Download Chapter 1 Review of Real Numbers and Problem Solving

MTH070
Elementary Algebra
Chapter 1
Review of Real Numbers
and Problem Solving
Copyright © 2010 by Ron Wallace, all rights reserved.
What is Algebra?
What is Algebra?
• A generalization of arithmetic.
Mathematics Dictionary, 4th Ed.
by James/James
Therefore, you need to know
and understand arithmetic!
What is Algebra?
• A generalization of arithmetic in
which symbols, usually letters of
the alphabet, represent numbers
or members of a specified set of
numbers and are related by
operations that that obey
specified laws.
The American Heritage Dictionary, 2nd College Edition
What is Algebra?
• A branch of mathematics in which
symbols, usually letters of the
alphabet, represent numbers or
members of a specified set and are
used to represent quantities and to
express general relationships that
hold for all members of the set.
www.dictionary.com
Terminology & Notation
• Mathematical Vocabulary
• Precise Communication
• Notation
• Symbols instead of words
• Semantics & Syntax
• Multiplication & Division in Algebra?
Yes, you are expected to be able to read, understand, and
use correct mathematical terminology and notation.
Terminology & Notation
Example …
Terminology
• Variable
x
• A symbol used to represent a value.
• Usually a letter (Latin or Greek)
• The value is …

– unknown
– to be determined
– assigned different quantities
17
Terminology
• Constant
• A specific value.
–
–
–
–
Integers
Fractions
Decimals
Special

7.53 7.53
21
5
Terminology
5  3x
• Algebraic Expression
• A legal combination of variables,
constants, operators, and grouping
symbols.
• AKA: Expression
3( x  7)
2
Terminology
• Equivalent Expressions
• Expressions that are equal for all
values of their variables.
• Simplifying: Changing expressions
into simpler equivalent expressions.
2( x  3)
2x  6
Terminology
3x
• Term
• A constant, variable, or product of a
constant and one or more variables.
• When grouping symbols are removed, all
expressions are a sum of terms.
• Coefficient – The constant part of a term
• Linear Term – A constant times a variable.
x = 1x
-x = -1x
Terminology
• Factor
7 x  5x
• The constants or variables that are
multiplied together in a term.
• Common Factor – A constant or
variable that is a factor of each term of
an expression.
3x 12
3x  3(4)
Terminology
• Identities
• additive identity: 0
• multiplicative identity: 1
x0  x
1x  x
Terminology
• Inverses
• Opposite – The negative of the number.
– aka: Additive inverse
• Reciprocal - One divided by the number.
– aka: Multiplicative inverse
x  ( x)  0
1
x  1
x
Terminology
• Commutative
• Order of addition and multiplication
may be reversed.
5 x  x 5
5  x  x  5
5  ( x)
x (3)  3 x
Terminology
• Associative
• Sums and products may be grouped
in any way.
( x  5)  7  x  (5  7)  x  5  7
2(7 x)  (2  7) x  2  7  x
What about subtraction and division?
Terminology
• Distributive (involves both operations)
3( x  5)  3x  3  5  3 x  15
“3 is distributed over the sum x+5.”
4 x  7 x  (4  7) x  11x
What about subtraction and division?
Terminology
• Equation
• A mathematical statement that two
expressions are equal.
• Three possibilities …
2x  3x  5x
– always true  “identity”
– always false  “fallacy”
– neither  “conditional”
x5  x
2x  5  7
Terminology
• Inequality
• A mathematical statement that one
expression is ...
>
<
≥
≤
2x  5  7
… a second expression.
Terminology
• Solution
• A value for the variable that makes an
equation or inequality a true
statement.
• Solution Set: All such values.
The solution of 2 x  5  7
is x  1.
Terminology
• Absolute Value
x
• How far a number is from zero.
• AKA: Magnitude
 x if x  0
x 
 x if x  0
Terminology
R
• Real Numbers
• The set of all numbers that
correspond to the points on a number
line.
– Can be thought of as directed distances
from zero.
y
0
1
x
Terminology
• Integers
Z
• The set of positive and negative
counting numbers and zero.
...
 3,  2, 1, 0, 1, 2, 3, ...
Terminology
• Positive Integers
Z
• The set of counting numbers.
• AKA: Natural Numbers
1,
2, 3, ...

