EXAMPLE 2
any two fractions a/b and c/d, (a/b)(c/d) = ac/bd.
One number is called the reciprocal of another if their product is
1. To find the reciprocal of a fraction, we invert its numerator and
denominator.
Objective 4: Build Equivalent
Fractions
n 
The two rectangular regions on the right
are identical. The first one is divided into
10 equal parts. Since 6 of those parts
are red, 6/10 of the figure is shaded.
n 
The second figure is divided into 5 equal parts. Since 3 of
those parts are red, 3/5 of the figure is shaded.
n 
We can conclude that 6/10 = 3/5 because 6/10 and 3/5
represent the same shaded portion of the figure.
Why This is the rule for multiplying two fractions.
Solution:
¨  We
say that 6/10 and 3/5 are equivalent fractions. Two
fractions are equivalent if they represent the same
number.
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Objective 4: Build Equivalent
Fractions
n 
n 
Equivalent Fractions: Two fractions are equivalent if
they represent the same number. Equivalent fractions
represent the same portion of a whole.
Writing a fraction as an equivalent fraction with a larger
denominator is called building the fraction.
¨  Since
any number multiplied by 1 remains the same
(identical), 1 is called the multiplicative identity element.
¨  Multiplication property of 1: The product of 1 and any
number is that number.
¨  For any number,
EXAMPLE 4
Write 3/5 as an equivalent fraction
with a denominator of 35.
Strategy We will compare the given
denominator to the required denominator and
ask, “By what must we multiply 5 to get 35?”
Why The answer to that question helps us
determine the form of 1 to be used to build an
equivalent fraction.
1a = a and a1=a
EXAMPLE 4
Write 3/5 as an equivalent fraction
with a denominator of 35.
Solution
Objective 4: Build Equivalent
Fractions
n 
Building Fractions : To build a fraction, multiply it
by 1 in the form of c/c, where c is any nonzero
number.
n 
The Fundamental Property of Fractions: If the
numerator and denominator of a fraction are
multiplied by the same nonzero number, the
resulting fraction is equivalent to the original
fraction.
We need to multiply the denominator of 3/5 by 7 to obtain a
denominator of 35. It follows that 7/7 should be the form of 1 that
is used to build 3/5.
Multiplying 3/5 by 7/7 changes its appearance but does not
change its value, because we are multiplying it by 1.
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Objective 5: Simplify Fractions
n 
Every fraction can be written in infinitely many
equivalent forms. For example, some equivalent
forms of 10/15 are:
n 
Simplest form of a fraction: A fraction is in
simplest form, or lowest terms, when the
numerator and denominator have no common
factors other than 1.
EXAMPLE 5
Simplify each fraction, if possible:
a. 63/42; b. 33/40
Strategy We will begin by prime factoring the
numerator and denominator of the fraction.
Then, to simplify it, we will remove a factor
equal to 1.
Why We need to make sure that the
numerator and denominator have no
common factors other than 1. If that is the
case, then the fraction is in simplest form.
Objective 5: Simplify Fractions
n 
To simplify a fraction, we write it in simplest form by removing a
factor equal to 1. For example, to simplify 10/15, we note that the
greatest factor common to the numerator and denominator is 5 and
proceed as follows:
n 
To simplify 10/15, we removed a factor equal to 1 in the form of 5/5 .
The result,2/3, is equivalent to 10/15.
We can easily identify the greatest common factor of the numerator
and the denominator of a fraction if we write them in prime-factored
form.
n 
EXAMPLE 5
Simplify each fraction, if possible:
a. 63/42; b. 33/40
Solution
a. After prime factoring 63 and 42, we see that the greatest common
factor of the numerator and the denominator is 3  7 = 21.
b. To attempt to simplify the fraction, Prime factor 33
and 40. Since the numerator and the denominator have
no common factors other than 1, the fraction is in
simplest form (lowest terms).
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Objective 5: Simplify Fractions
n 
Simplifying fractions:
Objective 6: Add and Subtract Fractions
n 
¨  Factor (or prime factor) the numerator and denominator to
¨  For
example, we can add dollars to dollars, but we
cannot add dollars to oranges. This concept is important
when adding or subtracting fractions.
determine their common factors.
¨ 
¨ 
Remove factors equal to 1 by replacing each pair of factors
common to the numerator and denominator with the
equivalent fraction 1/1.
n 
Consider the problem 2/5 + 1/5. When we write it in
words, it is apparent we are adding similar objects.
n 
Because the denominators of 2/5 and 1/5 are the
same, we say that they have a common denominator.
Multiply the remaining factors in the numerator and in the
denominator.
Objective 6: Add and Subtract Fractions
n 
In algebra as in everyday life, we can only add or
subtract objects that are similar.
To add (or subtract) fractions that have the same
denominator, add (or subtract) their numerators
and write the sum (or difference) over the
common denominator.
¨  For
¨  a/d
any fractions a/d and b/d,
+ b/d = (a+b)/d and
a/d − b/d = (a−b)/d.
Objective 6: Add and Subtract Fractions
n 
Now we consider the problem 2/5 + 1/3.
¨  Since
the denominators are not the same, we cannot
add these fractions in their present form.
n 
To add (or subtract) fractions with different
denominators, we express them as equivalent
fractions that have a common denominator.
¨  The
smallest common denominator, called the least or
lowest common denominator, is usually the easiest
common denominator to use.
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Objective 6: Add and Subtract Fractions
n 
Objective 6: Add and Subtract Fractions
Least Common Denominator (LCD): The least or
lowest common denominator (LCD) for a set of
fractions is the smallest number each denominator will
divide exactly (divide with no remainder).
¨ 
The denominators of 2/5 and 1/3 are 5 and 3. The numbers 5 and
3 divide many numbers exactly (30, 45, and 60, to name a few),
but the smallest number that they divide exactly is 15. Thus, 15 is
the LCD for 2/5 and 1/3.To find 2/5 + 1/3, we find equivalent
fractions that have denominators of 15 and we use the rule for
adding fractions.
Objective 6: Add and Subtract Fractions
EXAMPLE 7
When adding (or subtracting) fractions with
unlike denominators, the least common
denominator is not always obvious. Prime
factorization is helpful in determining the LCD.
n  Finding the LCD Using Prime Factorization:
Strategy We will begin by expressing each fraction
as an equivalent fraction that has the LCD for its
denominator. Then we will use the rule for
subtracting fractions with like denominators.
n 
¨  Prime
factor each denominator.
