Intermediate Algebra
Prerequisite Topics Review
• Quick review of basic algebra skills that
you should have developed before taking
this class
• 18 problems that are typical of things you
should already know how to do
Order of Operations
• Many math problems involve more than one
math operation
• Operations must be performed in the following
order:
–
–
–
–
Parentheses (and other grouping symbols)
Exponents
Multiplication and Division (left to right)
Addition and Subtraction (left to right)
• It might help to memorize:
– Please Excuse My Dear Aunt Sally
Example of Order of Operations
(A fraction bar is a grouping symbol; top and bottom should be simplified separately )
• Evaluate the following expression:
7  315  3  4  6  23
2
38  5
7 9 68
27
7  315  12  6  23
2
33
16  6  8
27
7  33  6  23
2
33
22  8
27
7  33  6  8
39
14
27
Problem 1
• Perform the indicated operation:
30 1  2
 93   2  12 2
• Answer:
30
7
Terminology of Algebra
• Expression – constants and/or variables
combined in a meaningful way with one or more
math operation symbols for addition, subtraction,
multiplication, division, exponents and roots
Examples of expressions:
23
5 x
10
4
n
y  9 w
2
• Only the first of these expressions can be
simplified, because we don’t know the numbers
represented by the variables
Terminology of Algebra
• If we know the number value of each variable in
an expression, we can “evaluate” the expression
• Given the value of each variable in an
expression, “evaluate the expression” means:
– Replace each variable with empty parentheses
– Put the given number inside the pair of parentheses
that has replaced the variable
– Do the math problem and simplify the answer
Example
• Evaluate the expression for x  3, y  4 :
12  2 y  x
12  2   
2

2
12  24   3
12  8  9
13
2
Problem 2
• Evaluate for x = -2, y = -4 and z = 3
3x  y
 4z
2
2
• Answer:
1
3
Like Terms
• Recall that a term is a _________
constant , a
product of a ________
________,
constant
variable or a _______
variables
and _________
• Like Terms: terms are called “like terms”
if they have exactly the same variables
with exactly the same exponents, but may
have different coefficients
• Example of Like Terms:
2
3x y and
 7x y
2
Simplifying Expressions by
Combining Like Terms
• Any expression containing more than one
term may contain like terms, if it does, all
like terms can be combined into a single
like term by adding or subtracting as
indicated by the sign in front of each term
• Example: Simplify: 4 x  19 y  6 x  2 y  x
Middle two steps can
be done in your head!
4 x  6 x  x  19 y  2 y
4  6 1x  19  2y
9 x  17 y
Simplifying an Expression
• Get rid of parentheses by multiplying or
distributing
• Combine like terms
• Example:
 3 x  5x  2  2 2 x  4  x
3x  5x 10  4x  4  x
3x 14
Problem 3
• Simplify:
72m  3  28m  4
• Answer:
 2m  29
Linear Equations
• Linear equation – an equation where, after
parentheses are gone, every term is either a
constant, or of the form: cx where c is a
constant and x is a variable with exponent1
Linear equations never have a variable in a
denominator or under a radical (square root
sign)
• Examples of Linear Equations:
4
x

5

13
.
3x  7  x 1
3
2 x  3  x
5
1
.72 x  6  x  83  x 
2
Solving Linear Equations
• Simplify each side separately
– Get rid of parentheses
– Multiply by LCD to get rid of fractions and decimals
– Combine like terms
• Get the variable by itself on one side by adding
or subtracting the same terms on both sides
• If the coefficient of the variable term is not 1,
then divide both sides by the coefficient
Determine if the equation is linear.
If it is, solve it: Is it linear? Yes
 8  6x  5  7  2x  4
 8  6x  30  7  2x  8
6x  38  2x 1
6x  2x  38  2x  2x 1
8x  38  1
8x  38  38  1  38
8x  37
8 x 37
37

x
8
8
8
Problem 4
• Solve:
5x  9  2x  3  2x  7
• Answer:
x  19
Linear Equations with No Solution
or All Real Numbers as Solutions
• Many linear equations only have one number as
a solution, but some have no solution and others
have all numbers as solutions
• In trying to solve a linear equation, if the variable
disappears (same variable & coefficient on both
sides) and the constants that are left make a
statement that is:
– false, the equation has “no solution” (no number can
replace the variable to make a true statement)
– true, the equation has “all real numbers” as solutions
(every real number can replace the variable to make
a true statement)
Solve the Linear Equation
2x  x  3  x  7
2x  x  3  x  7
x 3  x 7
x  x 3  x  x 7
37
False!
Equation has no solution
Solve the Linear Equation
x  2  7 x  2x  21  3x
x  2  7 x  2x  2  6x
8x  2  8x  2
8 x  8x  2  8 x  8 x  2
 2  2
True!
All real numbers are solutions
Problem 5
• Solve:
m m m
 
