hma13_Chapter01

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INTRODUCTORY MATHEMATICAL ANALYSISFor Business, Economics, and the Life and Social Sciences

2011 Pearson Education, Inc.

Chapter 1Applications and More Algebra


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To model situations described by linear orquadratic equations.

To solve linear inequalities in one variable and

to introduce interval notation. To model real-life situations in terms of

inequalities.

To solve equations and inequalities involvingabsolute values.

To write sums in summation notation and

evaluate such sums.

Chapter 1: Applications and More Algebra

Chapter Objectives


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Chapter Outline

Applications of Equations

Linear Inequalities

Applications of Inequalities

Absolute Value

Summation Notation

1.1)1.2)

1.3)

1.4)

1.5)


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Modeling:Translating relationships in theproblems to mathematical symbols.


1.1 Applications of Equations

A chemist must prepare 350 ml of a chemical

solution made up of two parts alcohol and three

parts acid. How much of each should be used?

Example 1 - Mixture


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Solution:

Let n = number of milliliters in each part.

Each parthas 70 ml.Amount of alcohol = 2n = 2(70)= 140 mlAmount of acid = 3n = 3(70)= 210 ml

705

350

3505

35032

n

n

nn



Example 1 - Mixture


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Fixed cost is the sum of all costs that are

independent of the level of production.

Variable cost is the sum of all costs that aredependent on the level of output.

Total cost = variable cost + fixed cost

Total revenue= (price per unit) x(number of units sold)

Profit= total revenue total cost




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The Anderson Company produces a product for

which the variable cost per unit is $6 and the fixed

cost is $80,000. Each unit has a selling price of

$10. Determine the number of units that must besold for the company to earn a profit of $60,000.



Example 3 Profit



Solution:

Let q = number of sold units.

variable cost = 6q

totalcost = 6q+ 80,000

total revenue = 10q

Since profit = total revenue total cost

35,000 units must be sold to earn a profit of $60,000.

q

q

qq

000,35

4000,140

000,80610000,60



Example 3 Profit

Ch t 1 A li ti d M Al b



A total of $10,000 was invested in two businessventures, A and B. At the end of the first year, A and

B yielded returns of 6%and 5.75 %, respectively, onthe original investments. How was the original

amount allocated if the total amount earned was$588.75?



Example 5 Investment




Solution:

Letx = amount ($) invested at 6%.

$5500 was invested at 6%$10,000$5500 = $4500 was invested at 5.75%.

5500

75.130025.0

75.5880575.057506.0

75.588000,100575.006.0

x

x

xx

xx



Example 5 Investment






Example 7 Apartment Rent

A real-estate firm owns the Parklane GardenApartments, which consist of 96 apartments. At$550per month, every apartment can be rented.However, for each $25per month increase, there

will be three vacancies with no possibility of fillingthem. The firm wants to receive $54,600per monthfrom rent. What rent should be charged for each

apartment?




Solution 1:

Let r = rent ($) to be charged per apartment.

Total rent = (rent per apartment) x

(number of apartments rented)







Solution 1 (Cont):

Rent should be $650 or $700.

256756

500,224050

32000,365,13440504050

0000,365,140503

34050000,365,1

2534050600,54

25

165032400600,54

25

550396600,54

2

2

r

rr

rr

rr

rr

rr







Solution 2:

Let n = number of $25 increases.

Total rent = (rent per apartment) x

(number of apartments rented)





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1.2 Linear Inequalities

Supposing a and b are two points on the real-number line, the relative positions of two pointsare as follows:


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Rules for Inequalities:

1. If a < b, then a + c < b + c anda c < b c.

2. If a < b and c > 0, then ac < bcand a/c < b/c.

3. If a < b and c < 0, then a(c) > b(c) and a/c > b/c.

4. If a < b and a = c, then c < b.

5. If 0 < a < b or a < b < 0, then 1/a > 1/b .

6. If 0 < a < b and n > 0, then an< bn.

If 0 < a < b, then .



nn ba






Linear inequalitycan be written in the form

ax+ b< 0where a and b are constants and a 0

To solve

an inequality involving a variable is tofind all values of the variable for which theinequality is true.