LCD is a product of prime factors, where each
factor is used the greatest number of times it
appears in any one factorization found in step 1.
¨  The
Subtract: 3/10 − 5/28.
Why To add or subtract fractions, the fractions
must have like denominators.
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EXAMPLE 7
Subtract: 3/10 − 5/28.
Solution
EXAMPLE 7
Subtract: 3/10 − 5/28.
Solution
To find the LCD, we find the prime factorization of both denominators
and use each prime factor the greatest number of times it appears in
any one factorization.
Since 140 is the smallest number that 10 and 28 divide exactly, we
write 3/10 and 5/28 as fractions with the LCD 140.
Objective 6: Add and Subtract Fractions
n 
Adding and subtracting fractions that have
different denominators:
¨  Find
the LCD.
each fraction as an equivalent fraction with
the LCD as the denominator. To do so, build each
fraction using a form of 1 that involves any factors
needed to obtain the LCD.
¨  Add or subtract the numerators and write the sum or
difference over the LCD.
¨  Simplify the result, if possible.
Objective 7: Simplify Answers
n 
When adding, subtracting, multiplying, or dividing
fractions, remember to express the answer in
simplest form.
¨  Rewrite
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EXAMPLE 8
Perform the operations and simplify:
a. 45(4/9); b. 5/12 + 3/2 − 1/4.
Strategy We will perform the indicated
operations and then make sure that the
answer is in simplest form.
EXAMPLE 8
Perform the operations and simplify:
a. 45(4/9); b. 5/12 + 3/2 − 1/4.
Solution (a)
Why Fractional answers should always be
given in simplest form.
EXAMPLE 8
Perform the operations and simplify:
a. 45(4/9); b. 5/12 + 3/2 − 1/4.
Solution (b)
Since the smallest number that 12, 2, and 4 divide exactly is 12,
the LCD is 12.
Objective 8: Compute with Mixed Numbers
n 
A mixed number represents the sum of a whole
number and a fraction.
¨  For
example, 5¾ means 5 + ¾ and 179 15/16 means
179+15/16.
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EXAMPLE 9
Divide: 5¾ ÷ 2.
Strategy We begin by writing the mixed
number 5¾ and the whole number 2 as
fractions. Then we use the rule for dividing
two fractions.
EXAMPLE 9
Solution
Why To multiply (or divide) with mixed
numbers, we first write them as fractions,
and then multiply (or divide) as usual.
Objectives
Section 1.3
The Real Numbers
n  Define
the set of integers.
the set of rational numbers.
n  Define the set of irrational numbers.
n  Classify real numbers.
n  Graph sets of real numbers on the number
line.
n  Find the absolute value of a real number.
n  Define
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Objective 1: Define the Set of Integers
Objective 1: Define the Set of Integers
n  A
n 
set is a collection of objects, such as a
set of golf clubs or a set of dishes.
n 
The natural numbers, together with 0, form the
set of whole numbers.
¨  The
Natural numbers are the numbers that we use for
counting. To write this set, we list its members (or
elements) within braces { }.
¨  The set of natural numbers is {1, 2, 3, 4, 5, ...}
set of whole numbers is {0, 1, 2, 3, 4, 5, ...}
Read as “the set containing one, two, three, four, five, and so on.”
Objective 1: Define the Set of Integers
n 
Whole numbers are not adequate for describing many
real-life situations.
¨ 
n 
For example, if you write a check for more than what’s in your
account, the account balance will be less than zero.
We can use the number line below to visualize numbers
less than zero.
A number line is straight and has uniform markings. The
arrowheads indicate that it extends forever in both directions.
¨  For each natural number on the number line, there is a
corresponding number, called its opposite, to the left of 0.
Objective 1: Define the Set of Integers
n 
n 
¨  The
¨ 
n 
In the diagram, we see that 3 and (negative three) are opposites, as
are (negative five) and 5. Note that 0 is its own opposite.
Two numbers that are the same distance from 0 on the
number line, but on opposite sides of it, are called
opposites.
The whole numbers, together with their opposites, form
the set of integers.
n 
set of integers is { ... , −4, −3, −2, −1, 0, 1, 2, 3, 4, ... }
On the number line, numbers greater than 0 are to the
right of 0. They are called positive numbers.
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Objective 1: Define the Set of Integers
¨  Positive
n 
n 
numbers can be written with or without a
positive sign +.
Numbers less than 0 are to the left of 0 on the number
line. They are called negative numbers.
¨  Negative numbers are always written with a negative
sign −.
Positive and negative numbers are called signed
numbers.
Objective 2: Define the Set of Rational
Numbers
n 
Fractions such as 1/4 and 2/3, that are
quotients of two integers, are called
rational numbers.
n 
Negative fractions are also rational numbers.
n 
Positive and negative mixed numbers are also rational numbers
because they can be expressed as fractions.
n 
Any natural number, whole number, or integer can be expressed as a
fraction with a denominator of 1.
¨ 
¨ 
¨ 
¨ 
A rational number is any number that can be expressed as a fraction
(ratio) with an integer numerator and a nonzero integer denominator.
For any numbers a and b, where b is not 0, −(a/b) = (−a/b) = (a/−b).
For example, 75/8 = 61/8.
For example, 5 = 5/1, 0 = 0/1, and −3 = −3/1.
Therefore, every natural number, whole number, and integer is also a
rational number.
Objective 2: Define the Set of Rational
Numbers
n 
Many numerical quantities are written in decimal
notation. For instance, a candy bar might cost
$0.89, a dragster might travel at 203.156 mph, or
a business loss might be −$4.7 million. These
decimals are called terminating decimals
because their representations terminate (stop).
¨ 
n 
To find the decimal equivalent for a fraction, we divide its numerator
by its denominator.
¨ 
For example, to write 1/4 and 5/22 as decimals, we proceed as follows:
Terminating decimals can be expressed as fractions: 0.89 = 89/100,
203.156 = 203156/1000 = 203,156/1,000. Therefore, terminating decimals are rational
numbers.
n 
Decimals such as 0.3333... and 2.8167167167... , which have a digit (or block
of digits) that repeats, are called repeating decimals.
n 
The set of rational numbers cannot be listed as we listed other sets in this
section. Instead, we use set-builder notation.
¨ 
Objective 2: Define the Set of Rational
Numbers
Since any repeating decimal can be expressed as a fraction, repeating decimals
are rational numbers.
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Objective 2: Define the Set of Rational
Numbers
n 
n 
The decimal equivalent of 1/4 is 0.25
and the decimal equivalent of 5/22 is
0.2272727...
