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Transcript
Chapter 1
Fundementals of Algebra
Michael Giessing
[email protected]
University of Utah
Fundementals of Algebra – p.1/21
Distance
1. Absolute value (| |)gives returns a positive
number.
2. | − 4| = 4, |3| = 3, |a| = a if a ≥ 0 and |a| = −a
if a < 0.
3. The distance between a and b is |a − b|.
4. The distance between 200 and −1.5 is
|200 − (−1.5)| = |200 + 1.5| = |201.5| = 201.5.
Fundementals of Algebra – p.2/21
Adding Mixed Numbers
1. To add mixed numbers first we add the whole
parts.
2. Now we need to add the fractional part.
3. To add fractions they need to be of the same
denomonation.
Fundementals of Algebra – p.3/21
Fractions of the same donomination
1. All denominators must match. How many halves,
thirds, or Catholics.
2. To change the denomonator without changing the
fraction multiply the numerator and the
denomonator by the same number
3. This can always be accomplished by multiplying
the the denomonators by eachother.
4. is best to find the least common denomonator
(LCD)
Fundementals of Algebra – p.4/21
Addition Example
Add 1 19 + 10 17
1
1
1 1
1 + 10 = 1 + 10 + +
9
7
9 7
1 1
= 11 + +
9 7
1×7 1×9
= 11 +
+
9×7 7×9
9
7
+
= 11 +
63 63
7+9
= 11 +
63
16
= 11
63
Fundementals of Algebra – p.5/21
Subtraction Example
1 19 − 10 17
1
1
1 1
1 − 10 = 1 − 10 + −
9
7
9 7
1 1
= −9 + −
9 7
1×7 1×9
= −9 +
−
9×7 7×9
9
7
−
= −9 +
63 63
7−9
= −9 +
63
2
= −9
63
Fundementals of Algebra – p.6/21
Multiplication of Mixed Numbers
1. Write the mixed number as a fraction
2. Multiply the numerators and the denominator
Fundementals of Algebra – p.7/21
Division of Mixed Numbers
1. Write the mixed number as a fraction
2. Cross multiply
3. Example
7 4
1 4
=
÷
3 ÷
2 9
2 9
7×9
=
2×4
63
7
=
=7
8
8
Fundementals of Algebra – p.8/21
Properties of Real Numbers
Fundementals of Algebra – p.9/21
Order of Operations
Please
Parethesis
excuse Exponenents
my
Multiplication
dear
division
aunt
Addition
Sally
Subtraction
Work from left to right.
Fundementals of Algebra – p.10/21
The Pemdas Way
P
Parethesis
e Exponenents
m Multiplication
d
division
a
Addition
s
Subtraction
Fundementals of Algebra – p.11/21
Commutative Property
Multiplication ab = ba (example 3 × 2 = 2 × 3)
Addition a + b = b + a (example 3 + 2 = 2 + 3)
Subtraction is not commutative 2 − 3 6= 3 − 2
Division is not commutative 2/3 6= 3/2
To use the commutative property write everything
in terms of addition and multiplication
6. Think of the work commuter to remember what
the commutative property is about.
1.
2.
3.
4.
5.
Fundementals of Algebra – p.12/21
Associative Property
1.
2.
3.
4.
Mulitplication is associative (ab)c=a(bc)
Addition is associative (a+b)+c=a+(b+c)
Subtraction and Division are not associative
The paranthesis associate numbers together.
Fundementals of Algebra – p.13/21
Distributive property
Multiplication distributes accross addition and
subtraction
a(b + c) = ab + ac
a(b − c) = ab − ac
Every body gets an a!
Fundementals of Algebra – p.14/21
Identity and Inverses
•
a+0=a
a×1=a
a + (−a) = a − a = 0
•
a×
•
•
1
a
=1
Fundementals of Algebra – p.15/21
Algebraic Expressions
Fundementals of Algebra – p.16/21
Expressions, terms and Coefficients
Expression Terms Coefficients Variables
5x − 4
5x, −4
5, −4
x
+,-,×, ÷ only ×
Known
Unknown
Fundementals of Algebra – p.17/21
Simplifying
Use the properties of real numbers to modify
an expression into something simpler.
Example:
Simplify
5(x
5(x − 3) = 5x − 5 × 3 = distributive
= 5x − 15
−
3)
Fundementals of Algebra – p.18/21
Harder Example
Simplify
(x
−
3)/2
−
6x
(x − 3)/2 − 6x = x/2 − 3/2 − 6x distributive
= x/2 + (−3/2) + (−6)x
definition of sub
= x/2 + (−6)x + (−3/2)
commutative
= (1/2)x + (−6)x + (−3/2)
definition of sub
= (1/2 − 6)x + (−3/2)
distributive
= ( 12 − 12
common denomi
2 )x + (−3/2)
= −11
subtraction
2 x + −(3/2)
Fundementals of Algebra – p.19/21
Translation
Key Word
Addition
Description
Expressio
The sum of 5 and x
5+x
seven more than a number
7+y
Subtraction
b is subtracted from 4
4−b
Three less than a number
z−3
Multiplication
two times x
2x
300% of a number
3.00x
x
Division
The ratio of x and 8
8
x
half of a number
2
Fundementals of Algebra – p.20/21
Contructing Expressions
One oreo cookies contains 55 calories. A
cookie jar contains x cookies Write an
expression for the numeber of calories in the
jar.
Number of calories = number of cookies × number of
caleries per cookie.
Calories = x × 55
Fundementals of Algebra – p.21/21