2 3 6
• Answer:
All Real Numbers, -, 
Problem 6
• Solve:
3x   2x  6  4x  4  x
• Answer:
No Solution, 
Problem 7
• Solve:
5 x 2 x

3
6
• Answer:
8
x
3
Formulas
• A “formula” is an equation containing
more than one variable
• Familiar Examples:
A  LW
(Area of a Rectangle)
P  2L  2W (Perimeter of a Rectangle)
1
(Area of a Triangle)
A  bh
2
P  a  b  c (Perimeter of a triangle)
Solving Formulas
• To solve a formula for a specific variable means
that we need to isolate that variable so that it
appears only on one side of the equal sign and
all other variables are on the other side
• If the formula is “linear” for the variable for which
we wish to solve, we pretend other variables are
just numbers and solve as other linear equations
(Be sure to always perform the same operation
on both sides of the equal sign)
Example
• Solve the formula for B:
1
2
A  B  A
2
3
1
1
2
A B  A
2
2
3
1  2 
1
6 A  B   6 A 
2  3 
2
3A  3B  4 A
3A  3A  3B  4 A  3A
3B  A
3B A

3
3
A
B
3
Problem 8
• Solve for n:
A  P1  ni 
• Answer:
A P
n
Pi
Steps in Solving
Application Problems
• Read the problem carefully trying to understand what the
unknowns are (take notes, draw pictures, don’t try to
write equation until all other steps below are done )
• Make word list that describes each unknown
• Assign a variable name to the unknown you know the
least about (the most basic unknown)
• Write expressions containing the variable for all the other
unknowns
• Read the problem one last time to see what information
hasn’t been used, and write an equation about that
• Solve the equation (make sure that your answer makes
sense, and specifically state the answer)
Example of Solving an Application
Problem With Multiple Unknowns
• A mother’s age is 4 years more than twice her
daughter’s age. The sum of their ages is 76.
What is the mother’s age?
• List of unknowns
– Mother’s age
2x  4
– Daughter’s age x
Which do we know least about? Daughter' s age
• What else does the problem tell us that we
haven’t used?
Sum of their ages is 76
• What equation says this?
x  2x  4  76
Example Continued
• Solve the equation:
x  2x  4  76
3x  4  76
3x  4  4  76  4
3x  72
x  24
• Answer to question?
Mother’s age is 2x + 4:
224  4  52
Example of Solving an Application
Involving a Geometric Figure
• The length of a rectangle is 4 inches less
than 3 times its width and the perimeter of
the rectangle is 32 inches. What is the
length of the rectangle?
• Draw a picture & make notes:
Length is 4 inches less than 3 times width
Nothing know about widt h
Perimeter is 32 inches
• What is the rectangle formula that applies
P  2L  2W
for this problem?
Geometric Example Continued
• List of unknowns:
– Length of rectangle:
– Width of rectangle:
3x  4 Length is 4 inches less than 3 times width
x This is the most basic unknown
• What other information is given that hasn’t
been used? Perimeter is 32 inches
• Use perimeter formula with given
perimeter and algebra names for
P  2L  2W
unknowns:
32  23x  4  2 x
Geometric Example Continued
• Solve the equation: 32  23x  4  2 x
32  6x  8  2x
32  8x  8
40  8x
5 x
• What is the answer to the problem?
The length of the rectangle is:
3x  4  35  4  11
Problem 9
• The perimeter of a certain rectangle is 16 times
the width. The length is 12 cm more than the
width. Find the width.
• Answer:
w  2 cm.
Inequalities
• An “inequality” is a comparison between
expressions involving these symbols:
<

