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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=lxwxh
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
320 x 106 (cu mi)
(US Geol. Survey)
Introduction

Arithmatic
• 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
1. Four more that 5 times a number
2. 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
• 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
-3
-2
-1
0
1
2
3
Number Line

Which of the following is true?
1. -3 < -2
2. -2 > -3
-3
-2
-1
0
1
2
3
Using Inequalities
to Describe a set of Numbers

All positive numbers
-3
-2
-1
0
1
2
3
• 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
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.
1. {x | x is a natural number less than 3}
2. {x | x is integer and -3 < x <= 2}
3. {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
-3
-2
-1
0
1
2
3
Subtraction with Real Numbers




a
5
5
5
–
–
–
–
b = a + (-b)
(3) = 5 + (-3)
(-3) = 5 + (3)
(-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
1.
2.
3.
4.
-4 – (-3)
2 – (-3 – 1)
(-1)2 – 1
(-2/3)2 – (-5/9)
• Solutions
1.
2.
3.
4.
-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
• Deals with numbers, their operations,
and their relationships

Geometry
• Deals with shapes, figures, and their
relationships

Coordinate Geometry
• Combines algebra and geometry
Plotting Points
3
A(4, 3)
2
1
C(-3, 1)
B(2, 1)
0
-4
-3
-2
-1
1
-1
D(-1, -2)
-2
2
3
4
Plotting Points
3
x
2
y
1
3
1.5
4
2
2
C(4, 2)
B(3, 1.5)
1
A(2, 1)
0
-4
-3
-2
-1
1
-1
-2
2
3
4
Graphs of Equations
y = -2x
x
-5
15
y
10
0
0
2
-4
4.5
-9
5
-15
10
5
0
-20
-15
-10
-5
5
-5
-10
10
15
20
Graphs of Equations
y = x2 - 3
15
10
5
0
-20
-15
-10
-5
5
-5
-10
x
y
-3
6
-1
-2
0
-3
2
1
2.2
1.44
4
13
10
15
20
Applications
Temperature
T = 1.5H + 10
F = -5S2 + 10
Frequency
Hours
Score
Your Turn

Plot the following equations
•y
•y
•y
•y
=
=
=
=
2x
absolute value of (x)
2x2 – 1
1/x
Your Turn
1. Plot the equation.
y = 2x – 4
2. 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
1. 3x4 ∙ 2x2
2. (4x5y4)(20x7y8)
3. 15x9/3x4

Solutions
1. 3x4 ∙ 2x2 = 6x6
2. (4x5y4)(20x7y8) = 80x11y12
3. 15x9/3x4 = 5x5
Your Turn (cont.)

Write with positive exponents only.
1. (-5)-2
2. -5-2
3. 1/5-3

Solutions
1. (-5)-2 = 1/(-5)2 = 1/25
2. -5-2 = -1/52 = -1/25
3. 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
1. 32,000
2. -1,500
3. 0.0027

Express in normal decimal notation
1. 3.2 x 103
2. 2.7 x 10-5
3. 2.4 x 100