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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
-10 -9   -8   -7 -6   -5   -4   -3   -2   -1   0   1   2   3   4   5   6   7   8   9   10   11

Origin
Graph and label 5.5, ½, –4.25, –9/4

Positive numbers: Points to the right of zero                                               Ex: 4.3
Negative numbers: Points to the left of zero                                                Ex: -1.6
Zero: Neither positive nor negative                                                         Ex: 0
Integers: Negative and positive numbers & zero Abbr: Z                                      Ex:-6, 4, 0
Real numbers: All integers and all numbers in Abbr: R                                       Ex:-2.8, -4,
between (fractions and decimals)
0, 7, 8/3
II. Comparing Real Numbers
• We compare numbers in order by their location
on the number line.

• Graph –4 and –5 on the number line. Then write
two inequalities that compare the two numbers.

-10 -9      -8   -7 -6    -5   -4   -3    -2    -1   0       1       2       3   4       5   6       7   8   9     10    11

Since –5 is farther left, we say                                                 –4 > –5                     or             –5 < –4

• Put –1, 4, –2, 1.5 in increasing order
-10 -9   -8    -7 -6   -5   -4   -3    -2    -1    0       1       2       3   4       5   6       7   8   9    10    11

Left to right                              –2, –1, 1.5, 4
II. Comparing Real Numbers

• Write the following set of numbers in increasing
order:

-10 -9   -8    -7 -6   -5   -4   -3   -2   -1   0   1   2   3   4   5   6   7   8   9   10   11

–2.3, –4.8, 6.1, 3.5, –2.15, 0.25, 6.02

–4.8, –2.3, –2.15, 0.25, 3.5, 6.02, 6.1
III. Fraction Time!!!
• To convert a fraction to a decimal,
Divide the numerator by the denominator

• Write each set of numbers in increasing order.
a. 2 , 0.3,  5 , 0, 0.25, 2   b. 5 1 ,  3.8, 9 , 4.8,  6
3        2          9               2       2        5
5    2            2                       6 9         1
 , 0, , 0.25, 0.3, ,               3.8,  , , 4.8, 5
2    9            3                       5 2         2
• YOU TRY a, c, and d!
1 2    5   8 7
c. –3, -3.2, -3.15, -3.001, 3         ,
d. 3 7 ,  ,  ,
3   3 9
3.2,  3.15,  3.001,  3, 3          8   5 2 1 7
 ,  , , ,
3   3 7 3 9
IV. Opposites

• Opposites: Two points (numbers) that are the same
distance from the origin but on opposite
sides of the origin

8 units away                      8 units away
from origin                       from origin

-10 -9   -8   -7 -6    -5   -4   -3   -2   -1   0   1   2    3    4   5   6   7   8   9   10   11

• Name the opposite of the following numbers:

a. 10                          b. -½                           c. 4.5                        d. 0
–10                               1/2                            –4.5                          0
V. Absolute Value
• What is absolute value?
– The distance a number is from origin                  .
– Distance is ALWAYS     Non-negative         !
• What does it look like?               We say “the absolute value of x”

Think: “How many
x                   Think: “How many
units away from 5                                         units away from -5
Absolute
to zero?                                                   to zero?
value bars

5 5                                                       5  5

Find the absolute value of the following:
4     4
6 6        0 0      4  4               
3     3
• Tricky Case: How are these different from each other?
From the ones on the previous slide?

 4  –4                         4  –4
VI. Solving Absolute Value Equations Using Mental Math

• Ex. x  3             x    3 or –3
Think, “What
number(s)                                                    So… How many
is/are 3 units   • Ex.      x 7 x            7 or –7           answers can
away from                                                      these have?
zero?

