1.4 Absolute Values
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Transcript 1.4 Absolute Values
1.4 Absolute Values
Solving Absolute Value Equations
By putting into one of 3 categories
What is the definition of “Absolute Value”?
__________________________________
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a if a 0
Mathematically, a
a if a 0
___________________
For example, in x 3, where could x be?
-3
0
3
To solve these situations,
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Consider 3x 2 10
________________________
0
That was Category 1
Category 1 : ____________________________
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x a
Example
x5 3
_____________
Case 1
Case 2
Absolute Value Inequalities
Think logically about another situation.
What does x a mean?
For instance, in the equation x 6 5,
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0
How does that translate into a sentence?
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Now solve for x.
This is Category 2: when x is less than a
number
x a ___________
Absolute Value Inequalities
What does x a mean?
In the equation 2x 1 11 ,
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0
x a
____________
x a ____________
Less than = And statement
x a
_____________
Greater than = Or statement
Note: is the same as ; is the same as ; just have the sign in
the rewritten equation match the original.
Isolate Absolute Value
• _______________________________________
3 x 5 12
• ______________________________________
x5 4
• ______________________________________
When x is on both sides
3x 4 x 3
• ______________________________________
________________________________________
• ______________________________________
Case 1
Case 2
Example inequality with x on other side
2x 1 5 x
Case 1
Case 2
Examples
1. 2x 4 24
Case 1
Case 2
2. x 6 9
Case 1
Case 2
3. 2x 1 11
Case 1
Case 2
4. 2 5 x 5 27
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Whats wrong with this?
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5. 3x 9 10
1
6. -4 x 2 8
2
7. 3x 4 2x 6
8. x 2 x 7
1-4 Compound Absolute Values
Equalities and Inequalities
More than one absolute value in
the equation
Some Vocab
• Domain- _________________________
• Range- __________________________
• Restriction- _______________________
x2
5
x4
• ______________________________________
____________________________
Find a number that works.
x 2 x 1 6
We will find a more methodical approach to find all
the solutions.
In your approach, think about the values of this
particular mathematical statement in the 3 different
areas on a number line.
2x 2 x 5 9
-_________________________________________
-________________________________________
- ________________________________________
It now forms 3 different areas or cases on the
number line.
2x 2 x 5 9
•______________________________
______________________________
•___________________________________
Case 1
Domain x 5
________________________
____________________________________________________
_________________________________
_________________________________
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Case 2
Domain
1 x 5
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Case 3
Domain
x 1
_____________________________
Final Answers
•______________________________
•______________________________
• _____________________________
-2
2
5
If you get an answer in any of the cases where the
variable disappears and the answer is TRUE
________________________________
________________________________
________________________________
________________________________
3x 6 x 2 12
x 2
x4
5
1-5 Exponential Rules
You know these already
Review of the Basics
m
1. a a ____
a
5. ____
b
2. am an ____
6. a ____
3. (a ) ____
7. a ____
m n
m n
4. (ab) ____
n
0
1
8. a
n
____
Practice problems
Simplify each expression
9
x
; 3
x
1. x x
9
3
2. 4 4
n
n 7
3
2
3.
5
3
x
; 9
x
1
4.
3
3
2
-3
2
3x y
5. 1 2 5
2 x y
6. (3) (2)
2
3
7. (2x y) (3x y )
3
4
2
3 2
(x y )
8.
5 3 3
(x y )
2
2 3
8
9. 7(x 3y) (12y 4x)
25x-9
10. 5x 12
2
11. 4 2 4 2
x
x
Do NOT use calculators on the
homework, please!!
28
(x 3y)7
8
9. 7(x 3y) (12y 4x)
10. 4 2 4 2
x
11. 3 3
x
x 4
x
81
2 2 2 2
2
x
2x 4
3
2
3
4
x
2
x 3
x 0
Rule:If bases the same set exponents equal
to each other
1-6 Radicals (Day 1) and
Rational Exponents (Day 2)
What is a Radical?
In simplest term, it is a square root (
= ________________
)
The Principle nth root of a:
0 0
1.
n
•
If a < 0 and n is odd then n a is a
negative number ___________________
•
If a < 0 and n is even, _____________
__________
Vocab
n
a
Lets Recall
x __
x __
2
3
x __
3
5
x __
5
3
5
x __
4
20
x __
3
x __
x __
5
x __
6
10
33
60
Rules of Radicals
1.
2.
3.
n
a b _______
n
a
n
b
n
m n
___
a ____
Practice problems
Simplify each expression
1. 20
5.
2.
3
81
6.
3.
