Intermediate Algebra – 1.3
•Operations with
Real Numbers
Three people were at
work on a construction
site. All were doing the
same job, but when each
was asked what the job
was, the answers varied.
“Breaking rocks,” the first
replied. “Earning my living,” the
second said.”Helping to build a
cathedral,” said the third.” – Peter
Schultz, German businessman
Procedure - Addition
• Adding numbers with the same sign
• To add two numbers that have
the same sign, add their
absolute values and keep the
same sign
Procedure - Addition
• Adding numbers with different
signs
• To add two numbers that have
different signs, subtract their
absolute values and keep the sign
of the number with the greater
absolute value.
Procedure - Subtraction
• For any real number a
• a – b = a + (-b)
Distance on number line
• The distance between two
points a and b is
• d = |a – b| = |b – a|
Procedure - Multiplying
• When multiplying two real
numbers that have different
signs, the product is negative
Procedure - Multiplying
• When multiplying two numbers that have
the same sign, the product is positive
Procedure - multiplying
• The product of an even number
of negative factors is positive,
• The product of an odd number
of negative factors is negative.
Division
• Division by Zero is undefined.
• 4/0 is undefined
• 0/4 = 0
n
a
Procedure - Division
a
b

a
b

a
b
Definition Square Root
• For all real numbers a and b, if
b a
2
then b is a square root of a
Def: radicand
• The number or expression
under the radical symbol
2
x3
Def: Index of radical
• The index is n
n
a
3
x
b
Calculator Keys
• [+], [*], [/], [-], [^]
• [ENTER]
[2ND][ENTRY]
ND
• [2 ] [QUIT]
[x,t,n]
• [MODE]
• [MATH][NUM][1:abs( ]
Norman Vincent Peale:
• “What seems impossible one
minute becomes, …, possible
the next.
Section 1.4
• Intermediate Algebra
• Properties of Real numbers
(9)
Commutative for Addition
• a+b=b+a
• 2+3=3+2
Commutative for
Multiplication
• ab = ba
•2x3=3x3
•2*3=3*2
Associative for Addition
• a + (b + c) = (a + b) + c
• 2 + (3 + 4) = (2 + 3) + 4
Associative for Multiplication
• (ab)c = a(bc)
• (2 x 3) x 4 = 2 x (3 x 4)
Distributive
multiplication over addition
• a(b + c) = ab + ac
• 2(3 + 4) = 2 x 3 + 2 x 4
• X(Y + Z) = XY +XZ
Additive Identity
•a+0=a
•3+0=3
•X+0=X
Multiplicative Identity
•ax1=a
•5x1=5
•1x5=5
•Y*1=Y
Additive Inverse
• a(1/a) = 1 where a not equal to 0
• 3(1/3) = 1
George Simmel - Sociologist
• “He is educated who
knows how to find out
what he doesn’t know.”
Section 1.4
Intermediate Algebra
• Apply order of operations
• Please Excuse My Dear Aunt
Sally.
• P – E – M – D – A- S
The order of operations
• Perform within grouping symbols – work
innermost group first and then outward.
• Evaluate exponents and roots.
• Perform multiplication and division left to
right.
• Perform addition and subtraction left to
right.
Grouping Symbols
•
•
•
•
•
•
Parentheses
Brackets
Braces
Radical symbols
Fraction symbols – fraction bar
Absolute value
Algebraic Expression
• Any combination of numbers, variables,
grouping symbols, and operation symbols.
• To evaluate an algebraic expression, replace
each variable with a specific value and then
perform all indicated operations.
Evaluate Expression by
Calculator
•
•
•
•
•
Plug in
Use store feature
Use Alpha key for formulas
Table
Program - evaluate
The Pythagorean Theorem
• In a right triangle, the sum of the square of
the legs is equal to the square of the
hypotenuse.
a b c
2
2
2
Equation
• A statement that two expression
have the same value
Intermediate Algebra – 1.5
• Walt Whitman – American Poet
• “Seeing, hearing, and
feeling are miracles,
and each part and tag
of me is a miracle.”
1.5 – Simplifying Expressions
• Term – An expression that is separated by
addition
• Numerical coefficient – the numerical factor
in a term
• Like Terms – Variable terms that have the
same variable(s) raised to the same
exponential value
Combining Like Terms
• To combine like terms, add or
subtract the coefficients and
keep the variables and their
exponents the same.
example
7  3  4   x  2    7  3  4  x  2 
  11  3 x
H. Jackson Brown Jr. Author
•“Let your
performance do the
thinking.”
Integer Exponents
• For any real number b and any natural
number n, the nth power of b o if found by
multiplying b as a factor n times.
b  b b b 
n
N times
b
Exponential Expression – an
expression that involves
exponents
• Base – the number being multiplied
• Exponent – the number of factors of the
base.
Calculator Key
• Exponent Key
^
Sydney Harris:
• “When I hear somebody
sigh,’Life is hard”, I am
always tempted to ask,
“Compared to what?”
Intermediate Algebra 1.5
•Introduction
•To
•Linear Equations
Def: Equation
• An equation is a
statement that two
algebraic expressions
have the same value.
Def: Solution
• Solution: A replacement for the
variable that makes the equation
true.
• Root of the equation
• Satisfies the Equation
• Zero of the equation
Def: Solution Set
• A set containing all the
solutions for the given
equation.
• Could have one, two, or many elements.
• Could be the empty set
• Could be all Real numbers
Def: Linear Equation in One
Variable
• An equation that can be written in
the form ax + b = c where a,b,c are
real numbers and a is not equal to
zero
Linear function
• A function of form
• f(x) = ax + b where a and b
are real numbers and a is not
equal to zero.
Def: Identity
• An equation is an identity if every
permissible replacement for the variable is a
solution.
• The graphs of left and right sides coincide.
• The solution set is R
R
Def: Inconsistent equation
• An equation with no solution is an
inconsistent equation.
• Also called a contradiction.
• The graphs of left and right sides never
intersect.
• The solution set is the empty set.

