1. MA 103: College Algebra
Chapter 1

2. 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

3. 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)

4. 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)

5. 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

6. 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

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

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

9. Real Numbers (1.1)

10. 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, …}

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

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

13. 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

14. 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

15. 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}

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

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

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

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

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

21. 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

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

23. 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
C(4, 2)

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

25. Graphs of Equations y = x2 - 3 x y -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

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

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

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

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

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

31. 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

32. 1.4 Solving Linear Equations
Modeling college cost
T = 385 x where T is the average total cost at public 4-year college; x is the number of years after Thus, in year 2000, T = In what year will T be over $10,000? Solve for x in: = 385x
Linear equation in one variable
ax + b = 0

33. Solving Linear Equation
Solve: 2x + 3 = x = 17 – x = x = 7
Solve: 2x – 7 + x = 3x x x + x – 3x – 2x = x = x = -4

34. 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

35. 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).

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

37. 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)

38. 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

39. Negative Exponent

40. Zero as Exponent

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

42. 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

43. Your Turn (cont.)

44. 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

45. 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)

46. 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