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.

Gateway Quiz Try #1• 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:

• 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 from 8:00 a.m. – 7:30 p.m. in Room 203 JHSW.

Quiz coming up next week:

Quiz coming up the class session after Gateway

Quiz 1 on sections 1.2 through 1.8• 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:• 15 questions; 50 minute time limit; can take it anywhere• Worth 6 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. M – Th: 8:00 a.m. – 7:30 p.m.

Next Class Session:

1. HW 1.2/1.8 is due at the start of the next class session.

2. Gateway HW #3 also due next class session (paper worksheet required!)

3. Review lecture for Gateway Quiz 1 (this online quiz will be given at the next class session after this review lecture)

4. Take a Preview Gateway Quiz during last half of class session (worth 5 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

combinedComments 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.

• 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.)

2– 2 0 1 3 4 5– 1– 3– 4– 5

Negative numbers

Positive numbers

Check on page 11 of your textbook (either the online version or the paper copy) to see this diagram that shows the relationship between all of the sets of numbers:

(It would be worth copying this into your class notebook and studying it as you prepare for Quiz 1.)

Accessing the online textbook, power point lecture slides, and the two homework assignments due at the next class session:

Answer: a

Answer: e

• 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.

Answer: True

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.

Answer: <

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:

Complete each statement using the associative property:

(x + 16) + 2y = __________ Answer: x + (16 + 2y)

4·(xy) = _______ Answer: (4x)·y

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

Answer: s

• 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

Answer: -63x - 13

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)

Answer: 2(x +y)

REMINDERS:HW 1.2/1.8 AND Gateway HW # 3 (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 LAPTOPSand begin working on the

homework assignment.