• Ex. x  2            x  No solution

• Ex.     x 0          x    0

Don’t confuse that “No solution” and “x = 0” are different!!!
Summary
1. Graph the real numbers on the number line.
2. Compare the real numbers (rewrite the real numbers in
increasing/decreasing order). The number at the LEFT is
smaller than the one at the RIGHT.
3. When converting a fraction to a decimal, just divide the
numerator by the denominator.
4. Opposites are the two real numbers “mirror” each other.
5. Absolute value is the distance a number on the number line
from the origin.
Can you compare and order real numbers and find the opposite and absolute
value of numbers??
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 

Let‟s see if we all get the same answer for
4 + 3 – 2 + (-5) + 7 – 1 + 3 + (-4) - 5= _____

-11 -10 -9   -8   -7 -6   -5   -4   -3   -2    -1   0   1    2   3   4    5   6   7   8   9   10   11
A few rules:

Properties of Addition                    Definition                          Example
Commutative                          a+b=b+a
4 + (-2) = (-2) + 4
Associative                 (a + b) +c=a +(b + c)
[(-4) + 3] + (-2) = (-4) + [3 + (-2)]
Identity                          a+0=a
(-2) + 0 = (-2)
Inverse                          a + (-a) = 0
7 + (-7) = 0
Procedures for Adding Real Numbers
• Adding Same Signs:
• Ex: -5 + -6 =                  Ex: 5 + 6 =

• Rule:   Signs alike  Add and keep the sign
Same Signs  Do “Sum” (Add) and keep the sign

• Adding Different Signs:
• Ex: -4 + 9 =            Ex: 4 + (-9) =

• Rule: Different Signs  Do “Diff ” (Subtract) and keep
the sign of what you have more of  ASK!!
Decimals and Fractions

• Decimals:
Ex: 4.03 + 3.142 =              Ex: -2.15 + 4.2
Line up the decimals and fill in 0s to make the numbers
Rule:   have the same length. Be careful with subtract – The
number with the larger absolute value must go first!!

Fractions:
2  1                     4  3
Ex)     1            Ex)       
9  3                     3  4
Rule:   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. 
So… What do we do with subtraction?
Ex 1) 12  3                          Ex 2)  6  ( 4)

1 4
Ex 3)  18.3  ( 4.514)               Ex 4) 
5 15

• Rule:   1) Change subtraction to „add the opposite‟. then
follow the rules of addition.
2) Directly subtract.

**Cross the line change the sign.**
**Make a “Smiling” face.**
Now you try
Ex 1)  14  3                        Ex 2)  8  ( 8)

2 5
Ex 3) 23.3  ( 4.514)               Ex 4) 
3 12

• Rule:   1) Change subtraction to „add the opposite‟. then
follow the rules of addition.
2) Directly subtract.

**Cross the line change the sign.**
**Make a “Smiling” face.**
Evaluate the function when x = –2, –1, 0, 1, 2 and y = x
–6

x       y

–2     –8       y = –2 – 6 = –2 + (–6) = –8

–2     –7       y = –1 – 6 = –1 + (–6) = –7

0      –6       y = 0 – 6 = 0 + (–6) = –6

1      –5       y = 1 – 6 = 1 + (–6) = –5

2      –4       y = 2 – 6 = 2 + (–6) = –4
Summary
Adding

Fractions                                             Decimals
Whole Numbers

Get Common                                              Line Up the
Denominators!                                            Decimals

Same Sign           Opposite Signs

Subtract
Add                Keep the sign
of what you have
Keep the sign            more of
Subtracting

Change to      Cross the Line,   Two Negative
Addition      Change the Sign       signs
Make a Positive

Then Follow Your
Addition Rules!
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, 3 x 3
 4 6            1
 6 2 1
A                              B   2 0           8 
2 0 5
3rows
      
2 rows                            
 9 3
                12 

3 columns
3 columns
equal matrices – matrices with equal entries in corresponding
positions  What does this mean??

2      5          
Ex)                   
6      1           

2     5  x      y 
Ex 1)           w        x = 2, y = –5 , w = 6, k = 1
6     1          k
 4 c  b         12
Ex 2) H            3        a = 3, b = –4 , c = 12, d = 5
a     5          d 

 4      12 
What is H = 
3        5 
**State law prohibits you from adding or
subtracting matrices of different sizes.**
Procedure: Add corresponding entries
 2 3  5 1                  7     4 
Ex:      1 4   12 2             11    6 
                                    
 0  2  3 16 
                            3
      14 

3x2           3x2              3x2

Ex. Find the values for all variables.

