MA 103: College AlgebraMA 103: College Algebra

Chapter 1Chapter 1

Estimate the area of earth’s surface Estimate the area of earth’s surface which is covered by waterwhich is covered by water

A: surface area of spherer: radius of earth

A = 4 ∏ r∏ r22

r = 4000 mi= 4000 mi

A = 4 (3.14) (4000)A = 4 (3.14) (4000)2 2

≈ 12 (4 x 1033)22

≈ 12 (16 x 10 ≈ 12 (16 x 1066)) ≈ 192 x 10 ≈ 192 x 106 6 (sq mi)(sq mi)

W: area covered by waterW: area covered by water

W = 70% of AW = 70% of A = 0.7 x A = 0.7 x A ≈ 134 x 10≈ 134 x 106 (6 (cu mi)cu mi)

IntroductionIntroduction

ArithematicArithematic• Manipulation of numbers using Manipulation of numbers using

arithmetic operations-- +, - , x, ÷arithmetic operations-- +, - , x, ÷• 2 + 3 = 52 + 3 = 5

AlgebraAlgebra• Use of symbols as variables and Use of symbols as variables and

constants -- x, y, ∏constants -- x, y, ∏• Use of formulas for express relationshipsUse of formulas for express relationships

1.1 Algebraic Expression1.1 Algebraic Expression

A number increased by sixA number increased by six Nine less than a numberNine less than a number Four more than five times a numberFour more than five times a number Two more than the quotient of five Two more than the quotient of five

and a numberand a number

1.1 Evaluating Expressions1.1 Evaluating Expressions

8 + 6x, for x = 58 + 6x, for x = 5 xx22 – 4(x – y), for x = 8 and y = 3 – 4(x – y), for x = 8 and y = 3

Real Numbers (1.1)Real Numbers (1.1)

Real NumbersReal Numbers• Set of all rational or irrational numbersSet of all rational or irrational numbers

Rational NumbersRational Numbers• Whole numbersWhole numbers• FractionsFractions

Irrational NumbersIrrational Numbers• All other numbersAll other numbers• Cannot be expressed a ratio of whole numbersCannot be expressed a ratio of whole numbers

∏ ∏ = 3.14159265 … = 3.14159265 … e = 2.71828 18284 59045 23536 …e = 2.71828 18284 59045 23536 …

Number LineNumber Line

Number Line to represent real Number Line to represent real numbersnumbers

-1 21-2 3-3 0

InequalitiesInequalities

All positive numbersAll positive numbers

• Set of all x such that x >= 0Set of all x such that x >= 0• {x | x > 0}{x | x > 0}

All negative numbersAll negative numbers

-1 21-2 3-3 0

InequalitiesInequalities

Illustrate the following set of Illustrate the following set of numbers usingnumbers using• A) number lineA) number line• B) set notationB) set notation

All non-positive numbersAll non-positive numbers All numbers between 12 and 20All numbers between 12 and 20 All numbers less than -10All numbers less than -10

Operations with Real Numbers Operations with Real Numbers (1.2)(1.2)

Absolute value of xAbsolute value of x• Distance between 0 and xDistance between 0 and x• |x| = x, if x >= 0|x| = x, if x >= 0

-x, if x < 0 -x, if x < 0

Subtraction with Real NumbersSubtraction with Real Numbers

a – b = a + (-b)a – b = a + (-b) 5 – (3) = 5 + (-3)5 – (3) = 5 + (-3) 5 – (-3) = 5 + (3)5 – (-3) = 5 + (3) 5 – (-3)5 – (-3)22 = 5 – (-3)(-3) = 5 – (-3)(-3)

= 5 – 9 = 5 – 9 = - 4 = - 4

Exponential ExpressionsExponential Expressions

EvaluateEvaluate• A) (-2)A) (-2)22

• B) -2B) -222

• C) (-5)C) (-5)33

• D) (2/3)D) (2/3)44

• E) (-2/3)E) (-2/3)44

Dividing Real NumbersDividing Real Numbers a ÷ b or a/b a ÷ b or a/b

meansmeans

a . a . 11 b b

5 ÷ 35 ÷ 3 = 5 = 5 . . 11 3 3 = = 55 3 3 = 1.666666… = 1.666666…

Practice ExercisesPractice Exercises(Try odd-numbered problems)(Try odd-numbered problems)

Absolute Values (p 23)Absolute Values (p 23)• 1 - 10 1 - 10

SubtractionSubtraction• 35 – 4535 – 45

MultiplicationMultiplication• 47 – 5747 – 57

Order of operationOrder of operation• 83 - 9383 - 93

1.3 Graphing Equations1.3 Graphing Equations

AlgebraAlgebra• Deals with Deals with numbersnumbers and their and their

operations their relationshipsoperations their relationships GeometryGeometry

• Deals with shapes and figuresDeals with shapes and figures Coordinate GeometryCoordinate Geometry

• Combines algebra and geometryCombines algebra and geometry

Plotting PointsPlotting Points

1

-2

2

3

2 3

1

-1 4

-1

-3 -2-4

0

B(2, 1)

D(-1, -2)

C(-3, 1)

A(4, 3)

Plotting PointsPlotting Points

C(4, 2)

1

-2

2

3

2 3

1

-1 4

-1

-3 -2-4

0

A(2, 1)

B(3, 1.5)

x y 2 1

3 1.5

4 2

Graphs of EquationsGraphs of Equations

5

-10

10

15

10 15

5

-5 20

-5

-15 -10-20

0

y = -2x

x y

-5 10

0 0

2 -4

4.5 -9

5 -15

Graphs of EquationsGraphs of Equations

5

-10

10

15

10 15

5

-5 20

-5

-15 -10-20

0

y = x2 - 3

x y

-3 6

-1 -2

0 -3

2 1

2.2 1.44

4 13

ApplicationsApplications

Tem

pera

ture

Hours

T = 1.5H + 10

Frequency

Score

F = -5S2 + 10

Your TurnYour Turn

Plot the following equationsPlot the following equations• y = 2xy = 2x• y = absolute value of (x)y = absolute value of (x)• y = 2xy = 2x22 – 1 – 1• y = 1/xy = 1/x

Exercises (p. 33)Exercises (p. 33)(Try odd-numbered problems)(Try odd-numbered problems)

Graph equationsGraph equations• 1-261-26• 39-4139-41• 43-4543-45

Graphical DescriptionGraphical Description

An airplane flew from San Francisco to San Jose

Pla

ne’s

h

eig

ht

Time after takeoff

Graphical DescriptionGraphical Description

Measurements are taken from a person’s height from birth to age 100

Heig

ht

Age

Graphical DescriptionGraphical Description

You begin your bike ride by riding down a hill. Then you ride up another hill. Finally, you ride along a level surface before coming to a stop.

Sp

eed

Time

1.4 Solving Linear Equations1.4 Solving Linear Equations

1.5 Problem Solving and Using 1.5 Problem Solving and Using FormulasFormulas

1.5 Properties of Integral 1.5 Properties of Integral ExponentsExponents

1.7 Scientific Notation1.7 Scientific Notation