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MA 103: College Algebra Chapter 1

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Estimate the area of earth’s surface which is covered by water A: surface area of sphere r: radius of earth ∏ r 2 = 4000 mi A = 4 ∏ r 2 r = 4000 mi A = 4 (3.14) (4000) 2 32 ≈ 12 (16 x 10 6 ) ≈ 192 x 10 6 (sq mi) A = 4 (3.14) (4000) 2 ≈ 12 (4 x 10 3 ) 2 ≈ 12 (16 x 10 6 ) ≈ 192 x 10 6 (sq mi) W: area covered by water W = 70% of A = 0.7 x A ≈ 134 x 10 6 ( cu mi)

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Introduction Arithematic Arithematic Manipulation of numbers using arithmetic operations-- +, -, x, ÷Manipulation of numbers using arithmetic operations-- +, -, x, ÷ 2 + 3 = 52 + 3 = 5 Algebra Algebra Use of symbols as variables and constants -- x, y, ∏Use of symbols as variables and constants -- x, y, ∏ Use of formulas for express relationshipsUse of formulas for express relationships

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1.1 Algebraic Expression A number increased by six A number increased by six Nine less than a number Nine less than a number Four more than five times a number Four more than five times a number Two more than the quotient of five and a number Two more than the quotient of five and a number

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1.1 Evaluating Expressions 8 + 6x, for x = 5 8 + 6x, for x = 5 x 2 – 4(x – y), for x = 8 and y = 3 x 2 – 4(x – y), for x = 8 and y = 3

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Real Numbers (1.1) Real Numbers Real Numbers Set of all rational or irrational numbersSet of all rational or irrational numbers Rational Numbers Rational Numbers Whole numbersWhole numbers FractionsFractions Irrational Numbers Irrational 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 …

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Number Line Number Line to represent real numbers Number Line to represent real numbers 21 -2 3 -3 0

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Inequalities All positive numbers All 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 All negative numbers 21 -2 3 -3 0

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

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Operations with Real Numbers (1.2) Absolute value of x Absolute value of x Distance between 0 and xDistance between 0 and x |x| = x, if x >= 0 -x, if x = 0 -x, if x < 0

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Subtraction 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) 2 = 5 – (-3)(-3) = 5 – 9 = - 4 5 – (-3) 2 = 5 – (-3)(-3) = 5 – 9 = - 4

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

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

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Practice Exercises (Try odd-numbered problems) Absolute Values (p 23) Absolute Values (p 23) 1 - 101 - 10 Subtraction Subtraction 35 – 4535 – 45 Multiplication Multiplication 47 – 5747 – 57 Order of operation Order of operation 83 - 9383 - 93

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

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Plotting Points 1 -2 2 3 23 1 4 -3-2-4 0 B(2, 1) D(-1, -2) C(-3, 1) A(4, 3)

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Plotting Points C(4, 2) 1 -2 2 3 23 1 4 -3-2-4 0 A(2, 1) B(3, 1.5) x y 2 1 3 1.5 4 2

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Graphs of Equations 5 -10 10 15 1015 5 -520 -5 -15-10-20 0 y = -2x x y -5 10 0 2 -4 4.5 -9 5 -15

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Graphs of Equations 5 -10 10 15 1015 5 -520 -5 -15-10-20 0 y = x 2 - 3 x y -3 6 -1 -2 0 -3 2 1 2.2 1.44 4 13

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Applications Temperature Hours T = 1.5H + 10 Frequency Score F = -5S 2 + 10

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

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Exercises (p. 33) (Try odd-numbered problems) Graph equations Graph equations 1-261-26 39-4139-41 43-4543-45

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

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

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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

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1.4 Solving Linear Equations

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1.5 Problem Solving and Using Formulas

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1.5 Properties of Integral Exponents

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1.7 Scientific Notation

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