3
17
3
4. 4 4
3
3
3
64
50 72
7. 3 27x 7 12x
2
8.
3
1 3
6
81x
24
2
Class Work problems
Simplify each expression
1. 20
2.
3
81
3.
3
17
3
2 5
5.
33 3
6.
3
53
3
2
64
2
50 72 11 2
23x 3
7. 3 27x 2 7 12x 2
4. 4 4 2 2
3
3
3
3
1-6 Radicals (Day 1) and
Rational Exponents (Day 2)
What is a Rational Exponent?
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Don’t be overwhelmed by fractions!
These problems are not hard, as long as you remember
what each letter means. Notice I used “p” as the
numerator and “r” as the denominator.
p= _____________________________
r= _____________________________
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________________________________________
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Practice problems
Simplify each expression
1.
9
1
2
27
3. 4
1
2.
36
1
3
3
2
16x 4
5. 8
y
4. ( 8)
3
4
2
3
6. 243
2
5
1
2
The last part of this topic
What is wrong with this number?
1
3
1
2
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Rationalizing
If you see a single radical in the denominator
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7.
6
8
8.
6
3
2
Rationalizing
If you see 1 or more radicals in a binomial, what
can we do?
10.
11.
7
6 5
5
5 18
What is a Conjugate?
The conjugate of a + b ______. Why?
The conjugate _______________________
___________________________________
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Rationalizing
Multiply top and bottom by the conjugate!
Its MAGIC!!
10.
7
6 5
11.
5
5 18
1.7 Fundamental Operations
Terms
• Monomial ______________________________
• Binomial _______________________________
• Trinomial ______________________________
Standard form:
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x 4x x 7
3
2
Collecting Like terms
5x 7 x x 3 4 x x x 6
3
2
3
2
F ____________
O ____________
I ____________
L ____________
(3 x 2)(2 x 5)
Examples
( x 2 3x 5)(2 x3 x 2 3x)
1-8 Factoring Patterns
What is the first step??
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Why?
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Perfect Square Trinomial
Factors as
Difference of squares
Sum/Difference of cubes
3 terms but not a pattern?
This is where you use combinations of
the first term with combinations of the
third term that collect to be the middle
term.
6 x 10 x 4
2
4 or more terms?
x 3x 8x 24
4
3
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Examples
9x 4 y
2
2
( x y) 1
3
x y
6
6
Solve using difference of Squares
5x 7
16 y 5 x
2
Solve using sum of cubes
27 x 1
Examples of Grouping
a 2b 2a ab
2
a 4a 4 y
2
2
a 3 6a 2 12a 8
4 x5 x3 108 x 2 27
Examples of Grouping
a 2b 2a ab
2
a 4a 4 y
2
2
4 x5 x3 108 x 2 27
1.9 Fundamental Operations
What are the Fundamental
Operations?
Addition, Subtraction, Multiplication
and Division
We will be applying these fundamental
operations to rational expressions.
This will all be review. We are working on
the little things here.
Simplify the following
Hint: __________________________________
_______________________________________
_______________________________________
x 9x 20
1.
x 4
2
2. What is the domain
of problem 1?
3x 10x 3 3x 17x 28
5.
2
2
3x 13x 12
x 49
2
2
x 1
x 1
6. 3
2
x 1 x x 1
2
4
7
7. 3
2
x 1 x 1
x 3x 2x 2
9.
x 1
4
2
_______________________________
_______________________________
Lets Watch.
Polynomial Long Division
x 1 x 0x 3x 2x 2
4
3
2
Polynomial Long Division
2x 1 4x 0x 0x 1
3
2
1-10 Introduction to Complex
Numbers
What is a complex number?
To see a complex number we have
to first see where it shows up
Solve both of these
x 81 0
2
x 81 0
2
Um, no solution????
x 81
does not have a real
answer.
It has an “imaginary” answer.
To define a complex number ____________
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This new variable is “ i “
Definition: i 1
Note: __________________
1
So, following this definition:
______________________________
______________________________
______________________________
______________________________
And it cycles….
i 1
i 1
2
i i
3
i 1
4
Do you see a pattern yet?
What is that pattern?
We are looking at the remainder when the
power is divided by 4.
Why?
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Try it with
Hints to deal with i
1. __________________________________
2. __________________________________
____________________________________
3. _________________________________
____________________________________
Examples
1.
36 81
2.
36 81
OK, so what is a complex number?
______________________________
______________________________
A complex number comes in the form
a + bi
Lets try these 4 problems.
1. (8 3i) (6 2i)
2. (8 3i) (6 2i)
3. (8 3i) (6 2i)
5. 6i
5
6- i
5
6.
4
2i