Example 1 Solving a Linear Inequality

Solve 2(x 3)< 4.Solution:Replace inequality by equivalent inequalities.

5

2

10

2

2

102

64662

462432

x

x

x

x

xx






Example 3 Solving a Linear Inequality

Solve(3/2)(s 2) + 1 > 2(s 4).

7

20

207

16443

442232

422122

32

42122

3

s

s

ss

ss

s-s

ss

The solution is ( 20/7 ,).

Solution:





1.3 Applications of Inequalities

Example 1 - Profit

Solving word problems may involve inequalities.

For a company that manufactures aquariumheaters, the combined cost for labor and material is

$21per heater. Fixed costs (costs incurred in agiven period, regardless of output) are $70,000. If

the selling price of a heater is $35, how many mustbe sold for the company to earn a profit?



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Solution:

profit = total revenue total cost

5000

000,7014

0000,702135

0costtotalrevenuetotal

q

q

qq

Let q = number of heaters sold.



Example 1 - Profit



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After consulting with the comptroller, the presidentof the Ace Sports Equipment Company decides to

take out a short-term loan to build up inventory. The

company has current assets of $350,000 and

current liabilities of $80,000. How much can thecompany borrow if the current ratio is to be no less

than 2.5? (Note: The funds received areconsidered as current assets and the loan as a

current liability.)

p pp g


Example 3 Current Ratio



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Solution:

Letx = amount the company can borrow.

Current ratio = Current assets / Current liabilities

We want,

The company may borrow up to $100,000.

x

x

xx

x

x

000,100

5.1000,150

000,805.2000,350

5.2000,80

000,350

p pp g


Example 3 Current Ratio



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On real-number line, the distance ofx from 0 iscalled the absolute value ofx, denoted as|x|.

DEFINITION

The absolute valueof a real numberx, written |x|,

is defined by

0if,

0if,

xx

xxx

p pp g

1.4 Absolute Value



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p pp g

1.4 Absolute Value

Example 1 Solving Absolute-Value Equations

a. Solve |x 3| = 2

b. Solve |7 3x| = 5

c. Solve |x 4| = 3



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Solution:

a.x 3 = 2 or x 3 = 2x = 5 x = 1

b. 7 3x = 5 or 7 3x = 5

x = 2/3 x = 4

c. The absolute value of a number is nevernegative. The solution set is .

g

1.4 Absolute Value

Example 1 Solving Absolute-Value Equations



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Absolute-Value Inequalities

Summary of the solutions to absolute-valueinequalities is given.

1.4 Absolute Value


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Properties of the Absolute Value

5 basic properties of the absolute value:

Property 5 is known as the triangle inequality.

baba

aaa

abba

b

a

b

a

baab

.5

.4

.3

.2

.1

1.4 Absolute Value



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1.4 Absolute Value

Example 5 Properties of Absolute Value

323251132g.

222f.

5

3

5

3

5

3e.

3

7

3

7

3

7;

3

7

3

7

3

7d.

77c.

24224b.

213737-a.

-

xxx

xx

Solution:



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1.5 Summation Notation

DEFINITIONThe sum of the numbers ai, with isuccessivelytaking on the values mthrough nis denoted as

nmmm

n

mi

i aaaaa ...

21



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Evaluate the given sums.

a. b.

Solution:

a.

b.


Example 1 Evaluating Sums

7

3

25n

n

6

1

2 1j

j

115

3328231813

275265255245235257

3

n

n

97

3726171052

1615141312111222222

6

1

2

j

j



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To sum up consecutive numbers, we have

where n= the last number.

2

1

1

nni

n

i




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Evaluate the given sums.

a. b. c.

Solution:

a.

b.

c.

550,2510032

10110053535

100

1

100

1

100

1

kkk

kk

300,180,246

401201200999

200

1

2200

1

2

kk

kk


Example 3 Applying the Properties of Summation Notation

2847144471

1

100

30

ij

100

1

35k

k

200

1

29k

k

100

30

4j

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