We can use an overbar to write
repeating decimals in more compact
form. 0.2272727 . . .=0.227.
Objective 3: Define the Set of Irrational
Numbers
n 
Not all numbers are rational numbers.
¨ 
¨ 
¨ 
¨ 
¨ 
n 
Objective 4: Classify Real Numbers
n 
The set of real numbers is formed by combining the set
of rational numbers and the set of irrational numbers.
Every real number has a decimal representation.
If it is rational, its corresponding decimal terminates or repeats. If
it is irrational, its decimal representation is nonterminating and
nonrepeating.
¨  A real number is any number that is a rational number or an
irrational number.
¨ 
¨ 
One example is the square root of 2, written √2. It is the number
that, when multiplied by itself, gives 2: √2 × √2 = 2.
It can be shown that √2 cannot be written as a fraction with an
integer numerator and an integer denominator. Therefore, it is
not rational; it is an irrational number.
It is interesting to note that a square with sides of length 1 inch
has a diagonal that is √2 inches long.
The number represented by the Greek letter π (pi) is another
example of an irrational number. A circle, with a 1-inch diameter,
has a circumference of π inches.
Expressed in decimal form, √2 = 1.414213562... and
π = 3.141592654...
These decimals neither terminate nor repeat.
An irrational number is a nonterminating, nonrepeating decimal. An
irrational number cannot be expressed as a fraction with an integer
numerator and an integer denominator.
Objective 4: Classify Real Numbers
n 
The following diagram shows how various sets of
numbers are related. Note that a number can belong to
more than one set. For example, −6 is an integer, a
rational number, and a real number.
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EXAMPLE 1
n 
Which numbers in the following set are natural numbers,
whole numbers, integers, rational numbers, irrational
numbers, real numbers?
{−3.4, 2/5, 0, −6, 1¾, π, 16}
EXAMPLE 1
Solution
Strategy We begin by scanning the given set, looking for any natural
numbers. Then we scan it five more times, looking for whole
numbers, for integers, for rational numbers, for irrational numbers,
and finally, for real numbers.
Why We need to scan the given set of numbers six times, because
numbers in that set can belong to more than one classification.
Objective 5: Graph Sets of Numbers on
the Number Line
n 
n 
Every real number corresponds to a point on the number line, and
every point on the number line corresponds to exactly one real
number.
As we move right on the number line, the values of the numbers
increase. As we move left, the values decrease.
¨ 
Objective 5: Graph Sets of Numbers on
the Number Line
n 
n 
On the number line below, we see that 5 is greater than −3, because 5
lies to the right of −3. Similarly, −3 is less than 5, because it lies to the left
of 5.
The inequality symbol > means “is
greater than.” It is used to show that
one number is greater than another.
The inequality symbol < means “is
less than.” It is used to show that one
number is less than another.
¨ 
n 
For example,
To distinguish between these inequality symbols, remember that
each one points to the smaller of the two numbers involved.
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EXAMPLE 2
Use one of the symbols > or < to make each statement true:
a. −4 ? 4, b. −2 ? −3, c. 4.47 ? 12.5, d. 3/4 ? 5/8
n 
Strategy To pick the correct inequality symbol to place
between a given pair of numbers, we need to determine
the position of each number on a number line.
Why For any two numbers on a number line, the number to
the left is the smaller number and the number to the right is
the larger number.
Objective 6: Find the Absolute Value of a
Real Number
n 
A number line can be used to measure the distance from one number
to another.
¨ 
For example, in the following figure we see that the distance from 0 to −4
is 4 units and the distance from 0 to 3 is 3 units.
EXAMPLE 2
Solution
a. Since −4 is to the left of 4 on the number line,
we have −4 < 4.
b. Since −2 is to the right of −3 on the number line, we have −2 > −3.
c. Since 4.47 is to the left of 12.5 on the number line, we have 4.47 < 12.5.
d. To compare fractions, express them in terms of the same denominator,
preferably the LCD. If we write 3/4 as an equivalent fraction with denominator
8, we see that 3/4 = (3×2)/(4×2) = 6/8. Therefore, 3/4 > 5/8.
To compare the fractions, we could also convert each to its decimal
equivalent. Since 3/4 = 0.75 and 5/8 = 0.625, we know that 3/4 > 5/8.
EXAMPLE 4
n 
Find each absolute value:
a. |18|, b. |−7/8|, c. |98.6|, d. |0|
Strategy We need to determine the distance that the
number within the vertical absolute value bars is from 0.
n 
To express the distance that a number is from 0
on a number line, we can use absolute values.
¨ 
¨ 
The absolute value of a number is its distance
from 0 on the number line.
To indicate the absolute value of a number, we
write the number between two vertical bars. From
the figure above, we see that |−4| = 4. It also
follows from the figure that |3| = 3.
Why The absolute value of a number is the distance
between 0 and the number on a number line.
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EXAMPLE 4
Solution
Section 1.4
a. Since 18 is a distance of 18 from 0 on the number line, |18| = 18.
Adding Real Numbers;
Properties of Addition
b. Since −7/8 is a distance of 7/8 from 0 on the number line, |−7/8| = 7/8.
c. Since 98.6 is a distance of 98.6 from 0 on the number line, |98.6| = 98.6.
d. Since 0 is a distance of 0 from 0 on the number line, |0| = 0.
Objectives
n  Add
two numbers that have the same sign
n  Add two numbers that have different signs
n  Use properties of addition
n  Identify opposites (additive inverses)
Objective 1: Add Two Numbers That Have
the Same Sign
n 
A number line can be used to explain the addition
of signed numbers.
For example, to compute 5 + 2, we begin at 0 and draw an
arrow five units long that points right. It represents 5.
¨  From the tip of that arrow, we draw a second arrow two units
long that points right. It represents 2.
¨  Since we end up at 7, it follows that 5 + 2 = 7.
¨  The numbers that we added, 5 and 2, are called addends,
and the result, 7, is called the sum.
¨ 
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Objective 1: Add Two Numbers That Have
the Same Sign
n 
n 
To compute −5 + (−2), we begin at
0 and draw an arrow five units long
that points left. It represents −5.
From the tip of that arrow, we draw
a second arrow two units long that
points left. It represents −2. Since
we end up at −7, it follows that
−5 + (−2) = −7.
Objective 1: Add Two Numbers That Have
the Same Sign
n 
n 
n 
When we use a number line to add numbers with
the same sign, the arrows point in the same
direction and they build upon each other.
