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Any questions on
today’s homework?
(Sections 1.6/1.7)
Reminder: You should be doing
this homework without using a
calculator, because calculators
can’t be used for Quizzes 1 or 2,
Test 1 or the Gateway Quiz.
Quiz coming up next week:
Gateway Quiz Try #1
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8 questions
Online quiz with paper worksheet
no calculator
Worth 80 points (only your best score of the semester counts)
You have had a number of practice Gateway questions
mixed in with your regular homework assignments, but
you should also take Practice Gateway Quiz 1 before the
next class session:
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8 questions on the practice Gateway, same type as on the actual Gateway
You can take the practice Gateway as many times as you want, and only
your best score counts for points. (Practice Gateway is worth 8 points)
Additional paper versions of practice Gateways can be picked up in the
open lab from any TA or teacher.
REMEMBER: Gateway study help available every MTWTh
in Room 203 JHSW.
Quiz coming up the class session after Gateway
Quiz 1 on sections 1.2 through 1.8
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15 questions; 50 minute time limit
Taken in class, but via the online course web site
Worth 60 points
1 attempt
No calculators allowed
Practice Quiz 1 should be taken at least 2-3 times before the
quiz day:
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19 questions; 50 minute time limit; can take it anywhere
Worth 10 points
Unlimited attempts, only best score counts
Each time you take it, you will get a different set of questions, so the more
times you take it, the more chances you’ll have of seeing all the types of
questions that might be on the real quiz.
Don’t use a calculator, since you can’t on the real quiz.
REMEMBER: Come to the open lab in 203 if you need help.
Next Class Session:
Two assignments due at the start of the next
class session:
1. HW 1.2/1.8
2. Gateway HW #2 (paper worksheet required!)
In Class:
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Lecture: Review for Gateway Quiz 1 (this
online quiz will be given at the next class session after
this review lecture)
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Practice: Take a Preview Gateway Quiz
during last half of class session (worth 8 points,
for extra practice).
NOW
CLOSE
YOUR LAPTOPS
(You may reopen them when I finish the
lecture, at which time you can start this
homework assignment.)
Section 1.2
Sets of numbers:
• Natural numbers: {1, 2, 3, 4, 5, 6 . . .}
(These are also called the “Counting Numbers” –
think about how you count out loud: 1, 2, 3,…
• Whole numbers: {0, 1, 2, 3, 4 . . .}
(Just add the number zero to the natural numbers)
• Integers: {. . . –3, -2, -1, 0, 1, 2, 3 . . .}
(All positive and negative counting numbers and
zero)
More sets of numbers:
• Rational numbers – the set of all numbers that can be
expressed as a ratio (or quotient) of integers, with
denominator  0. (In other words, any number that can
be written as a fraction or a ratio.)
Comments on the set of rational numbers:
• All integers are also rational numbers, because to make them
into a fraction, you just write 1 on the bottom (the bottom
number is called the denominator, top number is called the
numerator.)
• If you divide the numerator by the denominator on your
calculator, you will get a decimal number that either ENDS or
REPEATS.
Examples:
1 = 0.25
8 = 1.6 -2 = -0.666
2 = 0.181818…
4
5
3
11
More sets of numbers:
• Irrational numbers – the set of all numbers that
can NOT be expressed as a ratio of integers
(“irrational” literally mean “NOT rational”
_
Examples: Π , √2
These numbers CAN’T be expressed as a fraction,
and their decimal form never ends and never
repeats. (Try entering these numbers on a scientific
calculator and you’ll see this behavior.)
And one last set of numbers (the big one…):
• Real numbers – the set of all rational and
irrational numbers combined
Comments about the real numbers:
• When you draw a number line, every point on the line is
associated with a real number – the number tells how far
the point is from zero, the middle of the number line.
• A negative number means the point is to the LEFT of zero
on a number line; positive numbers are to the right of zero.
–5 –4 –3 –2 –1
0
1
2
3
4
5
Positive
Negative
numbers
numbers
• The actual DISTANCE of a point from zero is called the
ABSOLUTE VALUE of the number, and it’s always
positive (except of course the number zero, whose absolute
value is just zero – more on this later today.)
Page 11 in your online textbook (same in hardcopy version) provides a helpful
diagram of all these number sets and their relationships to each other.
Underneath this diagram on page 11 are some example problems (EXAMPLE
5) that will be useful in preparing to do the homework problems.
