introduction + chapter no 1

8/9/2019 Introduction + Chapter No 1

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IntroductionBUSINESSMATHEMATICS

BA(M) 531

MBA - I


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Important Notes

Maintain a separate register forMathematics

Probability of having a Quiz in eachclass is very high

No late assignments will be

accepted

Switch off cell phones

Late comers will be marked absent


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Division of Marks [ 40 ]

Internal finals = 20

Hourly Exam = 8 (average)

Quiz = 2 Assignment = 2

Attendance = 2

Past papers = 4 Register = 2


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Basic Algebra

Rules

1. - , - = +

2. + , - = -3. - , + = -

4. + , + = +


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Some Preliminaries

Chapter No 1

Pg no 4


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Terms

A variable is a letter which represents anunknown number. Any letter can be used asa variable. such as x , y,z

An algebraic expression contains at leastone variable.

Examples: a, x+5, 3y 2z

An equation is a sentence that states thattwo mathematical expressions are equal.

Example: 2x-16=18


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FirstDegree Equations in

One Variable Three types of equations

Identity equation

Conditional equation False statement


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Steps to Solving Equations

Simplifyeach side of the equation, if needed, by

distributing or combining like terms.

Move variables to one side of the equation by using

the opposite operation of addition or subtraction.

Isolate the variable by applying the opposite

operation to each side.

First, use the opposite operation of addition or subtraction.

Second, use the opposite operation of multiplication ordivision.

Checkyour answer.


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Examples

y is the variable.

Add 6 to each side to isolatethe variable.

Now divide both sides by 3.

The answer is 5.

Check the answer bysubstituting it into theoriginal equation.

3 6 9

6 6

3 15

3 3

5

Check: 3(5)-6=9

15-6=9

y

y

y

!

!

!

!

!


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Try this. . .

Did you getx = - 4?You were right!

4 8 24x !


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Second Degree Equations inone Variable

Quadratic Equations

Generalized form

ax2 +bx + c

Where a,b and c are constantsand

a 0


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Solving QuadraticEquations

Types of roots No real root

One real root

Two real roots

Discriminant = b2 4ac if b2 4ac < 0 no real root

if b2 4ac = 0 One real root if b2 4ac > 0 Two real roots


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Two methods

Factoring Method

Example

X2 -4x =0

X( x 4 ) = 0

Either x =0 or x -4=0x = 4

Quadratic Formula


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An inequality is like an equation, but instead of

an equal sign ( ) it has one of these signs:

< : less than : less than or equal to

> : greater than

: greater than or equal to

Section 1.3 Inequalities


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Types of Inequalities

Absolute Inequality

Which is always true

Conditional Inequality Is true under certain conditions x >

100

Double Inequality

Is an interval 0< y < 100


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Interval Notation

Open interval (a,b)

a < x < b

Closed interval [a,b]

a x b

Half open interval (a,b] or [a,b)


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-5 50 10-10

A number line is a line with marks on it that are placed at

equal distances apart.

One mark on the number line is usually labeled zero and then

each successive mark to the left or to the right of the zero

represents a particular unit such as 1 or .

On the number line above, each small mark represents

unit and the larger marks represent 1 unit.


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-5 50 10-10

Number lines can be used to represent:

A. Whole numbers The set {0, 1, 2, 3, }.

B. Positive numbers any number that is greater than

zero.

C. Negative numbers any number that is less than zero.

D. Integers The set of numbers represented as

{, -3, -2, -1, 0, 1, 2, 3, }

The arrows at the ends of the number line show that the

number line continues in both directions without ending.


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-5 50 10-10

A number can be graphed on a number line by placing a point

at the appropriate position on the number line.

Example

a) {4} (blue point)

b) {integers between 10 and 5} (purple)


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x < 5

means that whatevervalue x has, it must be

less than 5.

Try to name ten numbersthat are less than 5!

line and Inequality


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Numbers less than 5 are to the left

of 5 on the number line.

0 5 10 15-20 -15 -10 -5-25 20 25

If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.