Furthermore, the answer has the same sign as the
numbers that we added. These observations
suggest the following rules.
Adding Two Numbers That Have the Same (Like)
Signs:
¨  1.
To add two positive numbers, add them as usual. The
final answer is positive.
¨  2. To add two negative numbers, add their absolute values
and make the final answer negative.
EXAMPLE 1
n 
Add:
a. −20 + (−15), b. −7.89 + (−0.6), c. −1/3 + (−1/2)
EXAMPLE 1
Solution
Strategy We will use the rule for adding two numbers that
have the same sign.
Why In each case, we are asked to add two negative
numbers.
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Objective 2: Add Two Numbers That Have
Different Signs
n 
n 
n 
n 
To compute 5 + (−2), we begin at 0 and draw an
arrow five units long that points right.
From the tip of that arrow, we draw a second arrow
two units long that points left.
Since we end up at 3, it follows that
5 + (−2) = 3.
In terms of money, if you won $5 and then lost $2,
you would have $3 left.
Objective 2: Add Two Numbers That Have
Different Signs
n 
When we use a number line to add numbers with different
signs, the arrows point in opposite directions and the longer
arrow determines the sign of the answer.
If the longer arrow represents a positive number, the sum is
positive.
¨  If it represents a negative number, the sum is negative.
Objective 2: Add Two Numbers That Have
Different Signs
n 
n 
n 
n 
To compute −5 + 2, we begin at 0 and draw an arrow
five units long that points left.
From the tip of that arrow, we draw a second arrow
two units long that points right.
Since we end up at 3, it follows that 5 + (−2) = −3.
In terms of money, if you lost $5 and then won
$2, you have lost $3.
EXAMPLE 2
n 
Add:
a. −20 + 32, b. 5.7 + (−7.4), c. −19/25 + 2/5
¨ 
n 
These observations suggest the following rules.
Adding Two Numbers That Have Different (Unlike) Signs:
To add a positive number and a negative number, subtract the
smaller absolute value from the larger.
¨  1. If the positive number has the larger absolute value, the final
answer is positive.
¨  2. If the negative number has the larger absolute value, make the
final answer negative.
Strategy We will use the rule for adding two numbers
that have different (unlike) signs.
Why In each case, we are asked to
add a positive number and a
negative number.
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EXAMPLE 2
EXAMPLE 2
Solution
Solution
c. Since 2/5 = 10/25, the fraction −19/25 has the larger absolute
value. To find −19/25 +2/5, we subtract the smaller absolute
value from the larger:
a.
b. To find 5.7 + (−7.4), subtract the smaller
absolute value, 5.7, from the larger, 7.4.
Since the negative fraction −19/25 has the larger absolute value,
make the final answer negative: −19/25 + 10/25 = −9/25.
Since the negative decimal, −7.4, has the
larger absolute value, make the final answer
negative: 5.7 + (−7.4) = −1.7.
Objective 3: Use Properties of Addition
n 
The addition of two numbers can be done in
any order and the result is the same.
¨  For
n 
n 
Objective 3: Use Properties of Addition
n 
In the following example, we add −3 + 7 + 5 in two ways.
¨ 
example, 8 + (−1) = 7 and −1 + 8 = 7.
This example illustrates
that addition is commutative.
n 
The Commutative Property of Addition:
¨  Changing
the order when adding does not affect
the answer.
For any real numbers a and b, a + b = b + a.
n 
We will use grouping symbols ( ), called parentheses, to
show this. Standard practice requires that the operation within
the parentheses be performed first.
It doesn’t matter how we group the numbers in this addition;
the result is 9. This example illustrates that addition is
associative.
The Associative Property of Addition:
¨ 
Changing the grouping when adding does not affect the answer.
For any real numbers a, b, and c, (a + b) + c = a + (b + c).
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EXAMPLE 4
Find the sum:
98 + (2 + 17)
Strategy We will use the associative
property to group 2 with 98. Then, we
evaluate the expression by following
the rules for the order of operations.
Objective 4: Identify Opposites
(Additive Inverses)
n 
Recall that two numbers that are the same distance from 0
on a number line, but on opposite sides of it, are called
opposites. To develop a property for adding opposites, we
will find −4 + 4 using a number line.
We begin at 0 and draw an arrow four units long that points
left, to represent −4.
¨  From the tip of that arrow, we draw a second arrow, four units
long that points right, to represent 4.
¨  We end up at 0; therefore, −4 + 4 = 0.
¨ 
Why It is helpful to regroup because 98 and 2 are a
pair of numbers that are easily added.
Solution
Objective 4: Identify Opposites
(Additive Inverses)
n 
This example illustrates that when we add opposites, the
result is 0.
n 
Also, whenever the sum of two numbers is 0, those
numbers are opposites. For these reasons, opposites are
also called additive inverses.
Addition Property of Opposites (Inverse Property of
Addition):
¨ 
n 
¨ 
Therefore, 1.6 + (−1.6) = 0 and −3/4 + 3/4 = 0.
EXAMPLE 6
Add:
12 + (−5) + 6 + 5 + (−12)
Strategy Instead of working from left to right, we will use
the commutative and associative properties of addition to
add pairs of opposites.
Why Since the sum of a number and its opposite is 0, it
is helpful to identify such pairs in an addition.
Solution
The sum of a number and its opposite (additive inverse) is 0.
For any real number a and its opposite or additive inverse −a,
a + (−a) = 0.
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Objectives
Section 1.5
Subtracting Real
Numbers
Objective 1: Use the Definition of
Subtraction
n 
A minus symbol − is used to indicate subtraction.
¨ 
However, this symbol is also used in two other ways, depending
on where it appears in an expression.
n  Use
the definition of subtraction.
n  Solve application problems using
subtraction.
Objective 1: Use the Definition of
Subtraction
n 
This observation illustrates the following rule.
¨  Opposite
of an Opposite: The opposite of the
opposite of a number is that number.
For any real number a, −(−a) = a.
Read as “the opposite of the opposite of a is a.”
n 
In −(−5), parentheses are used to write the opposite of a
negative number.
¨ 
When such expressions are encountered in computations, we
simplify them by finding the opposite of the number within the
parentheses.
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EXAMPLE 1
n 
Simplify each expression:
a. −(−45), b. −(−h), c. −|−10|
Strategy To simplify each expression, we will use the
concept of opposite.
EXAMPLE 1
Solution
a. The number within the parentheses is −45. Its opposite is 45.
