MA 103: College Algebra Chapter 1

Estimate the number of ping pong balls that can be packed in this room V: volume of room v: volume of ping pong ball n: number of v in V V = l x w x h V ≈ (10’)(10’)(10’) = 103 (ft3) ≈ 103 x (12 in)3 ≈ 103 x (10 in)3 ≈ 103 x (10)3 (in3) = 106 (in3) v ≈ 1” x 1” x 1” = 1 (in3) n = V/n ≈ 106 (in3)/1 (in3) = 106

Estimate the area of earth’s surface which is covered by water A: surface area of sphere r: radius of earth A = 4 ∏ r2 r = 4000 (mi) A = 4 (3.14) (4000)2 ≈ 12 (4 x 103)2 ≈ 12 (16 x 106) ≈ 192 x 106 (sq mi) ≈ 200 x 106 (sq mi)

Estimate the area of earth’s surface which is covered by water Aw: area covered by water Aw = 70% of A = 0.7 x (200 x 106 ) ≈ 140 x 106 (sq mi) ≈ 100 x 106 (sq mi) d: average depth of ocean ≈ 2.61 (mi) ≈ 3 (mi) V: volume of ocean water V = Aw x d V ≈ (100 x 106 ) x 3 ≈ 300 x 106 (cu mi) Actual volume of ocean water (US Geol. Survey) 320 x 106 (cu mi)

Introduction Arithmatic Algebra Manipulation of numbers using arithmetic operations-- +, - , x, ÷ 2 + 3 = 5 Algebra Use of symbols as variables and constants -- x, y, ∏ Use of formulas to express relationships among quantities – A = 4 ∏ r2

1.1 Algebraic Expression A number increased by six -- x + 6 Nine less than a number -- x – 9 Four more than five times a number -- 5x + 4 Two more than the quotient of five and a number -- 5/x + 2

Your Turn Express as algebraic expressions Four more that 5 times a number Two less than the quotient of five and a number

Evaluating Expressions 8 + 6x, for x = 5 8 + 6(5) = 8 + 11 = 19 x2 – 4(x – y), for x = 8 and y = 3 (8)2 – 4(8 – 3) = 64 – 4(5) = 64 – 20 = 44

Real Numbers (1.1)  

Real Number Sets Natural Numbers Whole Numbers Integers N = {1, 2, 3, 4, 5 …} (roster method) N = {x | x = 1, 2, 3, …} (set-builder not.) Whole Numbers W = {0, 1, 2, 3 …} W = {x | x = 0, 1, 2, 3 …} Integers I = {…-3, -2, -1, 0, 1, 2, 3, …}

Number Line Number Line – to graph real numbers -1 2 1 -2 3 -3

Number Line Which of the following is true? -3 < -2 -2 > -3 -1 2

Using Inequalities to Describe a set of Numbers All positive numbers Set of all x such that x >= 0 {x | x > 0} All negative numbers (your turn) All numbers less than 12 or greater than 65 -1 2 1 -2 3 -3

Using Inequalities Illustrate the following set of numbers using A) number line B) set notation All non-positive numbers All numbers between 12 and 20 All numbers less than -10

Roster Method to Describe a Set For each case, use a roster method to list the elements of the set. {x | x is a natural number less than 3} {x | x is integer and -3 < x <= 2} {x | x odd number and x > -3}

Operations with Real Numbers (1.2) Absolute value of x |x| = x, if x >= 0 -x, if x < 0 Represents distance between 0 and x |3| = 3 |-3| = 3 -1 2 1 -2 3 -3

Subtraction with Real Numbers a – b = a + (-b) 5 – (3) = 5 + (-3) 5 – (-3) = 5 + (3) 5 – (-3)2 = 5 – (-3)(-3) = 5 – 9 = - 4

Exponential Expressions Evaluate A) (-2)2 B) -22 C) (-5)3 D) (2/3)4 E) (-2/3)4

Dividing Real Numbers a ÷ b or a/b means a . 1 b 5 ÷ 3 = 5 3 = 5 . (1/3) = 1.666666…

Your turn Evaluate Solutions -4 – (-3) 2 – (-3 – 1) (-1)2 – 1 (-2/3)2 – (-5/9) Solutions -4 + 3 = -1 2 – (-4) = 2 + 4 = 6 1 – 1 = 0 (-2/3)(-2/3) – (-5/9) = (-2)2/(3)2 + 5/9 = 4/9 + 5/9 = 1

