MA 103: College AlgebraMA 103: College Algebra

Chapter 1Chapter 1

Estimate the number of ping pong Estimate the number of ping pong balls that can be packed in this roomballs that can be packed in this room

V: volume of roomv: volume of ping pong balln: number of v in V

V = l x w x hV V ≈ (10’)(10’)(10’) = 10(10’)(10’)(10’) = 103 (ft (ft3))

≈ 10103 xx (12 in) (12 in)3

≈ 10103 xx (10 in) (10 in)3

≈ 10103 xx (10) (10)3 (in(in3) = 10) = 106 (in (in3))v ≈ 1” x 1” x 1” = 1 (in3)

n = V/nn = V/n

≈ 10106 (in (in3)/)/1 (in3) = 10106

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 ≈ 12 (16 xx 10 1066)) ≈ 192 ≈ 192 x x 10106 6 (sq mi)(sq mi) ≈ 200 ≈ 200 x x 10106 6 (sq mi)(sq mi)

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

AAww: : area covered by water

AAw w = 70% of A= 70% of A = 0.7 = 0.7 xx (200 x 10 (200 x 106 6 ) ) ≈ 140 ≈ 140 xx 10 106 6 (sq mi) ≈ 100 (sq mi) ≈ 100 xx 10 106 6 (sq mi) (sq mi)

d: d: average depth of ocean average depth of ocean ≈ 2.61 (mi) ≈ 3 (mi)≈ 2.61 (mi) ≈ 3 (mi)V: V: volume of ocean watervolume of ocean water

V V = = AAw w xx d dV ≈ (100 V ≈ (100 xx 10 1066 ) x 3 ) x 3 ≈ 300 ≈ 300 x x 10 1066 (cu mi) (cu mi)

Actual volume of ocean water Actual volume of ocean water (US Geol. Survey)(US Geol. Survey)

320 x 10320 x 1066 (cu mi) (cu mi)

IntroductionIntroduction

ArithmaticArithmatic• 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 to express relationships Use of formulas to express relationships

among quantities – A = 4 among quantities – A = 4 ∏∏ r r22

1.1 Algebraic Expression1.1 Algebraic Expression

A number increased by sixA number increased by six-- x + 6-- x + 6

Nine less than a numberNine less than a number-- x – 9 -- x – 9

Four more than five times a numberFour more than five times a number-- 5x + 4-- 5x + 4

Two more than the quotient of five Two more than the quotient of five and a numberand a number-- 5/x + 2-- 5/x + 2

Your TurnYour Turn

Express as algebraic expressionsExpress as algebraic expressions1.1. Four more that 5 times a numberFour more that 5 times a number

2.2. Two less than the quotient of five and a Two less than the quotient of five and a numbernumber

Evaluating ExpressionsEvaluating Expressions

8 + 6x, for x = 58 + 6x, for x = 5• 8 + 6(5) = 8 + 11 = 198 + 6(5) = 8 + 11 = 19

xx22 – 4(x – y), for x = 8 and y = 3 – 4(x – y), for x = 8 and y = 3• (8)(8)22 – 4(8 – 3) = 64 – 4(5) = 64 – 20 – 4(8 – 3) = 64 – 4(5) = 64 – 20

= 44 = 44

Real Numbers (1.1)Real Numbers (1.1)

Real Number SetsReal Number Sets Natural NumbersNatural Numbers

• N = {1, 2, 3, 4, 5 …} (roster method)N = {1, 2, 3, 4, 5 …} (roster method)• N = {x | x = 1, 2, 3, …} (set-builder not.)N = {x | x = 1, 2, 3, …} (set-builder not.)

Whole NumbersWhole Numbers• W = {0, 1, 2, 3 …}W = {0, 1, 2, 3 …}• W = {x | x = 0, 1, 2, 3 …}W = {x | x = 0, 1, 2, 3 …}

Integers Integers • I = {…-3, -2, -1, 0, 1, 2, 3, …}I = {…-3, -2, -1, 0, 1, 2, 3, …}

Number LineNumber Line

Number Line – to graph real numbersNumber Line – to graph real numbers

-1 21-2 3-3 0

Number LineNumber Line

Which of the following is true?Which of the following is true?1.1. -3 < -2-3 < -2