Therefore, −(−45) = 45.
b. The opposite of the opposite of h is h. Therefore, −(−h) = h.
c. The notation −|−10| means “the opposite of the absolute value
of negative ten.” Since |−10| = 10, we have:
Why In each case, the outermost − symbol is read as
“the opposite.”
Objective 1: Use the Definition of
Subtraction (continued)
n 
To develop a rule for subtraction,
we consider the following
illustration. It represents the
subtraction 5 − 2 = 3.
n 
The illustration above also represents the addition
5 + (−2) = 3. We see that:
Objective 1: Use the Definition of
Subtraction (continued)
n 
Subtraction of Real Numbers: To subtract two
real numbers, add the first number to the
opposite (additive inverse) of the number to be
subtracted.
For any real numbers a and b, a − b = a + (−b).
Read as “a minus b equals a plus the opposite of b.”
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Objective 2: Solve Application Problems
Using Subtraction
EXAMPLE 5
n 
n 
Subtraction finds the difference between two
numbers.
When we find the difference between the
maximum value and the minimum value of a
collection of measurements, we are finding the
range of the values.
n 
U.S. Temperatures
The record high temperature in the United States was 134°F in Death Valley,
California, on July 10, 1913.
The record low was −80°F at Prospect Creek, Alaska, on January 23, 1971.
Find the temperature range for these extremes.
Strategy We will subtract the lowest temperature from the
highest temperature.
Why The range of a collection of data indicates the
spread of the data. It is the difference between the
largest and smallest values.
Solution:
The temperature range for these extremes is 214°F.
Objectives
Section 1.6
Multiplying and Dividing Real
Numbers; Multiplication and
Division Properties
n  Multiply
signed numbers
properties of multiplication
n  Divide signed numbers
n  Use properties of division
n  Use
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Objective 1: Multiply Signed Numbers
n 
Multiplication represents repeated addition.
¨ 
For example, 4(3) is equal to the sum of four 3’s.
n 
This example illustrates that the product of two
positive numbers is positive.
n 
To develop a rule for multiplying a positive number and a negative
number, we will find 4(−3), which is equal to the sum of four −3’s.
¨  4(−3) = −3 + (−3) + (−3) + (−3) = −12.
We see that the result is negative. This example illustrates that the
product of a positive number and a negative number is negative.
n 
¨ 
¨ 
Multiplying Two Numbers That Have Different (Unlike) Signs: To
multiply a positive real number and a negative real number, multiply their
absolute values. Then make the final answer negative.
Multiplying Two Numbers That Have the Same (Like) Signs: To
multiply two real numbers that have the same sign, multiply their absolute
values. The final answer is positive.
EXAMPLE 1
Solution
EXAMPLE 1
n 
Multiply:
a. 8(−12), b. −151× 5, c. (−0.6)(1.2), d. 3/4(−4/15)
Strategy We will use the rule for multiplying two numbers
that have different signs.
Why In each case, we are asked to multiply a positive
number and a negative number.
Objective 2: Use Properties of
Multiplication
n  The
multiplication of two numbers can be
done in any order; the result is the same.
¨ For
example, −9(4) = −36 and 4(−9) = −36.
n  This
illustrates that multiplication is
commutative.
¨  The
Commutative Property of Multiplication:
Changing the order when multiplying does not affect
the answer. For any real numbers a and b, ab = ba.
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Objective 2: Use Properties of
Multiplication
n 
In the following example, we multiply –3 × 7 × 5 in two ways.
¨ 
n 
n 
Recall that the operation within the parentheses should be
performed first.
It doesn’t matter how we group the numbers in this
multiplication; the result is −105. This example illustrates that
multiplication is associative.
¨ 
EXAMPLE 3
Solution
Using the commutative and associative properties of multiplication, we
can reorder and regroup the factors to simplify computations.
b. −4(−3)(−2)(−1)
Strategy First, we will use the commutative and
associative properties of multiplication to reorder and
regroup the factors. Then we will perform the
multiplications.
Why Applying of one or both of these properties before
multiplying can simplify the computations and lessen
the chance of a sign error.
The Associative Property of Multiplication: Changing the
grouping when multiplying does not affect the answer. For any
real numbers a, b, and c, (ab)c = a(bc).
EXAMPLE 3
Multiply:
a. −5(−37)(−2),
Objective 2: Use Properties of
Multiplication (continued)
n 
Some more properties of multiplication:
¨ 
¨ 
¨ 
Multiplying Negative Numbers: The product of an even number of
negative numbers is positive. The product of an odd number of negative
numbers is negative.
Multiplication Property of 0: The product of 0 and any real number is
0. For any real number a, 0 × a = 0 and a × 0 = 0.
Multiplication Property of 1(Identity Property of Multiplication): The
product of 1 and any number is that number. For any real number a, 1 ×
a = a and a × 1 = a.
n 
¨ 
Since any number multiplied by 1 remains the same (is identical), the number
1 is called the identity element for multiplication.
Multiplicative Inverses (Inverse Property of Multiplication): The
product of any number and its multiplicative inverse (reciprocal) is 1. For
any nonzero real number a, a(1/a) = 1.
n 
Two numbers whose product is 1 are reciprocals or multiplicative inverses
of each other. For example, 8 is the multiplicative inverse of 1/8, and 1/8 is
the multiplicative inverse of 8.
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Objective 3: Divide Signed Numbers
n 
Every division fact can be written as an equivalent
multiplication fact.
¨ 
n 
For any real numbers a, b, and c, where b ≠ 0, a/b = c provided
Quotient  divisor = dividend
that c × b = a.
We can use this relationship between multiplication and
division to develop rules for dividing signed numbers.
¨ 
¨ 
For example, 15/5 = 3 because 3(5) = 15.
From this example, we see that the quotient of two positive
numbers is positive.
EXAMPLE 5
n 
Objective 3: Divide Signed Numbers
n  We
summarize this and other such rules
and note that they are similar to the rules
for multiplication.
¨ Dividing
Two Real Numbers: To divide two
real numbers, divide their absolute values.
1. The quotient of two numbers that have the
same (like) signs is positive.
2. The quotient of two numbers that have
different (unlike) signs is negative.
EXAMPLE 5
Solution
Divide and check the result:
a. −81/−9, b. 45/−9, c. −2.87 ÷ 0.7, d. −5/16 ÷ (−1/2)
Strategy We will use the rules for dividing signed
numbers. In each case, we need to ask, “Is it a quotient
of two numbers with the same sign or different signs?”
