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.
Chapter 1
Fundementals of Algebra
Michael Giessing
giessing@math.utah.edu
University of Utah
Fundementals of Algebra – p.1/16
Fundementals of Algebra – p.2/16
Fractions of the same donomination
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.3/16
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.4/16
Multiplication of Mixed Numbers
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.5/16
1. Write the mixed number as a fraction
2. Multiply the numerators and the denominator
Fundementals of Algebra – p.6/16
Division of Mixed Numbers
1. Write the mixed number as a fraction
2. Cross multiply
3. Example
Properties of Real Numbers
7×9
7 4
÷
=
2 9
2×4
7
63
=7
=
8
8
Fundementals of Algebra – p.7/16
Order of Operations
Fundementals of Algebra – p.8/16
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.
Please
Parethesis
excuse Exponenents
my
Multiplication
dear
division
aunt
Addition
Sally
Subtraction
Work from right to left.
Fundementals of Algebra – p.9/16
Associative Property
1.
2.
3.
4.
Fundementals of Algebra – p.10/16
Distributive property
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.
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.11/16
Fundementals of Algebra – p.12/16
Identity and Inverses
•
•
•
•
a+0=a
a×1=a
a + (−a) = a − a = 0
a×
1
a
Algebraic Expressions
=1
Fundementals of Algebra – p.13/16
Expressions, terms and Coefficients
Expression Terms Coefficients Variables
5x − 4
5x, −4
5, −4
x
+,-,×, ÷ only ×
Known
Unknown
Fundementals of Algebra – p.15/16
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 subtraction
= x/2 + (−6)x + (−3/2)
commutative
= (1/2)x + (−6)x + (−3/2)
definition of subtraction
= (1/2 − 6)x + (−3/2)
distributive
1
12
= ( 2 − 2 )x + (−3/2)
common denominator
= −11
x
+
−(3/2)
subtraction
2
11x+3
=− 2
distributive
Fundementals of Algebra – p.17/16
Fundementals of Algebra – p.14/16
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
−
3)
Fundementals of Algebra – p.16/16