2.2. -2 > -3-2 > -3

-1 21-2 3-3 0

Using Inequalities Using Inequalities to Describe a set of Numbersto Describe a set of Numbers

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 numbers (your turn)All negative numbers (your turn) All numbers less than 12 or greater All numbers less than 12 or greater

than 65than 65

-1 21-2 3-3 0

Using InequalitiesUsing Inequalities

Illustrate the following set of numbers Illustrate the following set of numbers usingusing

• A) number lineA) number line• B) set notationB) set notation

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

Roster Method Roster Method to Describe a Setto Describe a Set

For each case, use a roster method to For each case, use a roster method to list the elements of the set.list the elements of the set.1.1. {x | x is a natural number less than 3}{x | x is a natural number less than 3}

2.2. {x | x is integer and -3 < x <= 2}{x | x is integer and -3 < x <= 2}

3.3. {x | x odd number and x > -3}{x | x odd number and x > -3}

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

Absolute value of xAbsolute value of x• |x| = x, if x >= 0|x| = x, if x >= 0

-x, if x < 0 -x, if x < 0• Represents distance between 0 and xRepresents distance between 0 and x• |3| = 3|3| = 3

|-3| = 3|-3| = 3

-1 21-2 3-3 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 = = 55 3 3 = 5 = 5 . (1/3). (1/3)

= 1.666666… = 1.666666…

Your turnYour turn• EvaluateEvaluate

1.1. -4 – (-3)-4 – (-3)

2.2. 2 – (-3 – 1)2 – (-3 – 1)

3.3. (-1)(-1)22 – 1 – 1

4.4. (-2/3)(-2/3)22 – (-5/9) – (-5/9)

• SolutionsSolutions1.1. -4 + 3 = -1-4 + 3 = -1

2.2. 2 – (-4) = 2 + 4 = 62 – (-4) = 2 + 4 = 6

3.3. 1 – 1 = 01 – 1 = 0

4.4. (-2/3)(-2/3) – (-5/9) (-2/3)(-2/3) – (-5/9) = (-2)= (-2)22/(3)/(3)22 + 5/9 = 4/9 + 5/9 = 1 + 5/9 = 4/9 + 5/9 = 1

1.3 Graphing Equations1.3 Graphing Equations

AlgebraAlgebra• Deals with Deals with numbersnumbers, their operations,, their operations,

and their relationshipsand their relationships GeometryGeometry

• Deals with shapes, figures, and theirDeals with shapes, figures, and theirrelationshipsrelationships

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

Your TurnYour Turn1.1. Plot the equation.Plot the equation.

y = 2x – 4y = 2x – 4

2.2. Plot the equation.Plot the equation.y = -xy = -x22

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

Modeling college costModeling college cost• T = 385 x + 3129T = 385 x + 3129

where T is the average total cost at public where T is the average total cost at public 4-year college; x is the number of years 4-year college; x is the number of years after 2000. Thus, in year 2000, T = 3129.after 2000. Thus, in year 2000, T = 3129.In what year will T be over $10,000?In what year will T be over $10,000?Solve for x in: 10000 = 385x + 3129Solve for x in: 10000 = 385x + 3129

Linear equation in one variableLinear equation in one variable• ax + b = 0ax + b = 0

Solving Linear EquationSolving Linear Equation

Solve: 2x + 3 = 17Solve: 2x + 3 = 17 2x = 17 – 3 2x = 17 – 3 2x = 14 2x = 14 x = 7 x = 7

Solve: 2x – 7 + x = 3x + 1 + 2xSolve: 2x – 7 + x = 3x + 1 + 2x 2x + x – 3x – 2x = 1 + 7 2x + x – 3x – 2x = 1 + 7 -2x = 8 -2x = 8 x = -4 x = -4

Solve: 2(x – 1) + 3 = x – 3(x + 1)Solve: 2(x – 1) + 3 = x – 3(x + 1) 2x – 2 + 3 = x – 3x - 3 2x – 2 + 3 = x – 3x - 3 2x –x + 3x = -3 + 2 – 3 2x –x + 3x = -3 + 2 – 3 4x = -4 4x = -4 x = -1 x = -1

1.5 Problem Solving and Using 1.5 Problem Solving and Using FormulasFormulas

Fahrenheit and Celsius temperaturesFahrenheit and Celsius temperaturesF = (9/5)C + 32F = (9/5)C + 32• What is the Fahrenheit reading when the What is the Fahrenheit reading when the

room temperature is 20°C?room temperature is 20°C? C = (5/9)(F – 32)C = (5/9)(F – 32)

• What is the boiling point of water in C? What is the boiling point of water in C? (212 °F).(212 °F).