Quotient  divisor = dividend
Quotient  divisor = dividend
Why The signs of the numbers that we are dividing
determine the sign of the result.
Quotient  divisor = dividend
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Objective 4: Use Properties of Division
n 
Whenever we divide a number by 1, the quotient is
that number.
n 
Furthermore, whenever we divide a nonzero
number by itself, the quotient is 1.
n 
These observations suggest the following
properties of division.
¨  Therefore,
¨  Therefore,
12/1 = 12, −80/1 = −80, and 7.75/1 = 7.75.
35/35 = 1, −4/−4 = 1, and 0.9/0.9 = 1.
¨  Any
number divided by 1 is the number itself.
number (except 0) divided by itself is 1.
For any real number a, a/1 = a and a/a = 1 (where a ≠ 0).
¨  Any
Objective 4: Use Properties of Division
n 
To examine division by zero, let’s look at 2/0
and its related multiplication statement.
n 
There is no number that can make 0(?) = 2 true
because any number multiplied by 0 is equal to
0, not 2. Therefore, 2/0 does not have an
answer.
¨  2/0
¨  We
= ? because ?(0) = 2
say that such a division is undefined.
Objective 4: Use Properties of Division
n  Division
is not commutative. For example,
and 6/3 ≠ 3/6. In general, a/b ≠ b/a.
n  We will now consider division that involves
zero. First, we examine division of zero.
Let’s look at two examples.
¨ 0/2
= 0 because 0(2) = 0.
= 0 because 0(−5) = 0.
¨ 0/−5
n  These
examples illustrate that 0 divided by
a nonzero number is 0.
EXAMPLE 7
Find each quotient, if possible:
a. 0/8, b. −24/0
Strategy In each case, we need to determine if we
have division of 0 or division by 0.
Why Division of 0 by a nonzero number is defined,
and the result is 0. However, division by 0 is
undefined; there is no result.
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EXAMPLE 7
Find each quotient, if possible:
a. 0/8, b. −24/0
Solution
Section 1.7
Exponents and Order of
Operations
Objectives
n  Evaluate
exponential expressions
n  Use the order of operations rules
n  Evaluate expressions containing grouping
symbols
n  Find the mean (average)
Objective 1: Evaluate Exponential
Expressions
n 
In the expression 3 – 3 – 3 – 3 – 3, the number 3 repeats as a
factor five times. We can use exponential notation to write
this product in a more compact form.
Exponent and Base: An exponent is used to indicate repeated
multiplication. It is how many times the base is used as a factor.
¨  In the exponential expression 35,
the base is 3, and 5 is the
exponent. The expression is
called a power of 3.
¨  Some other examples of
exponential expressions are:
¨ 
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Objective 1: Evaluate Exponential
Expressions
n 
To evaluate (find the value of) an exponential expression, we write the
base as a factor the number of times indicated by the exponent. Then we
multiply the factors.
EXAMPLE 2
n 
Evaluate each expression:
a. 53, b. (−2/3)3, c. 101, d. (0.6)2, e. (−3)4, f. (−3)5
Strategy We will rewrite each exponential expression as a
product of repeated factors, and then perform the
multiplication. This requires that we identify the base and
the exponent.
Why The exponent tells the number
of times the base is to be written as
a factor.
EXAMPLE 2
EXAMPLE 2
Solution
Solution
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Objective 1: Evaluate Exponential
Expressions
n 
Objective 2: Use the Order of Operations Rules
n 
Even and Odd Powers of a Negative Number:
¨ 
¨ 
Suppose you have been asked to contact a friend if you see a Rolex
watch for sale when you are traveling in Europe.
¨ 
When a negative number is raised to an even power, the result is positive.
When a negative number is raised to an odd power, the result is negative.
¨ 
¨ 
Although the expressions (−4)2 and −42 look alike, they are not. When we find
the value of each expression, it becomes clear that they are not equivalent.
n 
Objective 2: Use the Order of Operations Rules
n 
For example, consider the expression 2 + 3 – 6.
We can evaluate this expression in two ways.
¨  We
can add first, and then multiply.
¨  Or we can multiply first, and then add. However, the
results are different.
n 
If we don’t establish a uniform order of operations,
the expression has two different values. To avoid
this possibility, we will always use the following set
of priority rules.
While in Switzerland, you find the watch and send the text message
shown on the left.
The next day, you get the response shown on the right.
Something is wrong. The first part of the response (No price too high!) says to
buy the watch at any price. The second part (No! Price too high.) says not to buy
it, because it’s too expensive.
The placement of the exclamation point makes us read the two parts of
the response differently, resulting in different meanings.
When reading a mathematical statement, the same kind of confusion is
possible.
Objective 2: Use the Order of Operations Rules
n 
Order of Operations:
1. Perform all calculations within parentheses and other
grouping symbols following the order listed in Steps 2–4 below,
working from the innermost pair of grouping symbols to the
outermost pair.
¨  2. Evaluate all exponential expressions.
¨  3. Perform all multiplications and divisions as they occur from
left to right.
¨  4. Perform all additions and subtractions as they occur from left
to right.
¨  When grouping symbols have been removed, repeat Steps 2–4
to complete the calculation.
¨  If a fraction is present, evaluate the expression above and the
expression below the bar separately. Then simplify the fraction,
if possible.
¨ 
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EXAMPLE 4
n 
EXAMPLE 4
Evaluate:
a. 3 – 23 − 4, b. −30 − 4 – 5 + 9, c. 24 ÷ 6 – 2,
d. 160 − 4 + 6(−2)(−3)
Strategy We will scan the expression to determine what
operations need to be performed. Then we will perform
those operations, one-at-a-time, following the order of
operations rules.
Solution
a. Three operations need to be performed to evaluate this expression:
multiplication, raising to a power, and subtraction. By the order of operations
rules, we evaluate 23 first.
b. This expression involves subtraction, multiplication, and addition. The order
of operations rule tells us to multiply first.
Why If we don’t follow the correct order of operations, the
expression can have more than one value.
EXAMPLE 4
Solution
c. Since there are no calculations within parentheses nor
are there exponents, we perform the multiplications and
divisions as they occur from left to right. The division occurs
before the multiplication, so it must be performed first.
Objective 3: Evaluate Expressions Containing
Grouping Symbols
n 
Grouping symbols serve as mathematical punctuation
marks. They help determine the order in which an
expression is to be evaluated.