1.3 Graphing Equations Algebra Geometry Coordinate Geometry Deals with numbers, their operations, and their relationships Geometry Deals with shapes, figures, and their relationships Coordinate Geometry Combines algebra and geometry

Plotting Points 1 -2 2 3 -1 4 -3 -4 B(2, 1) D(-1, -2) C(-3, 1) A(4, 3)

Plotting Points 1 -2 2 3 -1 4 -3 -4 A(2, 1) B(3, 1.5) x y 2 1 3 1.5 A(2, 1) B(3, 1.5) x y 2 1 3 1.5 4 2 C(4, 2)

Graphs of Equations y = -2x x y -5 10 0 0 2 -4 4.5 -9 5 -15 15 10 5 -5 10 0 0 2 -4 4.5 -9 5 -15 15 10 5 -20 -15 -10 -5 5 10 15 20 -5 -10

Graphs of Equations y = x2 - 3 x y -3 6 -1 -2 0 -3 2 1 2.2 1.44 4 13 -3 6 -1 -2 0 -3 2 1 2.2 1.44 4 13 15 10 5 -20 -15 -10 -5 5 10 15 20 -5 -10

Applications T = 1.5H + 10 Temperature F = -5S2 + 10 Hours Frequency Score

Your Turn Plot the following equations y = 2x y = absolute value of (x) y = 2x2 – 1 y = 1/x

Your Turn Plot the equation. y = 2x – 4 Plot the equation. y = -x2

Graphical Description An airplane flew from San Francisco to San Jose Plane’s height Time after takeoff

Graphical Description Measurements are taken from a person’s height from birth to age 100 Height Age

Graphical 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. Speed Time

1.4 Solving Linear Equations Modeling college cost T = 385 x + 3129 where T is the average total cost at public 4-year college; x is the number of years after 2000. Thus, in year 2000, T = 3129. In what year will T be over $10,000? Solve for x in: 10000 = 385x + 3129 Linear equation in one variable ax + b = 0

Solving Linear Equation Solve: 2x + 3 = 17 2x = 17 – 3 2x = 14 x = 7 Solve: 2x – 7 + x = 3x + 1 + 2x 2x + x – 3x – 2x = 1 + 7 -2x = 8 x = -4

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

1.5 Problem Solving and Using Formulas Fahrenheit and Celsius temperatures F = (9/5)C + 32 What is the Fahrenheit reading when the room temperature is 20°C? C = (5/9)(F – 32) What is the boiling point of water in C? (212 °F).

Volume of circular cylinder: V = ∏r2h Solve for h V = ∏r2h V/(∏r2) = (∏r2h)/(∏r2) h = V/(∏r2) r h

Amount(A) that a principal(P) invested at a given rate (r) after a number of years(t) A = P + Prt Solve the formula for P. A = P(1 + rt) P = A/(1 + rt)

1.5 Properties of Integral Exponents Using Product Rule b3 ∙ b5 = b(3 + 5) = b8 Using Quotient Rule b7 / b3 = b(7 – 3) = b4 Using Power Rule (b3)4 = b(3 ∙ 4) = b12

Negative Exponent  

Zero as Exponent  

Your Turn Multiply/divide to simplify expression Solutions 3x4 ∙ 2x2 (4x5y4)(20x7y8) 15x9/3x4 Solutions 3x4 ∙ 2x2 = 6x6 (4x5y4)(20x7y8) = 80x11y12 15x9/3x4 = 5x5

Your Turn (cont.) Write with positive exponents only. Solutions (-5)-2 -5-2 1/5-3 Solutions (-5)-2 = 1/(-5)2 = 1/25 -5-2 = -1/52 = -1/25 1/5-3 = 53 = 125

Your Turn (cont.)  

1.7 Scientific Notation National Debt: $18,000,000,000,000 = $1.8 x 1013 US Population: 340,000,000| = 3.4 x 108 Debt per person: ($1.8 x 1013) / (3.4 x 108) ≈$0.53 x 105 = $5.3 x 104

How long does it take for the light to travel 1 foot? d = s ∙ t t = d / s d = 1 ft ≈ 0.3 m = 3 x 10-1 s = 300,000,000 m/sec = 3.0 x 108 t = (3 x 10-1) / (3.0 x 108) = 1.0 x 10-9 sec (1 nanosec)

Your Turn Express in scientific notation 32,000 -1,500 0.0027 Express in normal decimal notation 3.2 x 103 2.7 x 10-5 2.4 x 100