B. Translating Written or Verbal Information into a Mathematical Model
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College Algebra G&M—
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Section R.1 Algebraic Expressions and the Properties of Real Numbers
EXAMPLE 2
ᮣ
Translating English Phrases into Algebraic Expressions
Assign a variable to the unknown number, then translate each phrase into an
algebraic expression.
a. twice a number, increased by five
b. eleven less than eight times the width
c. ten less than triple the payment
d. two hundred fifty dollars more than double the amount
Solution
ᮣ
a. Let n represent the number. Then 2n represents twice the number, and 2n ϩ 5
represents twice the number, increased by five.
b. Let W represent the width. Then 8W represents eight times the width, and
8W Ϫ 11 represents 11 less than eight times the width.
c. Let p represent the payment. Then 3p represents triple the payment, and
3p Ϫ 10 represents 10 less than triple the payment.
d. Let A represent the amount in dollars. Then 2A represents double the amount,
and 2A ϩ 250 represents 250 dollars more than double the amount.
Now try Exercises 15 through 28
ᮣ
Identifying and translating such phrases when they occur in context is an important
problem-solving skill. Note how this is done in Example 3.
EXAMPLE 3
ᮣ
Creating a Mathematical Model
The cost for a rental car is $35 plus 15 cents per mile. Express the cost of renting a
car in terms of the number of miles driven.
Solution
ᮣ
Let m represent the number of miles driven. Then 0.15m represents the cost for
each mile and C ϭ 35 ϩ 0.15m represents the total cost for renting the car.
B. You’ve just seen how
we can create mathematical
models
Now try Exercises 29 through 40
ᮣ
C. Evaluating Algebraic Expressions
We often need to evaluate expressions to investigate patterns and note relationships.
Evaluating a Mathematical Expression
1. Replace each variable with open parentheses ( ).
2. Substitute the values given for each variable.
3. Simplify using the order of operations.
In this process, it’s best to use a vertical format, with the original expression written
first, the substitutions shown next, followed by the simplified forms and the final result.
The numbers substituted or “plugged into” the expression are often called the input
values, with the result called the output value.
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College Algebra G&M—
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CHAPTER R A Review of Basic Concepts and Skills
EXAMPLE 4
ᮣ
Evaluating an Algebraic Expression
Evaluate the expression x3 Ϫ 2x2 ϩ 5 for x ϭ Ϫ3.
Solution
ᮣ
WORTHY OF NOTE
In Example 4, note the importance
of the first step in the evaluation
process: replace each variable with
open parentheses. Skipping this
step could easily lead to confusion
as we try to evaluate the squared
term, since Ϫ32 ϭ Ϫ9, while
1Ϫ32 2 ϭ 9. Also see Exercises 55
and 56.
For x ϭ Ϫ3:
x3 Ϫ 2x2 ϩ 5 ϭ 1 2 3 Ϫ 21 2 2 ϩ 5
ϭ 1Ϫ32 3 Ϫ 21Ϫ32 2 ϩ 5
ϭ Ϫ27 Ϫ 2192 ϩ 5
ϭ Ϫ27 Ϫ 18 ϩ 5
ϭ Ϫ40
replace variables with open parentheses
substitute Ϫ3 for x
simplify: 1Ϫ32 3 ϭ Ϫ27, 1Ϫ32 2 ϭ 9
simplify: 2192 ϭ 18
result
When the input is Ϫ3, the output is Ϫ40.
Now try Exercises 41 through 60
ᮣ
If the same expression is evaluated repeatedly, results are often collected and analyzed in a table of values, as shown in Example 5. As a practical matter, the substitutions
and simplifications are often done mentally or on scratch paper, with the table showing
only the input and output values.
EXAMPLE 5
ᮣ
Evaluating an Algebraic Expression
Evaluate x2 Ϫ 2x Ϫ 3 to complete the table shown. Which input value(s) of x
cause the expression to have an output of 0?
Solution
ᮣ
Input
x
Ϫ2
Ϫ1
Output
x2 ؊ 2x ؊ 3
1Ϫ22 2 Ϫ 21Ϫ22 Ϫ 3 ϭ 5
0
0
Ϫ3
1
Ϫ4
2
Ϫ3
3
0
4
5
The expression has an output of 0 when x ϭ Ϫ1 and x ϭ 3.
Now try Exercises 61 through 66
Figure R.1
Graphing calculators provide an efficient means of
evaluating many expressions. After entering the expression on the Y= screen (Figure R.1), we can set
up the table using the keystrokes 2nd
(TBLSET). For this exercise, we’ll put the table in the
“Indpnt: Auto Ask” mode, which will have the calculator “automatically” generate the input and output
values. In this mode, we can tell the calculator where
to start the inputs (we chose TblStart ϭ Ϫ2), and
have the calculator produce the input values using any
increment desired (we choose ¢Tbl ϭ 1), as shown
in Figure R.2. We access the completed table using
2nd
GRAPH (TABLE), and the result for Example 5 is
shown in Figure R.3.
For exercises that combine the skills from Examples 3 through 5, see Exercises 91 to 98.