¨ 
d. Although this expression contains parentheses, there are no operations to perform within
them. Since there are no exponents, we will perform the multiplications as they occur from
left to right.
n 
Examples of grouping symbols are parentheses ( ), brackets [ ],
braces { }, absolute value symbols | |, and the fraction bar —.
Expressions can contain two or more pairs of grouping
symbols.
¨ 
To evaluate the following expression, we begin within the
innermost pair of grouping symbols, the parentheses. Then we
work within the outermost pair, the brackets.
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EXAMPLE 6
Evaluate: −4[2 + 3(4 − 82)] − 2
EXAMPLE 6
Evaluate: −4[2 + 3(4 − 82)] − 2
Solution
Strategy We will work within the parentheses first and
then within the brackets. At each stage, we follow the
order of operations rules.
Why By the order of operations, we must work from the
innermost pair of grouping symbols to the outermost.
Objective 4: Find the Mean (Average)
n 
The arithmetic mean (or simply mean) of a set
of numbers is a value around which the values
of the numbers are grouped. The mean is also
commonly called the average.
¨  Finding
An Arithmetic Mean: To find the mean of a
set of values, divide the sum of the values by the
number of values.
EXAMPLE 9
n 
Hotel Reservations
In an effort to improve customer service, a hotel
electronically recorded the number of times the
reservation desk telephone rang before it was
answered by a receptionist. The results of the
week-long survey are shown in the table.
Find the average(mean) number of times the
phone rang before a receptionist answered.
Strategy First, we will determine the total number of times the
reservation desk telephone rang during the week. Then we will
divide that result by the total number of calls received.
Why To find the average value of a set of values, we divide the sum
of the values by the number of values.
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EXAMPLE 9
Hotel Reservations
Solution
To find the total number of rings, we multiply each number of rings (1, 2, 3, 4,
and 5 rings) by the respective number of occurrences and add those subtotals.
Section 1.8
Algebraic Expressions
Total number of rings = 11(1) + 46(2) + 45(3) + 28(4) + 20(5)
The total number of calls received was 11 + 46 + 45 + 28 + 20. To find the
average, we divide the total number of rings by the total number of calls.
Objectives
n  Identify
terms and coefficients of terms
n  Translate word phrases to algebraic
expressions
n  Analyze problems to determine hidden
operations
n  Evaluate algebraic expressions
Objective 1: Identify Terms and
Coefficients of Terms
n 
n 
Recall that variables and/or numbers can be combined with the
operations of arithmetic to create algebraic expressions.
¨ 
Addition symbols separate expressions into parts
called terms. For example, the expression x + 8 has
two terms.
¨ 
Since subtraction can be written as addition of the opposite, the expression
a2 − 3a − 9 has three terms.
In general, a term is a product or quotient
of numbers and/or variables. A single
number or variable is also a term.
¨ 
Examples of terms are: 4, y, 6r, −w3, 3.7x5,
3/n, −15ab2.
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Objective 1: Identify Terms and
Coefficients of Terms
n 
The numerical factor of a term is called the coefficient of
the term.
¨ 
¨ 
¨ 
n 
For instance, the term 6r has a coefficient of 6 because 6r = 6 – r.
The coefficient of −15ab2 is −15 because −15ab2 = −15 – ab2.
More examples are shown below.
A term such as 4, that consists of a single number, is called
a constant term.
EXAMPLE 1
n 
Identify the coefficient of each term in the
expression: 7x2 − x + 6.
Strategy We will begin by writing the subtraction as
addition of the opposite. Then we will determine the
numerical factor of each term.
Why Addition symbols separate expressions into terms.
EXAMPLE 1
Solution
If we write 7x2 − x + 6 as 7x2 + (−x) + 6, we see that it has three
terms: 7x2, −x, and 6. The numerical factor of each term is its
coefficient.
Objective 2: Translate Word Phrases to
Algebraic Expressions
n 
The four tables show how key phrases can
be translated into algebraic expressions.
The coefficient of 7x2 is 7 because 7x2 means 7 – x2.
The coefficient of −x is −1 because −x means −1 – x.
The coefficient of the constant 6 is 6.
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Objective 2: Translate Word Phrases to
Algebraic Expressions
EXAMPLE 3
n 
Write each phrase as an algebraic expression:
a. one-half of the profit P
b. 5 less than the capacity c
c. the product of the weight w and 2,000, increased by 300
Strategy We will begin by identifying any key phrases.
Why Key phrases can be translated to mathematical
symbols.
EXAMPLE 3
Solution
The algebraic expression is: ½P.
Objective 3: Analyze Problems to
Determine Hidden Operations
n  Many
applied problems require insight
and analysis to determine which
mathematical operations to use.
Sometimes thinking in terms of specific numbers makes translating easier.
Suppose the capacity was 100. Then 5 less than 100 would be 100 − 5. If the
capacity is c, then we need to make it 5 less. The algebraic expression is: c − 5.
In the given wording, the comma after 2,000 means w is first multiplied by
2,000; then 300 is added to that product. The algebraic expression is: 2,000w +
300.
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EXAMPLE 7
n 
Vacations
Disneyland, in California, was in operation 16 years before
the opening of Disney World in Florida. Euro Disney, in
France, was constructed 21 years after Disney World. Write
algebraic expressions to represent the ages (in years) of
each Disney attraction.
Strategy We will begin by letting x = the age of Disney
World.
Why The ages of Disneyland and Euro Disney are both
related to the age of Disney World.
EXAMPLE 7
Vacations
Solution
In carefully reading the problem, we see that Disneyland was built 16
years before Disney World. That makes its age 16 years more than
that of Disney World. The key phrase more than indicates addition.
x + 16 = the age of Disneyland
Euro Disney was built 21 years after Disney
World. That makes its age 21 years less than that
of Disney World. The key phrase less than
indicates subtraction.
x − 21 = the age of Euro Disney
Objective 4: Evaluate Algebraic
Expressions
n  To
evaluate an algebraic expression, we
substitute given numbers for each
variable and perform the necessary
calculations in the proper order.
EXAMPLE 10
n 
Evaluate each expression for x = 3 and y = −4:
a. y3 + y2, b. −y − x, c. |5xy − 7|, d. (y − 0)/[x − (−1)]
Strategy We will replace each x and y in the expression
with the given value of the variable, and evaluate the
expression using the order of operation rule.
Why To evaluate an expression means to find its numerical
value, once we know the value of its variable(s).