12 6  4 7   w x 
 4 1   3 2   y z 
                      
w = 8, x = 1, y = –7 , z = 1
You try these!!! Remember - lk to make sure
the matrices are the same size before you try to
add or subtract!!! Otherwise…. What do you
do??
Ex1) 1  5 3  4  10 2   5 -15             5

10  6  4 5 14                    1
Ex2)           3 2   2
5 0                              2 

4 6 
5     6 5           
Ex3)              1  1     No solution
 3   0  1
7 9 
      
  Give the dimensions of each matrix.  
So… What do we do with subtraction of matrices?

 1  7    6 
3  4 12  1 4 
Ex1)  5  2    3 
   
       Ex2)              
No
4 5  1 0 7 
solution
  2  0    2 
              

 1 2 7  1         8 3 
Ex3)                  10  7 
6 5   4 12               

Add/subtract the numbers/entries in the corresponding
Rule: positions.
Summary
1. The size of the matrix is described by row number x column number.
2. Equal matrices are the matrices with equal entries in
corresponding positions.
3. Adding/subtracting matrices of the same size is to add/subtract the
entries in corresponding positions.
4. It can add/subtract if two matrices of different size.
Assignment
p. 89 Q12 - 25 all
2.5 Multiplying Real Numbers
Objective: To multiply positives and negatives, including decimals and fractions!!

Multiplication Rules
                                 1. (6)(-4)
-24
2. -4c (5d)
-20cd

                                 3. (-1)(-12)                         4. 12          (-3)

                                 5. - 2
-12
   -11
-36

6. (7)(-7)
                                            22                                -49

Same signs  “+”                          7. 8  (-3)(2)                       8. -2          -6      -4
Diff. signs  “–”                                        -48                           -48
Even “-” signs  “+”
Odd “-” signs  “–”                       9. (-3b)(-4b)(-2z)(-5z)(-1)                  -120 b2z2
Does the number of negative values(signs) affect the sign of the product?
How?
    Multiplying Decimals                   
What is the rule for multiplying decimals?     Line up at the right and
multiply as whole number, when finished, place decimal point from right
for the total number of decimal digits.

Try these: Don’t forget the sign!!
10. (2.4)(-3.1)          11. 4.2(1.14)            12. (-5.3)(-0.04)

-7.44                 4.788                         0.212

13. (2.93)(0.012)                         14. (-10)(3.56)(-2)

0.03516                                     71.2
Multiplying Fractions
What is the rule for multiplying fractions? or vertically cancel the
Diagonally
greatest common factor between any numerator and denominate and then multiply
the numerators and denominates.

What should you do to make whole numbers look like
fractions? Just put a dummy denominate “1”.
Try these: Don’t forget the sign!!
 6  33                  7  6  3 
15.  2  3 
                  16.          
11  18 
17.        
                           12  7  4 
 3  5 
2                          1                            3
5                                                        8
3           1                           5 
18.   5   2              19.  6n        3c
5           2                          24c 
3                               15
n
4
Properties of Multiplication

Properties of Multiplication          Definition                          Example
Commutative                     a·b=b·a
4 · (-2) = (-2) · 4
Associative              (a · b) · c = a ·(b · c)
[(-4) · 3] · (-2) = (-4) · [ 3 · (-2)]
Identity                     a·1=a
(-2) · 1 = -2

Ex.       (2) (–x) = –2x                                    3 (–n)(–n) = 3n2

–(y)4 = –(y·y·y·y) = –y4
Summary
Multiplying

Fractions                                               Decimals
Whole Numbers

I

Straight Across                                          Ignore Decimals
Count Numbers
Behind Decimals
To Figure Out
Negative     Positive        Positive    Negative    Where Decimal
X           X                X           X             Goes
Negative     Positve         Negative    Positive

POSITIVE!                    NEGATIVE!
2.6 The Distributive Property
Objective: To use the distributive property and combine
like terms
The distributive property: The mailman property

a (b + c) = ab + ac                   -5 (x + 2) =

(b + c) a = ab + ac                    (x + 4) 8 =

a (b – c) = ab - ac                   -4 (x – 1) =

(b – c) a = ab - ac                    (x – 5) 9 =
You try these! 
1. 2(x + 5)= 2 x  10                 5. (x - 4)x= x 2  4 x