Example from homework:
Choose which group of sets the following number belongs
to. Be sure to account for ALL sets.
A.
B.
C.
D.
E.
2
9
Real numbers, rational numbers
Real numbers, irrational numbers
Real numbers, rational numbers, natural numbers
Rational numbers, natural numbers, integers
Irrational numbers, natural numbers
Answer: A
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A number line used to represent ordered real numbers
has negative numbers to the left of 0 and positive
numbers to the right of 0.
Order Property for Real Numbers indicates how to
use inequality signs (< , which means “less than”, and
>, meaning “greater than”).
If a and b are real numbers,
• a < b means a is to the left of b on a number line.
• a > b means a is to the right of b on a number line.
Examples:
•
Fill in the blank with either < or > :
3 ___ 10
-2 ___ 5
-2 ___ -5 (be careful on this one!)
-5 ___ -2
When in doubt, draw the two numbers on a number line.
If the first number is farther LEFT, put in the < sign.
If the first number is to the RIGHT of the second one,
put in the > sign.
The absolute value of a number is the distance of that
number away from 0.
a 0, since distances are non-negative.
Note:  means “greater than or equal to”, so “ 0” really just
means “not negative”, i.e. “either positive or zero”
Sample problems: find the absolute value:
|3| = 3
|-3| = 3
|0| = 0
|- ½ | = ½
•
-|-3| = -3
Fill in the blank with either < or > :
|3| ___ |10|
|-2| ___ | 5 |
|-2| ___ |-5|
|-5| ___ |-2| (be careful on this one!)
-|-5| ___ -|-2| (and this one!)
When in doubt, draw the two numbers on a number line.
If the first number is farther LEFT, put in the < sign.
If the first number is to the RIGHT of the second one,
put in the > sign.
Example from homework:
Rewrite the inequality so that the inequality symbol
points in the opposite direction and the resulting
statement has the same meaning as the given one.
6 < 16
Answer: 16 > 6
Section 1.8
Properties of real numbers
• Commutative property
• of addition: a + b = b + a
• of multiplication: a · b = b · a
Examples:
Complete each statement using the commutative property:
x + 16 = __________
Answer: 16 + x
xy = _______
Answer: yx
More properties of real numbers:
• Associative property
• of addition: (a + b) + c = a + (b + c)
• of multiplication: (a · b) · c = a · (b · c)
Examples:
Complete each statement using the associative property:
(x + 16) + 2y = __________
Answer: x + (16 + 2y)
4·(xy) = _______
Answer: (4x)·y
More examples:
Use the commutative and associative properties to simplify:
8 + (9 + b)
Solution: 8 + (9 + b) = (8 + 9) + b = 17 + b
2(42x)
Solution: 2(42x) = (2·42)x = 84 x
13 + (a + 13)
Solution: 13 + (a + 13) = 13 + (13 + a) =
(13 + 13) + a = 26 + a
• Distributive property of multiplication over addition
• a(b + c) = ab + ac
Examples:
Use the distributive property to write each statement without
parentheses, and then simplify the result where possible:
8(x + 2)
Solution: 8(x + 2) = 8·x + 8·2 = 8x + 16
-2(3x – 4y + 7)
Solution: -2(3x – 4y + 7) = -2·3x + -2·(-4y) + -2·7
= -6x + 8y - 14
1 (6x - 2)
4
Solution: 1 (6x - 2) = 1·6x - 1·(-2) = 1 · 6·x + 1 · (-2) = 3 x - 1
4
4
4
4 1
4 1
2
2
Example from homework:
Choose the property illustrated by the following statement.
2+ 𝑥+5 = 2+𝑥 +5
A.
B.
C.
D.
Associative property of addition
Commutative property of addition
Distributive property
Identity element of addition
Answer: A
More examples:
Use the distributive property to write each sum as a product.
4x + 4y
Solution: 4x + 4y = 4(x + y)
(-1)·5 + (-1)·x
Solution: (-1)· 5 + (-1) · x = (-1)(5 + x) or -(5 + x)
REMINDERS:
HW 1.2/1.8 AND Gateway HW # 2 (with
worksheet!) are due at the start of the
next class session.
You should also take the Practice
Gateway Quiz at least once before
the next class session.
You may now
OPEN
your LAPTOPS
and begin working on the
homework assignment.