>

“is less than”
“is less than or equal to”
“is greater than”
“is greater than or equal to”
Inequalities Involving Variables
• Inequalities involving variables may be true or
false depending on the number that replaces the
variable
• Numbers that can replace a variable in an
inequality to make a true statement are called
“solutions” to the inequality
• Example:
What numbers are solutions to: x  5
All numbers smaller than 5
Solutions are often shown in graph form:
)
0
5
Notice use of parenthesi s to mean less than
Graphing Solutions to
Inequalities
x  2
• Graph solutions to:
]
2
0
x  2
• Graph solutions to:
)
2
• Graph solutions to:
0
x  2
[
2
• Graph solutions to:
0
(
2
0
x  2
Linear Inequalities
• A linear inequality looks like a linear
equation except the = has been replaced
by:  ,  ,  , or 
• Examples:
3
x  2 x  3
4x  5  13
5
1
.72 x  6  x  83  x 
x  7  3x 1
2
• Our goal is to learn to solve linear
inequalities
Solving Linear Inequalities
• Linear inequalities are solved just like
linear equations with the following
exceptions:
– Isolate the variable on the left side of the
inequality symbol
– When multiplying or dividing by a negative,
reverse the sense of inequality
– Graph the solution on a number line
Example of Solving Linear
Inequality
2
8
x
2
2
x  7  3x 1
x  3x  7  3x  3x 1
 2x  7  1
x  4
 2x  7  7  1  7
 2x  8
(
4
0
Problem 10
• Solve and graph solution:
4x  3x  10  4x  7 x
• Answer:
x  5
)
5
0
Three Part Linear Inequalities
• Consist of three algebraic expressions compared with
two inequality symbols
• Both inequality symbols MUST have the same sense
(point the same direction) AND must make a true
statement when the middle expression is ignored
• Good Example:
1
 3   x  4   1
2
• Not Legitimate:
1
 3   x  4   1 Inequality Symbols Don' t Have Same Sense
2
.
1
 3   x  4   1 - 3 is NOT  -1
2
Expressing Solutions to Three
Part Inequalities
• “Standard notation” - variable appears alone
in the middle part of the three expressions
being compared with two inequality symbols:
2 x 3
• “Graphical notation” – same as with two part
inequalities:
2
3
(
]
• “Interval notation” – same as with two part
inequalities:
(2, 3]
Solving
Three Part Linear Inequalities
• Solved exactly like two part linear
inequalities except that solution is
achieved when variable is isolated in the
middle
Example of Solving
Three Part Linear Inequalities
1
 x  4   1
2
1
 3  x  2  1
2
3
 6  x  4  2
2 x  2
Standard Notation Solution
2
2
[
)
Graphical Notation Solution
[2, 2) Interval Notation Solution
Problem 11
• Solve:
 3  2m  1  4
• Answer:
3
2 m
2
2
[
3
2
]
Exponential Expression
a
n
• An exponential expression is:
where
is called the base and n is
called the exponent
• An exponent applies only to what it is
immediately adjacent to (what it touches)
• Example:
2
3x Exponent applies only to x, not to 3
4
 m Exponent applies only to m, not to negative
3
2x  Exponent applies to (2x)
a
Meaning of Exponent
• The meaning of an exponent depends on
the type of number it is
• An exponent that is a natural number
(1, 2, 3,…) tells how many times to
multiply the base by itself
2
3x  3  x  x
• Examples:
 m  1 m  m  m  m
3
3
2x   2x 2x 2x  8x
4
In the next section we will learn the meaning of any integer exponent
Rules of Exponents
• Product Rule: When two exponential
expressions with the same base are
multiplied, the result is an exponential
expression with the same base having an
exponent equal to the sum of the two
exponents
m
n
m n
a a  a
• Examples:
4 2
3 3  3  3
11
7
4
7 4
x x  x  x
4
2
6
Rules of Exponents
• Power of a Power Rule: When an
exponential expression is raised to a
power, the result is an exponential
expression with the same base having an
exponent equal to the product of the two
exponents
m n
mn
• Examples:
a   a
3   3
x   x
4 2
42
7 4
74
 3
28
 x
8
Rules of Exponents
• Power of a Product Rule: When a
product is raised to a power, the result is
the product of each factor raised to the
n
power
n n
• Examples:
ab 
a b
3x 
 3 x  9x
2 y 
 2 y  16 y
2
4
2
4
2
4
2
4
Rules of Exponents
• Power of a Quotient Rule: When a
quotient is raised to a power, the result is
the quotient of the numerator to the power
and the denominator to the power
n
• Example:
a
a
   n
b
b
2
2
3
3
   2 
x
 x
n
9
2
x
Using Combinations of Rules to
Simplify Expression with Exponents
• Examples:

  5  2 m p 5 16m p  80m p
 5x y    5  x y  125x y
2 x y   3x y   8x y  9x y  72x y
2 x y   8x y  8 x
9y
 3x y  9 x y
2
3 4
2
3 3
5 2m p
2
3 3
2
5 2
8
12
3
2
3 3
2
4
6
3 2
8
6
9
4
10
6
9
6
12
9
4
2
6
8
12
9
10 15
Integer Exponents
• Thus far we have discussed the meaning
of an exponent when it is a natural
(counting) number: 1, 2, 3, …
• An exponent of this type tells us how many
times to multiply the base by itself
• Next we will learn the meaning of zero and
negative integer exponents
0
• Examples:
5
2
3
Definition of Integer Exponents
• The following definitions are used for zero
and negative integer exponents:
a 1
0
a
n
1
 
