2.1 The Real Number Line

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2.1 The Real Number Line
Objective To be able to graph numbers on the
number line and to find the opposite and
absolute value of numbers.
Negative Numbers
Positive Numbers
Origin
Graph and label 5.5, ½, -4.25, -9/4
Positive numbers Ex Negative
numbers Ex Zero Ex Integers
Abbr Ex Real numbers
Abbr Ex
Points to the right of zero
Points to the left of zero
Neither positive nor negative
Negative and positive numbers zero
Z
All integers and all numbers in between
(fractions and decimals)
R
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II. Comparing Real Numbers

• Graph -4 and -5 on the number line. Then write
  two inequalities that compare the two numbers.
• Write the following set of numbers in increasing
  order
• 2.3, -4.8, 6.1, 3.5, -2.15, .25, 6.02

3
III. Fraction Time!!!

• To convert a fraction to a decimal,
• Write each set of numbers in increasing order.
• a. b.
• YOU TRY!
• c.3, -3.2, -3.15, -3.001, 3 d.

Divide the numerator by the denominator
4
IV. Opposites
Two points that are the same distance from the
origin but on opposite sides of the origin

• Opposites
• Name the opposite of the following numbers
• a. 10 b. -½ c. 4.5 d. 0

8 units away from origin
8 units away from origin
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V. Absolute Value

• What is absolute value?
• The distance a number is from .
• Distance is ALWAYS !
• What does it look like? We say the
  absolute value of x

Absolute value bars
6

• Tricky Case How are these different from each
  other? From the ones on the previous slide?

VI. Solving Absolute Value Equations Using
Mental Math
So How many answers can these have?
Can you compare and order real numbers and find
the opposite and absolute value of
numbers?? Homework 2.1 p. 67 s 4-15,
28-48E, 58-65, 83-85
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2.1 Solutions

• 12 58.
• 60.
• -.2, -.03, -.02, 0, 0.2, 2 62. 12,799
• -5.1, -5, , 3.4, , 4.1 64. -282
• -6.11, -6.1, -6.08, -6.02, 6.03,
  6.07 72. Sorenstam
• 3 74. Estill
• 2.5 Lidback
• 38. Pitcock
• 40. Sorenstam
• 4 84. C
• 44.
• 2
• 48.

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• 2.2-2.3
• Adding and Subtracting Real Numbers
• Objective To add and subtract real numbers and
  to review rules for addition and subtraction of
  fractions and decimals.
• ? Please, no calculators today ?

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Procedures for Adding Real Numbers

• Adding Same Signs
• Ex -5 -6 Ex 5 6
• Rule

Signs alike ? Add and keep the sign

• Adding Different Signs
• Ex -4 9 Ex 4 (-9)
• Rule

Signs Different ? Subtract and keep the sign of
what you have more of ? ASK!!
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Decimals and Fractions, Oh My!

• DecimalsEx 4.03 3.142 Ex -2.15 4.2
• Rule

Line up the decimals and fill in 0s to make the
numbers have the same length. Be careful with
subtract The number with the larger absolute
value must go first!!
Make common denominators ? Yes I need to see
your work. You add/subtract the numerators, but
keep the denominators the same. If you choose
the least common denominator you will do less
work. ?
11
So What do we do with subtraction?

• Rule
• Cross the line change the sign.

Change subtraction to add the opposite. Then
follow the rules of addition.
So can you add and subtract real
numbers? Assign p. 75 s 12-32 (x4), 33-38,
44,48, 58-60 p. 82 s 16-48 (x4), 59, 64, 72,
76-78
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Adding
Decimals
Fractions
Whole Numbers
Get Common Denominators!
Line Up the Decimals
Same Sign
Opposite Signs
Subtract Keep the sign Of what you have more of
Add Keep the sign
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Subtracting
Change to Addition
Cross the Line, Change the Sign
Two Negatives Make a Positive
Then Follow Your Addition Rules!
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2.2 and 2.3 Solutions

• 4 16. -5 72. -.06/gal
• 1 20. -19 76. 410 ft
• 20. 24. 2.7 78. 2325 ft
• -5 28. -1
• 0 32. 4
• -3.7 36. -1
• Commutative of Addition 40. -5
• Associative of Addition 44. 46
• -81.14 48. 7.9
• 7 50.
• -12 64.
• 88.
• 90. 92.