2. (15+6x) 1 x = 5x  2 x 2              1
6. 2 y(2 - 6y)= y  3y2
3

3. -3(x + 4)= 3x  12                7. (y + 5)(-4)= 4 y  20

4. -(6 - 3x)= 3x  6                  8.  2 x  3x  9   2 x 2  6 x
3

Terms in an expression that have the same variable
What are like terms?   raised to the same power.
2 2
Give some examples:     x y, 4 x 2 y, 0.035 yx 2 ,
3
Simplified Expression:        expression with no grouping
symbols and all like terms combined.

What does this mean you need to do?       combine all like
terms and possibly apply the distributive properties.

What are the like terms? Combine them.

1. 8x + 3x = 11x                   2. 4n2 + 2m2 – m2 =
4n 2  m 2
3.   2n2   – 5n + 7n = 2n 2  2n   4. 4a – a + 5y = 3a  5 y

5. -5x – 7x + 3x2 = 12 x  3x 2 6. x2 – xy + y2 = x 2  xy  y 2
What do you do if there are parenthesis?
1. 2a(a – 3) + 4(a – 2)                2. 3x – 2(5x + 1)
= 2a 2  2a  8                             = 7 x  2
3. 10 – (x – 3) + 4(x2 – 3x)           4. 2w(w – 5) – (w + 6) – 4w2
= 4 x 2  13x  13                       = 2w2  11w  6

You try these!!
5. –3(3m – 5) + 2m(m – 1)                 6. x(4x + 2) + 5x(x – 2)
= 2m2  11m  15                             = 9 x2  8x
Find the perimeter of a triangle (add all sides)
3x + 4
Perimeter = 3x + 4 + (2x – 3) + (x + 2)           x+2
= 6x + 3
2x - 3
Summary
1. Like Terms are the terms in an expression that have the same
variable raised to the same power.

2. Distributive property is the “Mailman” property. Every number
in the parenthesis must multiply the “Mailman” number.

3. After using the distributive property, be aware of the operations
between the terms.
2.7 Division of Real Numbers
Objective: To divide real numbers
Division uses the same rules to determine signs as for multiplication.

Need to know: Reciprocal: If the product of a nonzero number a
and another number b is 1, then a
and b are ~ to each other.
Note – the reciprocal of a number keeps the same sign.

Give the reciprocal of the following:
2                        5                           1
1)  2      2)          3) 1       4)          2   5) 0       6) 3

1 3      3                   2                           2 2
2         2
1               5     Undef.
7
To divide real numbers  Change the divisor to its reciprocal

Re-write each division problem as a multiplication problem: Should the answer be
positive or negative? How do you know?
2
7) 3   6 1
2                8) 6  5  -6/5   9)   1 1  3/10
3
5

2                                               2
4
10)    7
 -1/(14x)    11)        8/a    12)    3
 -1/18
4 x                         a
2                  12

32 x  8         9  27k
13)  39  ( 4 )     1
14)               15)
3
3
4
 9                         8x – 2           – 3 + 9x
YOU TRY THESE!!
Simplify each expression:
d  3  2d
1) 6t   12t
1
2            2) 56   2   - 20
4
5             3)      
4  8 3

8 x                          2w     49w        36
4)           -   x2     5)  7      -       6)          - 54
8
x                            7      2          2
3
Simplify for the given value:
2x  5                      2m  5n
1)        and x  10        2)           and m  1 and n  4
2
9                            m
20 - 5 15 5                  1 - 20    19
                        1
= - = -38
1
9     9 3
2       2

20  4 x                    x 2  16
3)          and x  3      4)          and x  4
x3                            x
20 + 12 32                    16 - 16 0
=   = undefined              = =0
-3 + 3   0                     -4    -4
Summary
1. What is the reciprocal of a nonzero number?

2. What conversions do we do when dividing a nonzero number?

3. Can you determine the sign of the answer?

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Jun Wang Dr
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