There are also numbers in between the integers, like2.5, 1/2, -7.9, etc.

The number 5 would notbe a correct answer,

though, because 5 is not less than 5.


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x -2

means that whatevervalue x has, it must begreater than or equal to

-2.

Try to name ten numbersthat are greater than or

-


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Numbers greater than -2 are to the

right of 5 on the number line.

0 5 10 15-20 -15 -10 -5-25 20 25

If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.

There are also numbers in between the integers,like -1/2, 0.2, 3.1, 5.5, etc.

The number -2 would alsobe a correct answer,

because of the phrase, or equal to.

-2


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k> 5

A. k+3 > 2

Subtract 3 from both sides.3 3

k+3 > 2

Solve and graph.

Example : Adding and Subtracting to Solve

Inequalities

5 0


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B.r 9 u 12

ru 21

Add 9 to both sides.r 9 + 9 u 12 + 9

r 9 u 12

21 2415

Example 1B: Adding and Subtracting to Solve

Inequalities Continued

Solve and graph.


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C.u 5 e 3

u e 8

Add 5 to both sides.u 5 + 5 e 3 + 5u 5 e 3

8 1050

Solve and graph.

Additional Example 1C: Adding and Subtracting to

Solve Inequalities Continued


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ce 4

D.

c

+ 6e

2

Subtract 6 from both sides.6 6c+ 6 e 2

0 47 4

Solve and graph.

Additional Example 1D: Adding and Subtracting to

Solve Inequalities Continued


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yu 8

A. y+ 7 u 1

Subtract 7 from both sides.7 7y+ 7 u 1

Try This: Example 1A

011 8

Solve and graph.


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5 1

5 is greater than 1.

Sometimes youmustmultiply or dividetoisolatethevariable. Multiplyingor dividingbothsidesofaninequality by anegativenumbergivesa

surprisingresult.

5 > 1

Multiplyboth sides by1.1 5 1 (1) > or < ?

You know 5 islessthan 1, so youshould use 1


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3 > 1

4 e 12

Multiply by 2

Divide by 4 1 u 3

6 < 2

MULTIPLYINGINEQUALITIESBYNEGATIVEINTEGERS

Words Original

Inequality

Multiply/

Divide

Result

Multiplyingordividingby anegative

numberreverses theinequalitysymbol.


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The directionoftheinequality changesonly ifthenumber youareusingtomultiply or divideby isnegative.

HelpfulHint


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A.3yu 15

ye 5

Divide each side by

3;u

changes toe

.

B.7 < 21

m < 3

Divide each side by7.

Additional Example 2: Multiplying and Dividing to

Solve Inequalities

Solve and graph.

0 33 5

3ye 153 3

7m < 21

7 7

7 5 0 4


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A. 8yu 24

ye 3

Divide each side by

8; changes > to 45

f> 5

Divide each side by9.

Try This: Example 2

8y u 248 8

9f> 459 9

3 07 4

0 5 10

Solve and graph.

t


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o v ng so ute a ue

Inequalities

Section 1.4 Solving absolute value inequalities is acombination of solving absolute value

equations and inequalities.

Rewrite the absolute value inequality.

For the first equation, all you have to do is drop

the absolute value bars.

For the second equation, you have to negate theright side of the inequality and reverse the

inequality sign.


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Solve: |2x + 4| > 12

2x+ 4 > 12 or 2x+ 4 < -12

2x> 82x< -16

x> 4 or x< -8

x< -8 orx> 4

0 4-8


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Solve: 2|4 -x| < 10

4 -x< 5 and 4 -x> -5

-x< 1 -x> -9x> -1 and x< 9

|4 -x| < 5

0 9-1

-1


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Assignment # 1

Section 1.1 [5,15,19 ]

Section 1.2 [ 9,27]

Section 1.3 [23,27,35,41 ] Section 1.4 [11,21,24,25,]

Section 1.5 [3,23 ]

Submission Date :Feb 08,2010,to be done on sheets

introduction + chapter no 1

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