Figure R.2
WINDOW
C. You’ve just seen how
we can evaluate algebraic
expressions
Figure R.3
ᮣ
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Section R.1 Algebraic Expressions and the Properties of Real Numbers
D. Properties of Real Numbers
While the phrase, “an unknown number times five,” is accurately modeled by the
expression n5 for some number n, in algebra we prefer to have numerical coefficients
precede variable factors. When we reorder the factors as 5n, we are using the commutative property of multiplication. A reordering of terms involves the commutative
property of addition.
The Commutative Properties
Given that a and b represent real numbers:
ADDITION: a ϩ b ϭ b ϩ a
MULTIPLICATION: a # b ϭ b # a
Terms can be combined in
any order without changing
the sum.
Factors can be multiplied in
any order without changing
the product.
Each property can be extended to include any number of terms or factors. While
the commutative property implies a reordering or movement of terms (to commute
implies back-and-forth movement), the associative property implies a regrouping
or reassociation of terms. For example, the sum 1 34 ϩ 35 2 ϩ 25 is easier to compute if we
regroup the addends as 34 ϩ 1 35 ϩ 25 2 . This illustrates the associative property of addition.
Multiplication is also associative.
The Associative Properties
Given that a, b, and c represent real numbers:
ADDITION:
EXAMPLE 6
ᮣ
MULTIPLICATION:
1a ϩ b2 ϩ c ϭ a ϩ 1b ϩ c2
1a # b2 # c ϭ a # 1b # c2
Terms can be regrouped.
Factors can be regrouped.
Simplifying Expressions Using Properties of Real Numbers
Use the commutative and associative properties to simplify each calculation.
Solution
ᮣ
WORTHY OF NOTE
Is subtraction commutative?
Consider a situation involving
money. If you had $100, you could
easily buy an item costing $20:
$100 Ϫ $20 leaves you with $80.
But if you had $20, could you buy
an item costing $100? Obviously
$100 Ϫ $20 is not the same as
$20 Ϫ $100. Subtraction is not
commutative. Likewise, 100 Ϭ 20 is
not the same as 20 Ϭ 100, and
division is not commutative.
a.
3
8
Ϫ 19 ϩ 58
a.
3
8
Ϫ 19 ϩ
5
8
ϭ Ϫ19 ϩ
b. 3Ϫ2.5 # 1Ϫ1.22 4 # 10
3
5
8 ϩ 8
1 38 ϩ 58 2
ϭ Ϫ19 ϩ
ϭ Ϫ19 ϩ 1
ϭ Ϫ18
b. 3Ϫ2.5 # 1Ϫ1.22 4 # 10 ϭ Ϫ2.5 # 3 1Ϫ1.22 # 10 4
ϭ Ϫ2.5 # 1Ϫ122
ϭ 30
commutative property (order changes)
associative property (grouping changes)
simplify
result
associative property (grouping changes)
simplify
result
Now try Exercises 67 and 68
ᮣ
For any real number x, x ϩ 0 ϭ x and 0 is called the additive identity since the
original number was returned or “identified.” Similarly, 1 is called the multiplicative
identity since 1 # x ϭ x. The identity properties are used extensively in the process of
solving equations.
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College Algebra G&M—
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CHAPTER R A Review of Basic Concepts and Skills
The Additive and Multiplicative Identities
Given that x is a real number,
xϩ0ϭx
Zero is the identity
for addition.
1#xϭx
One is the identity
for multiplication.
For any real number x, there is a real number Ϫx such that x ϩ 1Ϫx2 ϭ 0. The
number Ϫx is called the additive inverse of x, since their sum results in the additive identity. Similarly, the multiplicative inverse of any nonzero number x is 1x , since x # 1x ϭ 1
p q
(the multiplicative identity). This property can also be stated as q # p ϭ 1 1p, q 02 for
q
p
p
any rational number q. Note that q and p are reciprocals.
The Additive and Multiplicative Inverses
Given that p, q, and x represent real numbers 1p, q
02:
x ϩ 1Ϫx2 ϭ 0
p q
# ϭ1
q p
p
q
x and Ϫx are
additive inverses.
EXAMPLE 7
ᮣ
Determining Additive and Multiplicative Inverses
Replace the box to create a true statement:
# Ϫ3 x ϭ 1 # x
a.
b. x ϩ 4.7 ϩ
5
Solution
ᮣ
q
and p are
multiplicative inverses.
a.
b.
ϭx
5
5 # Ϫ3
, since
ϭ1
Ϫ3
Ϫ3 5
ϭ Ϫ4.7, since 4.7 ϩ 1Ϫ4.72 ϭ 0
ϭ
Now try Exercises 69 and 70
ᮣ
Note that if no coefficient is indicated, it is assumed to be 1, as in x ϭ 1x,
1x2 ϩ 3x2 ϭ 11x2 ϩ 3x2 , and Ϫ1x3 Ϫ 5x2 2 ϭ Ϫ11x3 Ϫ 5x2 2.
The distributive property of multiplication over addition is widely used in a
study of algebra, because it enables us to rewrite a product as an equivalent sum and
vice versa.
The Distributive Property of Multiplication over Addition
Given that a, b, and c represent real numbers:
a1b ϩ c2 ϭ ab ϩ ac
A factor outside a sum can be
distributed to each addend in
the sum.
ab ϩ ac ϭ a1b ϩ c2
A factor common to each addend
in a sum can be “undistributed”
and written outside a group.