Download Chapter 1 Section 4 - Lamar County School District

Chapter 1
Real Numbers and
Introduction to
Algebra
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Bellwork:
1. -14 + (-3) +11
3. |43 + (-73)| + |-20|
2. [-2 + (-7)] + [-11 + 22]
4. 6 – 2 × 2 +25
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
2
Bellwork:
1. -14 + (-3) +11
3. |43 + (-73)| + |-20|
-6
2. [-2 + (-7)] + [-11 + 22]
4. 6 – 2 × 2 +25
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
3
Bellwork:
1. -14 + (-3) +11
3. |43 + (-73)| + |-20|
-6
2. [-2 + (-7)] + [-11 + 22]
4. 6 – 2 × 2 +25
2
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
4
Bellwork:
1. -14 + (-3) +11
-6
2. [-2 + (-7)] + [-11 + 22]
3. |43 + (-73)| + |-20|
50
4. 6 – 2 × 2 +25
2
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
5
Bellwork:
1. -14 + (-3) +11
-6
2. [-2 + (-7)] + [-11 + 22]
2
3. |43 + (-73)| + |-20|
50
4. 6 – 2 × 2 +25
34
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
6
1.4
Adding Real Numbers
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Objectives:
Add
real numbers
Solve problems with addition
Find additive inverses
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8
Adding Real Numbers
Adding real numbers can be visualized on a
number line.
A positive number can be represented on the
number line by an arrow of appropriate length
pointing to the right, and a negative number
by an arrow of appropriate length pointing to
the left.
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Example 1
Add: ‒5 + 2
Start
End
‒5
2
–5 –4 –3 –2 –1
0
1
2
3
4
5
‒5 + 2 = ‒3
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10
Example 1
Add: ‒5 + 2
Start
End
‒5
2
–5 –4 –3 –2 –1
0
1
2
3
4
5
‒5 + 2 = ‒3
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
11
Example 1
Add: ‒5 + 2
Start
End
‒5
2
–5 –4 –3 –2 –1
0
1
2
3
4
5
‒5 + 2 = ‒3
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12
Adding Real Numbers
To add two real numbers
1. with the same sign, add their absolute values.
Use their common sign as the sign of the answer.
2. with different signs, subtract their absolute values.
Give the answer the same sign as the number with
the larger absolute value.
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13
Example 2
Add without using a number line:
a. (‒8) + (‒3) = ‒11
Same sign
b. (‒7) + 1 = ‒6
Different signs
c.
9  2 7
  
10  10  10
Different signs
d.
(‒12.6) + (‒1.7) = ‒14.3
Same signs
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14
Example 2
Add without using a number line:
a. (‒8) + (‒3) = ‒11
Same sign
b. (‒7) + 1 = ‒6
Different signs
c.
9  2 7
  
10  10  10
Different signs
d.
(‒12.6) + (‒1.7) = ‒14.3
Same signs
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
15
Example 2
Add without using a number line:
a. (‒8) + (‒3) = ‒11
Same sign
b. (‒7) + 1 = ‒6
Different signs
c.
9  2 7
  
10  10  10
Different signs
d.
(‒12.6) + (‒1.7) = ‒14.3
Same signs
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
16
Example 2
Add without using a number line:
a. (‒8) + (‒3) = ‒11
Same sign
b. (‒7) + 1 = ‒6
Different signs
c.
9  2 7
  
10  10  10
Different signs
d.
(‒12.6) + (‒1.7) = ‒14.3
Same signs
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
17
Example 2
Add without using a number line:
a. (‒8) + (‒3) = ‒11
Same sign
b. (‒7) + 1 = ‒6
Different signs
c.
9  2 7
  
10  10  10
Different signs
d.
(‒12.6) + (‒1.7) = ‒14.3
Same signs
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
18
Example 2
Add without using a number line:
a. (‒8) + (‒3) = ‒11
Same sign
b. (‒7) + 1 = ‒6
Different signs
c.
9  2 7
  
10  10  10
Different signs
d.
(‒12.6) + (‒1.7) = ‒14.3
Same signs
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
19
Example 2
Add without using a number line:
a. (‒8) + (‒3) = ‒11
Same sign
b. (‒7) + 1 = ‒6
Different signs
c.
9  2 7
  
10  10  10
Different signs
d.
(‒12.6) + (‒1.7) = ‒14.3
Same signs
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
20
Example 2
Add without using a number line:
a. (‒8) + (‒3) = ‒11
Same sign
b. (‒7) + 1 = ‒6
Different signs
c.
9  2 7
  
10  10  10
Different signs
d.
(‒12.6) + (‒1.7) = ‒14.3
Same signs
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
21
Example 2
Add without using a number line:
a. (‒8) + (‒3) = ‒11
Same sign
b. (‒7) + 1 = ‒6
Different signs
c.
9  2 7
  