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2.4 Adding Matrices
Objective To add/subtract matrices ?
Vocabulary What is a matrix?? matrix
(matrices) rectangular arrangement of numbers
into horizontal rows and vertical columns Entry
or element a number in a matrix ex 3, ½ ,
0, 8, -1, 2, 4, -2, 5, size of a matrix (
rows) x ( columns) (really important!!) ex
2 x 3
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equal matrices matrices with equal entries in
corresponding positions ? What does this
mean??
State law prohibits you from adding or
subtracting matrices of different
sizes.Procedure Add corresponding entries
Ex
3 x 2
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You try these!!! Remember - l??k to make sure
the matrices are the same size before you try to
add!!! Otherwise. What do you do??
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So What do we do with subtraction of
matrices?(Guess which one is my favorite. ?)
Rule
Can you add/subtract matrices? Do you know how
to determine the dimensions of a matrix?Assign
p. 89 s 12-25 all
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2.4 Solutions
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Quiz Review
Absolute Value ?What is an absolute
value??Where should it take place in order of
operations?? How many solutions can an absolute
value equation have? Be able to give examples of
each. ? ? Can you find the opposites of
numbers? ? Can you put fractions/decimals and
integers in ascending order? (What is
ascending order?) ? Can you write 2 inequalities
to compare 2 numbers? ? Do all absolute values
have positive answers? Prove it!
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Adding and Subtracting Real Numbers

• What are the rules for adding integers?
• What do you need to remember when adding
  fractions? Can you do this?
• What do you need to do when adding decimals?
• What change do you need to make when subtracting?
• Can you list the 4 properties discussed in
  sections 2.2 and 2.3 and give an example of each?

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2.4 Matrices

• How do you determine the dimensions of a matrix?
• How can you tell if 2 matrices are equal?
• Can you add and subtract matrices?
• Are you ready for a quiz tomorrow? ?

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Review 2.1-2.4

• -2 2. 3. -153.82
• -1.59 5. -5.6 6.
• .053 8. 9.
• -13.1 11. 103.5 12.
• 13. -4.65 14. 15.

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• 16. 10.68 17. 5.9 18.
• 19. 20. 21. 0
• 22a. b. -2.3, -2.1, -2.012, -2.01, -2
• 23a. 5 b. -15 c. 0 d. -5
• 24. 50 and -50
• 25a. b. c. no solution d. 0
• 26. 2, 1, or 0 solutions
• 27a. b. c. no solution
• d. e. f.

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Operation Dominos

• In your group
• Turn the dominos face up.
• Match up the dominos question end to the answer
  end. (There is no real beginning or ending.)

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2.5 Multiplying Real Numbers
Objective To multiply positives and negatives,
including decimals and fractions!!

• Multiplication Rules

Does the number of negative values affect the
sign of the product? How?
27
? Multiplying Decimals ?

• What is the rule for multiplying decimals?
  

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Multiplying Fractions

• What is the rule for multiplying fractions?
  
• What should you do to make whole numbers look
  like
• fractions?

Can you multiply real numbers? This includes
fractions and decimals. Assign p. 96 s
16-54E, 58-60, 64, 65, 72
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Multiplying
Decimals
Fractions
Whole Numbers
Straight Across
I Ignore Decimals Count Numbers Behind
Decimals To Figure Out Where Decimal Goes
Negative X Negative
Positive X Positve
Positive X Negative
Negative X Positive
POSITIVE!
NEGATIVE!
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2.5 Solutions

• 16. -24 36. 58. True
• 18. -1300 38. -b9 60. False
• 20. 8.4 40. -x 64. answers
• 22. -9 42. -48 will
• 24. -18 44. 22 vary
• 26. -70 46. -28 72. C
• 28. -72 48. -27
• 30. 3y 50. 94.51
• 32. -5a3 52. 644.80
• 34. b3 54. -23.36

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Warm-up ?
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2.7 Division of Real Numbers
Objective To divide real numbers
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YOU TRY THESE!! Simplify each expression
34
Simplify for the given value
Can you divide real numbers? Can you determine
the sign of the answer? Assign p. 111 s 6-13,
16-46E,52-55, 72,73
35
Warm-up
Simplify
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2.7 Solutions

• 6. 8. 10. 14 12. - 4
• 3 18. - 6 20. - 8 22.
• 135 26. 52 28. - 6 30. - 48
• 294y 34. - 145z 36. 38.
• 40. 14x 42. - 36x 44. 46. - y
• 6x 3 50. 52. 54. 12
• 72. C

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2.6 The Distributive Property

• Objective To use the distributive property and
  combine like terms.

The distributive property The mailman property
-5 (x 2) (x 4) 8 -4 (x 1) (x 5)
9
ab
ac
a (b c) (b c) a a (b c) (b c) a

ac
ab
- ac
ab
- ac
ab
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You try these! ?

• 2(x 5) 5. (x - 4)x
• (156x) x 6. y(2 - 6y)
• -3(x 4) 7. (y 5)(-4)
• -(6 - 3x) 8.

What are like terms? Give some examples
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Simplified Expression
expression with no grouping symbols and all like
terms combined
What does this mean you need to do?
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What do you do if there are parenthesis?