Terminology
• Negative Integers
Z

• The set of opposites of the positive
integers.
...
 3,  2, 1
Terminology
• Non-Negative Integers
Z
*
• The set of counting numbers and zero.
• AKA: Whole Numbers
0, 1,
2, 3, ...
Terminology
• Rational Numbers
Q
• Set of numbers that can be written as a
quotient of two integers.
• Decimal forms will either be terminating
decimals or repeating decimals.
a
where a, b  Z and b  0
b
Terminology
• Irrational Numbers
• All Real numbers that are not Rational.
2

0.1011011101111...
Properties of Zero
x0  x
x0  x
x  00
x
is "undefined"
0
0
 0 if x  0
x
ab  0  a  0 or b  0
“Rules” on Page 28.
Signed Number Arithmetic
Addition
Think Position on # Line
• Start @ 1st #
– Negative  to left of 0
– Positive  to right of 0
• Add 2nd #
– Negative  move left
– Positive  move right
• Are you moving towards
or away from zero?
Think Money in Your Pocket
• Start w/ 1st #
– Negative  in debt (owe)
– Positive  cash on hand
• Add 2nd #
– Negative  spending
– Positive  receiving
• Are you …
– increasing your debt?
– increasing your savings?
– reversing your state?
“Rule” on Page 32.
Signed Number Arithmetic
Subtraction
Think …
SEE
THINK
– Add the opposite!
7  12
a - b = a + (-b)
(7)  12
(7)  (12)
7  (12)
7  12
(7)  (12)
7  (12)
(7)  12
“Rules” on Page 37.
Signed Number Arithmetic
Multiplication
Multiplying by -1
 x  (1) x
 1 if n is even
n
(1)  
1 if n is odd
So …
– Multiply numbers
• ignore signs
– Even # of negatives
• Positive result
– Odd # of negatives
• Negative result
“Rules” on Page 38.
Signed Number Arithmetic
Division
Treat as a Fraction
a
a b 
b
Therefore … reduce!
a a

b b
Division is …
– multiplication by the
reciprocal
– sign of the result is
just like multiplication.
a a
a


b b
b
“Rules” on Page 38.
Signed Number Arithmetic
Division
Warning: Be careful w/ Cancelling
45
?
4
5x  2
?
3x
“Value of an Expression”
The Question…
Find the value of 4(2 x  3)  x when x  5.
--- OR --Evaluate 4(2 x  3)  x for x  5.
Solution
Replace x with its value and do the arithmetic!
(be careful when x is negative)
Order of Operations
• Grouping symbols
{ }, [ ], ( ), —
• may be nested
• Exponents
• Multiplication & Division
• left to right
• Addition & Subtraction
• left to right
Simplify!
• Remove Grouping Symbols
• Distributive & Associative Laws
a  b(c  d )  a  bc  bd
a  b(c  d )  a  bc  bd
a  b(c  d )  a  bc  bd
a  b(c  d )  a  bc  bd
a  (c  d )  a  c  d
a  (c  d )  a  c  d
a  (c  d )  a  c  d
a  (c  d )  a  c  d
Simplify!
• Remove Grouping Symbols
• Distributive & Associative Laws
• Group Like Terms
• Commutative & Associative Laws
• Combine Like Terms
• Distributive Law (i.e. add coefficients)
• Complete All Arithmetic
Table of Terms w/ Examples on Page 50.
Translate: Words  Algebra
• Addition
– added to; sum of;
plus; more than;
increased by
• Subtraction
– subtracted from;
difference of;
minus; less than;
decreased by
• Multiplication
– multiplied by;
product of; times;
twice; of
• Division
– divided by; quotient
of; divided into;
ratio of; per
… and combinations of these (e.g. twice the sum of …)
Problem Solving: The 4P’s
Problem Solving?
• Analyzing a situation
• Organizing information
• Choosing a strategy
• Implementing the strategy
The 4P’s
• Prepare
– Read (not a novel, it’s technical)
– What do you know?
– What are you trying to find?
• Plan (i.e. strategy)
– Guess & check?
– Patterns?
– Algebraic?
• Process (i.e. implementation)
• Ponder
– Did you answer the question?
– Does the solution make sense?
– Units?