10  10  10
Different signs
d.
(‒12.6) + (‒1.7) = ‒14.3
Same signs
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22
Additive Inverses
Opposites or Additive Inverses
Two numbers the same distance from 0 on the
number line, but lie on opposite sides of 0 are called
opposites or additive inverses of each other.
The opposite of 8 is ‒8.
The opposite of ‒2.9 is 2.9.
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23
Additive Inverses
Opposites or Additive Inverses
Two numbers the same distance from 0 on the
number line, but lie on opposite sides of 0 are called
opposites or additive inverses of each other.
The opposite of 8 is ‒8.
The opposite of ‒2.9 is 2.9.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
24
Additive Inverses
Opposites or Additive Inverses
Two numbers the same distance from 0 on the
number line, but lie on opposite sides of 0 are called
opposites or additive inverses of each other.
The opposite of 8 is ‒8.
The opposite of ‒2.9 is 2.9.
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25
Example 3
Find the opposite of each number.
a.
‒16 The opposite is 16.
b. 5
The opposite is ‒5.
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26
Example 3
Find the opposite of each number.
a.
‒16 The opposite is 16.
b. 5
The opposite is ‒5.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
27
Example 3
Find the opposite of each number.
a.
‒16 The opposite is 16.
b. 5
The opposite is ‒5.
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General Rules
If a is a number, then –(–a) = a.
The sum of a number a and its
opposite ‒a is 0.
meaning:
a + (‒a) = 0 Also, ‒a + a = 0.
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29
Example 4
Simplify each expression.
a.
‒(‒16) = 16
b. ‒(‒5x) = 5x
c. ‒|9| = ‒9
d.
‒|‒2| = ‒2
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30
Example 4
Simplify each expression.
a.
‒(‒16) = 16
b. ‒(‒5x) = 5x
c. ‒|9| = ‒9
d.
‒|‒2| = ‒2
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31
Example 4
Simplify each expression.
a.
‒(‒16) = 16
b. ‒(‒5x) = 5x
c. ‒|9| = ‒9
d.
‒|‒2| = ‒2
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32
Example 4
Simplify each expression.
a.
‒(‒16) = 16
b. ‒(‒5x) = 5x
c. ‒|9| = ‒9
d.
‒|‒2| = ‒2
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33
Example 4
Simplify each expression.
a.
‒(‒16) = 16
b. ‒(‒5x) = 5x
c. ‒|9| = ‒9
d.
‒|‒2| = ‒2
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34
Example 5
Evaluate 5x + y when x = 4 and y = ‒2.
5x + y = 5·4 + (‒2)
= 20 + (‒2)
= 18
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35
Example 5
Evaluate 5x + y when x = 4 and y = ‒2.
5x + y = 5·4 + (‒2)
Plug it in!
= 20 + (‒2)
= 18
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36
Example 5
Evaluate 5x + y when x = 4 and y = ‒2.
5x + y = 5·4 + (‒2)
Plug it in!
= 20 + (‒2)
= 18
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37
Example 5
Evaluate 5x + y when x = 4 and y = ‒2.
5x + y = 5·4 + (‒2)
Plug it in!
= 20 + (‒2)
= 18
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
38
Example 5
Evaluate 5x + y when x = 4 and y = ‒2.
5x + y = 5·4 + (‒2)
= 20 + (‒2)
Plug it in!
Simplify.
= 18
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39
Example 5
Evaluate 5x + y when x = 4 and y = ‒2.
5x + y = 5·4 + (‒2)
= 20 + (‒2)
Plug it in!
Simplify.
= 18
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40
Example 5
Evaluate 5x + y when x = 4 and y = ‒2.
5x + y = 5·4 + (‒2)
= 20 + (‒2)
Plug it in!
Simplify.
= 18
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41
Example 6
During a four day period, a share of Walmart stock
recorded the following gains and losses:
Tuesday
a loss of $2
Wednesday
a loss of $1
Thursday
a gain of $3
Friday
a gain of $3
Find the overall gain or loss for the stock for the four
days.
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42
Example 6
During a four day period, a share of Walmart stock
recorded the following gains and losses:
Tuesday
a loss of $2
Wednesday
a loss of $1
Thursday
a gain of $3
Friday
a gain of $3
Find the overall gain or loss for the stock for the four
days.
original price – 2 – 1 + 3 + 3 = + 3
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
43
Example 6
During a four day period, a share of Walmart stock
recorded the following gains and losses:
Tuesday
a loss of $2
Wednesday
a loss of $1
Thursday
a gain of $3
Friday
a gain of $3
Find the overall gain or loss for the stock for the four
days.
original price – 2 – 1 + 3 + 3 = + 3
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
44
Example 6
During a four day period, a share of Walmart stock
recorded the following gains and losses:
Tuesday
a loss of $2
Wednesday
a loss of $1
Thursday
a gain of $3
Friday
a gain of $3
Find the overall gain or loss for the stock for the four
days.
original price – 2 – 1 + 3 + 3 = + 3
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
45
Example 6
During a four day period, a share of Walmart stock
recorded the following gains and losses:
Tuesday
a loss of $2
Wednesday
a loss of $1
Thursday
a gain of $3
Friday
a gain of $3
Find the overall gain or loss for the stock for the four
days.
original price – 2 – 1 + 3 + 3 = + 3
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
46
Example 6
During a four day period, a share of Walmart stock
recorded the following gains and losses:
Tuesday
a loss of $2
Wednesday
a loss of $1
Thursday
a gain of $3
Friday
a gain of $3
Find the overall gain or loss for the stock for the four
days.
original price – 2 – 1 + 3 + 3 = + 3
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
47
Example 6
During a four day period, a share of Walmart stock
recorded the following gains and losses:
Tuesday
a loss of $2
Wednesday
a loss of $1
Thursday
a gain of $3
Friday
a gain of $3
Find the overall gain or loss for the stock for the four
days.
original price – 2 – 1 + 3 + 3 = + 3
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48
Closure:
How
do you add two numbers using a
number line?
What is the rule for adding numbers
based on their signs?
What is the additive inverse? Give an
example.
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49