• 2a(a 3) 4(a 2) 2. 3x 2(5x 1)
• 3. 10 (x 3) 4(x2 3x) 4. 2w(w 5)
  (w 6) 4w2

You try these!! 5. -3(3m 5) 2m(m 1) 6.
x(4x 2) 5x(x 2)
Can you distribute and combine like
terms? Assign p. 103 s 1-3,6-8,13-22, 27-69
(x3), 72-74, 79-82
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2.6 Solutions

• -6 and 7, - 3x2 and 9x2, 3y and - 4y 6. 7y
  133
• - 4u 8 14. 2x2 16. - 4x2 4x 8
• 18. - 9w 12 2x2 20. True 22. False
  9(13) 2(13)
• 30. -3r 24 36. - 7s s2 42. - 24t 2t2
• y3 y2 54. 4 2a 60. 5b 5
• -10y 8 72. A 74. 15,897.6 tons
• 80. 14x 12 82. 18x 36

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Warm Up Review for Quiz

• How should you determine the sign of the answer?
• What do you need to remember about dividing
  fractions?
• What do you need to do when you multiply decimals?

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2.8 Probability and Odds
Objective To understand the difference between
probability and odds and to be able to calculate
both!
Probability
The measure of how likely an event is to happen!
Favorable cases and Possible cases

• How many possible cases are there for
• Rolling a die? (2dice?) b) Spinning a
  spinner?
• c) tossing a coin? d) Choosing a card from a
  deck?
• So it really depends on

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• Spinners
• 1st Are all the sections the same size?
• Why is this important?
• 2nd How many sections are there?
• What is this number?
• P(A) 2) P(C or D) 3) P(Z) 4) P (letter)
  

A
D
B
C

• What is the largest probability you can get?
  What does this mean?
• 6) What is the smallest probability you can get?
  What does this mean?
• 7) Give an example of both cases.

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You try this one!
I have a bag full of sticks of gum. I have 15
Juicy Fruit pieces, 10 Big Red Pieces, and 11
pieces of Extra. P(Juicy Fruit) P(Extra)
P(not Big Red) P(gum)
Find the probability for a deck of 52 cards. (no
jokers) ? P(red 4) P(king ?) P(8 or 9)
P(not 2) P(not 2 or 3) P(green
8) P(?or ?) P(12?) P(4 or 9)

Just because the probability of tossing a coin
and getting tails is ½ , does that mean you will
definitely get tails half of the time?
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Odds are different from probability Explain this
difference
Examples

• A candy dish contains 12 miniature chocolate bars
  and 21 hard candies. What are the odds that a
  candy picked at random from the dish will be a
  hard candy?
• You randomly choose an integer from 0 through 9.
  What are the odds that the integer is 4 or more?

21
favorable outcomes unfavorable outcomes
odds are for you
12
favorable outcomes unfavorable outcomes
6
odds are for you
4
47

• You try these!
• You randomly choose an integer from 0 to 9.
  What are the odds that the integer is 1 or
  more???
• What are the odds of randomly choosing the letter
  i from the word Mississippi?
• 3) If the odds of winning are 45, what is the
  probability of winning? Probability
  of losing?

Do you understand the difference between
probability and odds? Can you calculate each for
a given situation? Assign p. 117 s 9 -
30,33,34
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2.8 Solutions

• 0.2 30. 3 to 652
• 0.25 34. C
• 1 to 3
• 1 to 7
• 1 to 5
• 1 to 1 or even
• 22.
• 0.91
• 0.15
• 28.

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WARM-UP
What set(s) of numbers do the following belong to?
50
Chapter 2 Review Solutions

• a. -4 b. -22 c. 5.01 d. -3.39
• a. 96 b. c. d. -5x4
• e. -12a3 f. 64 g. -9 h. -y4
• a. b. -18 c. d.
• e. 16 f. 7 g.
• a. 6x 10 b. -6x2 24x
• c. -12x 54y d. 8x2 20xy

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• 5a. 4x b. 6 4c c. -2x2 5x 8 d.
  4x 3
• 6a. 8x 9 b. -3x 10 c. -3x2 21 d.
  12 8x
• e. 10x 18 f. 6x 6xy g. -3x 28
• a. -3.8, -2.33, -2.1, 0
• b.
• 8a. 6 b. -32 c. -5 d.
• e. f. No solution g. 0 h.
• 9a. Commutative of b. Associative of
• d. Distributive d. Identity of

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• 10a. b. Not possible c.
• 11a. b. c. d. 10
• Natural counting numbers (N)
• Whole 0 and the counting numbers (W)
• Integers positive and negative whole numbers
  (Z)
• Rational decimals and fractions (Q)
• Irrational decimals that never end and never
  repeat (I)
• Reals All numbers (except imaginary) (R)
• 12 a. Z, Q, R e. I
• b. N, W, Z, Q, R f. N, W, Z, Q, R
• c. Q, R g. Q, R
• d. Z, Q, R h. Q, Z