Volume of circular cylinder:Volume of circular cylinder:V = V = ∏∏rr22hh

Solve for hSolve for h• V = V = ∏∏rr22hh

V/(V/(∏∏rr22) = () = (∏∏rr22h)/(h)/(∏∏rr22)) h = V/( h = V/(∏∏rr22) )

h

r

Amount(A) that a principal(P) Amount(A) that a principal(P) invested at a given rate (r) after a invested at a given rate (r) after a number of years(t)number of years(t)• A = P + PrtA = P + Prt

Solve the formula for P.Solve the formula for P.• A = P(1 + rt)A = P(1 + rt)

P = A/(1 + rt)P = A/(1 + rt)

1.5 Properties of Integral 1.5 Properties of Integral ExponentsExponents

Using Product RuleUsing Product Rule• bb3 3 ∙ b∙ b5 5 = b = b(3 + 5) (3 + 5) = b= b88

Using Quotient RuleUsing Quotient Rule• bb77 / b / b33 = b = b(7 – 3) (7 – 3) = b= b44

Using Power RuleUsing Power Rule• (b(b33))44 = b = b(3 ∙ 4) (3 ∙ 4) = b= b1212

Negative ExponentNegative Exponent

Zero as ExponentZero as Exponent

Your TurnYour Turn Multiply/divide to simplify expressionMultiply/divide to simplify expression

1.1. 3x3x4 4 ∙ 2x∙ 2x22

2.2. (4x(4x55yy44)(20x)(20x77yy88))

3.3. 15x15x99/3x/3x44

SolutionsSolutions1.1. 3x3x4 4 ∙ 2x∙ 2x2 2 = 6x= 6x66

2.2. (4x(4x55yy44)(20x)(20x77yy88) = 80x) = 80x1111yy1212

3.3. 15x15x99/3x/3x4 4 == 5x5x55

Your Turn (cont.)Your Turn (cont.) Write with positive exponents only.Write with positive exponents only.

1.1. (-5)(-5)-2-2

2.2. -5-5-2-2

3.3. 1/51/5-3-3

SolutionsSolutions1.1. (-5)(-5)-2 -2 = 1/(-5)= 1/(-5)2 2 = = 1/251/25

2.2. -5-5-2 -2 = -1/5= -1/522 = = -1/25-1/25

3.3. 1/51/5-3 -3 = 5= 533 = = 125125

Your Turn (cont.)Your Turn (cont.)

1.7 Scientific Notation1.7 Scientific Notation

National Debt: $18,000,000,000,000National Debt: $18,000,000,000,000= $1.8 x 10= $1.8 x 101313

US Population: 340,000,000|US Population: 340,000,000|= 3.4 x 10= 3.4 x 1088

Debt per person:Debt per person:($1.8 x 10($1.8 x 101313) / (3.4 x 10) / (3.4 x 1088))≈$0.53 x 10≈$0.53 x 1055 = $5.3 x 10 = $5.3 x 1044

How long does it take for the light to How long does it take for the light to travel 1 foot?travel 1 foot?

d = s ∙ td = s ∙ tt = d / st = d / s

d = 1 ft ≈ 0.3 m = 3 x 10d = 1 ft ≈ 0.3 m = 3 x 10-1-1

s = 300,000,000 m/sec = 3.0 x 10s = 300,000,000 m/sec = 3.0 x 1088

t = (3 x 10t = (3 x 10-1-1) / (3.0 x 10) / (3.0 x 1088) ) = 1.0 x 10 = 1.0 x 10-9-9 sec (1 nanosec) sec (1 nanosec)

Your TurnYour Turn

Express in scientific notationExpress in scientific notation1.1. 32,00032,000

2.2. -1,500-1,500

3.3. 0.00270.0027 Express in normal decimal notationExpress in normal decimal notation

1.1. 3.2 x 103.2 x 1033

2.2. 2.7 x 102.7 x 10-5-5

3.3. 2.4 x 102.4 x 1000