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B. Translating Written or Verbal Information into a Mathematical Model

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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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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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.



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