```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
“Breaking rocks,” the first
replied. “Earning my living,” the
second said.”Helping to build a
cathedral,” said the third.” – Peter
• Adding numbers with the same sign
• To add two numbers that have
absolute values and keep the
same sign
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
• The number or expression
2
x3
• 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)
• a+b=b+a
• 2+3=3+2
Commutative for
Multiplication
• ab = ba
•2x3=3x3
•2*3=3*2
• 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
• a(b + c) = ab + ac
• 2(3 + 4) = 2 x 3 + 2 x 4
• X(Y + Z) = XY +XZ
•a+0=a
•3+0=3
•X+0=X
Multiplicative Identity
•ax1=a
•5x1=5
•1x5=5
•Y*1=Y
• 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
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
• 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
“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.
• 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
• 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
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
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CIRCLE
CIRCUM
CONE
CYLINDER
PRISM
PYRAMID
TRAPEZOI
APPS-AreaForm
• “Spectacular achievement
is always preceded by
spectacular preparation.”
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