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EXAMPLE 10
EXAMPLE 10
Solution
Solution
Objectives
Section 1.9
Simplifying Algebraic
Expressions Using Properties
of Real Numbers
n  Use
the commutative and associative
properties.
n  Simplify products
n  Use the distributive property
n  Identify like terms
n  Combine like terms
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Objective 1: Use the commutative and
associative properties.
n 
The Commutative and Associative properties of
multiplication can be explained as follows.
n 
Commutative Properties
¨ 
Changing the order when adding or multiplying does not affect the
answer.
a+b=b+a
and
ab = ba
EXAMPLE 1
n 
Use the given property to complete each statement:
Strategy To fill in each blank, we will determine the way in
which the property enables us to rewrite the given
expression.
Objective 1: Use the commutative and
associative properties.
n 
Associative Properties
¨ 
Changing the grouping when adding or multiplying does not affect
the answer.
(a + b) + c = a + (b + c)
and
(ab) = a (bc)
¨ 
These properties can be applied when working with algebraic
expressions that involve addition and multiplication.
EXAMPLE 1
Solution
a. To commute means to go back and forth. The commutative
property of addition enables us to change the order of the
terms, 9 and x.
b. We change the order of the factors, t and 5.
Why We should memorize the properties by name
because their names remind us how to use them.
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EXAMPLE 1
Solution
c. To associate means to group together. The associative
property of addition enables us to group the terms in a
different way.
d. We change the grouping of the factors, 6, 10, and n.
EXAMPLE 2
n 
Multiply:
a. −9(3b), b. 15a(6), c. 3(7p)(−5),
d. (8/3)(3r/8), e. 35(4/5)x
Objective 2: Simplify Products
n 
The commutative and associative
properties of multiplication can be used to
simplify certain products.
¨ 
For example, let’s simplify 8(4x).
¨ 
We have found that 8(4x) = 32x. We say that 8(4x) and 32x are
equivalent expressions because for each value of x, they
represent the same number.
EXAMPLE 2
Solution
Strategy We will use the commutative and associative
properties of multiplication to reorder and regroup the
factors in each expression.
Why We want to group all of the numerical factors of an
expression together so that we can find their product.
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EXAMPLE 2
Solution
Objective 3: Use the Distributive Property
n 
To illustrate one use of the
distributive property, let’s
consider the expression 5(x + 3).
¨  Since
we are not given the value of
x, we cannot add x and 3 within the
parentheses.
¨  However,
we can distribute the multiplication by the factor
of 5 that is outside the parentheses to x and to 3 and add
those products.
Objective 3: Use the Distributive Property
n 
Another property that is often used to simplify algebraic expressions is
the distributive property.
¨ 
To introduce it, we will evaluate 4(5+3) in two ways.
¨ 
Each method gives a result of 32. This
observation suggests the following
property.
The Distributive Property:
For any real numbers a, b, and c,
a(b + c) = ab + ac.
Objective 3: Use the Distributive Property
n 
Since subtraction is the same as adding the opposite,
the distributive property also holds for subtraction.
n 
The distributive property can be extended to several
other useful forms. Since
multiplication is commutative, we have:
¨ 
n 
¨ 
a(b − c) = ab − ac
For situations in which there are more than two terms within
parentheses, we have:
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EXAMPLE 5
n 
Multiply:
a. (6x + 4)½, b. 2(a − 3b)8, c. −0.3(3a − 4b + 7)
EXAMPLE 5
Solution
Strategy We will multiply each term within the
parentheses by the factor (or factors) outside the
parentheses.
Why In each case, we cannot simplify the expression
within the parentheses. To multiply, we must use the
distributive property.
Objective 3: Use the Distributive Property
n 
We can use the distributive property to find the
opposite of a sum.
¨  For
example, to find −(x + 10), we interpret the symbol −
as a factor of −1, and proceed as follows:
n 
Objective 4: Identify Like Terms
n 
Before we can discuss methods for simplifying algebraic
expressions involving addition and subtraction, we need
to introduce some new vocabulary.
Like terms are terms containing exactly the same variables
raised to exactly the same powers. Any constant terms in an
expression are considered to be like terms.
¨  Terms that are not like terms are called unlike terms.
¨ 
In general, we have the following property of real
numbers.
¨  The
Opposite of a Sum: The opposite of a sum is the
sum of the opposites. For any real numbers a and b,
−(a + b) = −a + (−b).
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Objective 4: Identify Like Terms
EXAMPLE 7
n 
List the like terms in each expression:
a. 7r + 5 + 3r, b. 6x4 − 6x2 − 6x, c. −17m3 + 3 − 2 + m3
Strategy First, we will identify the terms of the expression.
Then we will look for terms that contain the same variables
raised to exactly the same powers.
Why If two terms contain the same variables raised to the
same powers, they are like terms.
EXAMPLE 7
Solution
Objective 5: Combine Like Terms
n 
To add or subtract objects, they must be similar.
For example, fractions that are to be added must have a common
denominator.
¨  When adding decimals, we align columns to be sure to add tenths
to tenths, hundredths to hundredths, and so on.
¨  The same is true when working with terms of an algebraic
expression. They can be added or subtracted only if they are like
terms.
¨ 
a. 7r + 5 + 3r contains the like terms 7r and 3r.
b. Since the exponents on x are different, 6x4 − 6x2 − 6x
contains no like terms.
c. −17m3 + 3 − 2 + m3 contains two pairs of like terms: −17m3
and m3 are like terms, and the constant terms, 3 and −2, are
like terms.
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Objective 5: Combine Like Terms
n 
Recall that the distributive property can be written in the
following forms:
¨ 
¨ 
n 
Objective 5: Combine Like Terms
n 
We can simplify 15m2 − 9m2 in a similar way:
n 
In each case, we say that we combined like terms. These
examples suggest the following general rule.
(b + c)a = ba + ca
(b − c)a = ba − ca
We can use these forms of the distributive property in
reverse to simplify a sum or difference of like terms. For
example, we can simplify 3x + 4x as follows:
¨ 
EXAMPLE 8
n 
Simplify by combining like terms, if possible:
a. 2x + 9x, b. −8p + (−2p) + 4p, c. 0.5s3 − 0.3s3,
d. 4w + 6, e. (4/9)b + (7/9)b
Combining Like Terms: Like terms can be combined by adding
or subtracting the coefficients of the terms and keeping the same
variables with the same exponents.
EXAMPLE 8
Solution
Strategy We will use the distributive property in reverse to add
(or subtract) the coefficients of the like terms. We will keep the
same variables raised to the same powers.
Why To combine like terms means to add or subtract the like
terms in an expression.
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