a
n
• These definitions work for any base,
that is not zero:
3
5  1
0
1
1
2   
8
2
3
a,
Quotient Rule for Exponential
Expressions
• When exponential expressions with the same base are divided, the
result is an exponential expression with the same base and an
exponent equal to the numerator exponent minus the denominator
exponent
am
mn

a
an
Examples:
54
47
3
5

5

57
.
x12
12 4
8

x

x
x4
“Slide Rule” for Exponential
Expressions
• When both the numerator and denominator of
a fraction are factored then any factor may
slide from the top to bottom, or vice versa, by
changing the sign on the exponent
Example: Use rule to slide all factors to other
part of the fraction:
a mb  n
cr d s
 m n
r s
c d
a b
• This rule applies to all types of exponents
• Often used to make all exponents positive
Simplify the Expression:
(Show answer with positive exponents)
16
8 y y 
8 y 6 y 2
8  21
8 y 8
 1 4 1  1 3  3 8  11
1 4 1
y
2 y y
2 y y
y y
2 y
3
2
Problem 12
• Evaluate:
 2 4
• Answer:
1

16
Problem 13
• Evaluate:
 3
0
 3 1
• Answer:
4
3
Problem 14
• Use rules of exponents to simplify and use only
positive exponents in answer:
x  x y 
xy 
 3 2
1
2 2
• Answer:
x2 y6
2
Polynomial
• Polynomial – a finite sum of terms
• Examples:
6 x  5 x  4 How many terms ? 3
2
Degree of first term ? 2
Coefficien t of second term? - 5
3x y  5 x y
2
4
6
How many terms ? 2
Degree of second term? 10
3
Coefficien t of first term ?
Special Names for Certain
Polynomials
Number of Terms
Special Name
One term:
 9x y
Two terms:
3x y  5 x y
Three terms:
6 x  5x  4
2
2
2
monomial
4
6
binomial
trinomial
Adding and Subtracting
Polynomials
• To add or subtract polynomials
horizontally:
– Distribute to get rid of parentheses
– Combine like terms
• Example:
2x
2
 

 3x  1  x  x  3  3x  2
2
2 x 2  3x  1  x 2  x  3  3x  2
3x 2  5 x
Multiplying Polynomials
• To multiply polynomials:
– Get rid of parentheses by multiplying every
term of the first by every term of the second
using the rules of exponents
– Combine like terms
• Examples:
x  32 x 2  5x  4 
2x  35x  4 
2 x 3  5 x 2  4 x  6 x 2  15 x  12  2 x 3  x 2  11x  12
10 x 2  8 x  15 x  12 
10 x 2  7 x  12
Problem 15
• Multiply and simplify:
4x  3 y 2x  y 
• Answer:
8x 2  2 xy  3 y 2
Squaring a Binomial
• To square a binomial means to multiply it
by itself (the result is always a trinomial)
2x  32  2x  32x  3 
2
4 x 2  6 x  6 x  9  4 x  12 x  9
• Although a binomial can be squared by
foiling it by itself, it is best to memorize a
shortcut for squaring a binomial:
a  b 2 
2x  32 
a 2  2ab  b 2
4 x 2  12 x  9
first 2  2(first)(s econd)  second 2
Problem 16
• Multiply and simplify:
5 x  y 
2
• Answer:
25x 2  10 xy  y 2
Dividing a Polynomial by a
Monomial
• Write problem so that each term of the
polynomial is individually placed over the
monomial in “fraction form”
• Simplify each fraction by dividing out
common factors
8x y 12xy
3
3
2
2

 4 xy  2  2 xy
8x y 12 xy 4 xy 2



2 xy
2 xy 2 xy 2 xy
1
2
4x  6 y  2 
xy
Problem 17
• Divide:
8 y  6 y  4 y  10
2y
3
2
• Answer:
5
4 y  3y  2 
y
2
Dividing a Polynomial by a
Polynomial
• First write each polynomial in
descending powers
• If a term of some power is missing, write
that term with a zero coefficient
• Complete the problem exactly like a long
division problem in basic math
Example
 2x  3x 150 x  4
3x  2x  0x 150 x  0x  4
2
3
3
2
2
2
12 x  158
3x  2  2
x 4
x 2  0 x  4 3 x 3  2 x 2  0 x  150
 ( 3x 3  0 x 2  12 x )
 2 x 2  12 x  150
 ( 2x2  0x  8 )
12x 158
Problem 18
• Divide:
3
3x  4  x x  2
• Answer:
26
3 x  6 x  11 
x2
2