Def: Equivalent Equations
• Equivalent equations are equations that
have exactly the same solutions sets.
• Examples:
• 5 – 3x = 17
• -3x= 12
• x = -4
Addition Property of Equality
• If a = b, then a + c = b + c
• For all real numbers a,b, and c.
• Equals plus equals are equal.
Multiplication Property of
Equality
• If a = b, then ac = bc is true
• For all real numbers a,b, and c
where c is not equal to 0.
• Equals times equals are equal.
Solving Linear Equations
• Simplify both sides of the equation as
needed.
– Distribute to Clear parentheses
– Clear fractions by multiplying by the LCD
– Clear decimals by multiplying by a power of 10
determined by the decimal number with the
most places
– Combine like terms
Solving Linear Equations Cont:
• Use the addition property so that all variable
terms are on one side of the equation and all
constants are on the other side.
• Combine like terms.
• Use the multiplication property to isolate
the variable
• Verify the solution
Ralph Waldo Emerson – American essayist,
poet, and philosopher (1803-1882)
• “The world looks like a
multiplication table or a
mathematical equation,
which, turn it how you
will, balances itself.”
Problem Solving 1.6
• 1. Understand the Problem
• 2. Devise a Plan
– Use Definition statements
• 3. Carry out a Plan
• 4. Look Back
– Check units
Types of Problems
•
•
•
•
Number Problems
Angles of a Triangle
Rectangles
Things of Value
Les Brown
• “If you view all the things
that happen to you, both
good and bad, as
opportunities, then you
operate out of a higher level
of consciousness.”
Types of Problems Cont.
• Percentages
• Interest
• Mixture
• Liquid Solutions
• Distance, Rate, and Time
Albert Einstein
• “In the middle of
difficulty lies
opportunity.”
Ralph Waldo Emerson – American essayist,
poet, and philosopher (1803-1882)
• “The world looks like a
multiplication table or a
mathematical equation,
which, turn it how you
will, balances itself.”
Section 1.8
• Solve Formulas
• Isolate a particular variable in a formula
• Treat all other variables like constants
• Isolate the desired variable using the outline
for solving equations.
Know Formulas
• Area of a rectangle
A = LW
• Perimeter of a rectangle
• P = 2L + 2W
Formulas continued
• Area of a square
A s
2
• Perimeter of a square
P  4s
Formulas continued
• Area of Parallelogram
•A = bh
Formulas continued
• Trapezoid
A
1
2
 b1  b2  h
Formulas continued
• Area of Circle
A r
• Circumference of Circle
C  2 r
2
C d
Formulas continued:
• Area of Triangle
A
1
2
bh
Formulas continued
• Sum of measures of a triangle
o
0 8 1  3  m  2  m  1 m
Formulas continued
• Perimeter of a Triangle
P  s1  s 2  s 3
Formulas continued
• Pythagorean Theorem
a b c
2
2
2
Formulas continued:
• Volume of a Cube – all sides are equal
V  s
3
Formulas continued
• Rectangular solid
V  lw h
• Area of Base x height
Formulas continued
• Volume Right Circular Cylinder
V r h
2
Formulas continued:
• Surface are of right circular cylinder
S  2 rh  2 r
2
Formulas continued:
• Volume of Right Circular Cone
• V=(1/3) area base x height
V 
1
3
r h
2
Formulas continued:
• Volume Sphere
V 
4
3
r
3
Formulas continued:
•
•
•
•
•
General Formula surface area right solid
SA = 2(area base) + Lateral surface area
SA=2(area base) + LSA
Lateral Surface Area = LSA =
(perimeter)*(height)
Formulas continued:
• Distance, rate and Time
d = rt
Interest
I = PRT
Useful Calculator Programs
•
•
•
•
•
•
•
•
CIRCLE
CIRCUM
CONE
CYLINDER
PRISM
PYRAMID
TRAPEZOI
APPS-AreaForm
Robert Schuller – religious leader
• “Spectacular achievement
is always preceded by
spectacular preparation.”
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Chapter 1.3