Download Section 1.2 Exponents, Order of Operations, and Inequality

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Section 1.2
Exponents, Order of Operations, and Inequality
Exponent Notation
The base is _______ ; The exponent is _______.
Example 1
Find the value of the exponential expression.
(a)
(b)
(c)
Order of Operations
P
E
M
D
A
S
Example 2
Find the value of each expression.
(a)
(b)
(d) ( )
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Parentheses Exponents Multiplication Division Addition Subtraction
(c)
(d)
Inequalities
The alligator always chomps on the bigger number!
is less than
is greater than
is not equal to
Example 3
Determine whether each statement is true or false. If the statement is false, change the
inequality sign to make a correct statement.
(a)
(c)
(b)
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More Inequalities
is less than OR equal to
is greater than OR equal to
Example 4
Determine whether each statement is true or false. If the statement is false, change the
inequality sign to make a correct statement.
(a)
(c)
Example 5
Write each word statement in symbols.
(a) Nine is equal to eleven minus two.
(b) Fourteen is greater than twelve.
(c) Two is greater than or equal to two.
(d) Twelve is not equal to five.
(b)
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Section 1.3
Variables, Expressions, and Equations
Consider the following question:
There are 40 enrolled students. How many petitioners must there be if there
are 45 students in class on the first day?
We can represent this problem with an equation as follows:
Enrolled
students
Petitioners
Total
students
in class
In algebra, instead of drawing a box with a question mark, we use a letter of the alphabet to
represent the unknown number:
A __________________ is a symbol, usually a letter such as
unknown number.
or , used to represent any
An __________________ _________________ is a sequence of numbers, variables, operation
symbols (
), and/or grouping symbols that DOES NOT INCLUDE an equality (=) or
inequality (
) sign.
The following is an algebraic expression:
Example 1
Find the value of the algebraic expression
terms of the question posed above.
when
. Interpret your result in
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Use caution when writing expressions with exponents or finding their value.
is different than (
)
Exponent ONLY applies to
Example 2
Find the value of each algebraic expression if
Exponent applies to the entire
quantity in the parentheses
( )
.
(a)
(b) (
(c)
)
(
)
Example 3
Find the value of each algebraic expression if
(a)
(b)
and
.
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Example 4
Write each word phrase as an algebraic expression, using
as the variable.
(a) The sum of 3 and a number.
(b) A number minus 8.
(c) A number subtracted from 4.
(d) Eight subtracted from a number.
(e) The difference between a number and 2.
(f) The difference between 5 and a number.
(g) 14 times a number.
(h) Twelve divided by a number.
(i) The quotient of a number and 8.
(j) Nine multiplied by the sum of a number and 5.
(k) The product of 3 and five less than a number.
******************************************************************************
Let’s take another look at the original question posed at the beginning of this section.
“There are 40 enrolled students. How many petitioners must
there be if there are 45 students in class on the first day?”
We already found that we could represent the problem with the following equation:
where
represents the unknown number of petitioners.
An __________________ is a statement that two algebraic expressions are equal. An equation
ALWAYS INCLUDES an equality (=) sign.
To _______________ an equation means to find the values of the variable that make the
equation true. Such values of the variable are called the __________________ of the equation.
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Example 5
Decide whether the given number is a solution of the equation.
(a)
(b)
Example 6
Write the statement as an equation. Use
as the variable. Then find all solutions from the set
*
+
(a) The product of a number and 2 is 6.
(b) One less than twice a number is 15.
Example 7
Identify each as an expression or an equation.
(a)
(b)
(c)
(
)
Page 1 of 3
Section 1.4
Real Numbers and the Number Line
Natural Numbers:
Whole Numbers:
Integers:
Number Line:
0
Rational Numbers:
Irrational Numbers:
Real Numbers:
Real Numbers
Rational Numbers
Integers
Natural
Numbers
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Ordering of Real Numbers
For any two real numbers
and ,
is less than
if
is to the ____________ of .
Example 1
Determine whether the statement is true or false.
(a)
(b)
(c)
Additive Inverse
The _________________________________ of a number is the number that is the same
distance from 0 on the number line as , but on the opposite side of 0.
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Double Negative Rule
For any real number ,
Absolute Value
The absolute value,
of a real number is:
______________________________________________________________________________
*Distance is NEVER negative. Therefore, the absolute value of a number is never negative.
Example 2
Simplify.
(a)
(b)
(c)
Example 3
Decide whether each statement is true or false.
(a)
(b)
(c)
(d)
Page 1 of 2
Section 1.6
Multiplying & Dividing Real Numbers
Multiplication
For all real numbers ,
Rules of Signs:
( ) ( )
( ) ( )
( ) ( )
( ) ( )
Example 1
Find all integer factors of 50.
Division
Observe,
and
Any two numbers whose product is 1 are called reciprocals, or __________________________
_________________________.
The reciprocal of
is _______________. The reciprocal of
is ___________________.
What is the reciprocal of 0?? _________________________________________________.
******************************************************************************
Definition of Division: For any two real numbers and , (
)
In Words:
divided by equals times the reciprocal of .
******************************************************************************
Note:
Rules of Signs:
( )
( )
( )
( )
( )
( )
( )
( )
Page 2 of 2
Example 2
Perform the indicated operation(s).
(a) (
)( )
(b) (
)(
)
(c)
(
)
( )
(
(d)
(e)
(f)
(g)
(
(
Example 3
Evaluate. Let
)
)
)(
)
.
( )
( )
( )
( )
*******************************************************************
Words or phrases that mean MULTIPLY:
Product, times, double, twice, of, as much as
Words or phrases that mean DIVIDE:
Quotient, divided by, ratio of
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Section 1.7
Properties of Real Numbers
Commutative Properties
For all real numbers
,
(addition)
(multiplication)
Example 1
Use the commutative properties to complete the equality.
(a)
(b)
Associative Properties
For all real numbers
,
(addition)
(multiplication)
Example 2
Use the associative properties to complete the equality.
(a)
(b)
Identity Properties
For all real numbers
(addition)
(multiplication)
Example 3
Use the identity properties to complete the equality.
(a)
(b)
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Inverse Properties
For all real numbers
(addition)
(multiplication)
Example 4
Use the inverse properties to complete the equality.
(b) (
(a)
)
Distributive Property
For all real numbers
(addition)
(subtraction)
Example 5
Use the distributive property to complete the equality.
(a)
(b)
******************************************************************************
Example 6
Decide whether each statement is an example of the commutative, associative, identity, inverse
or distributive property.
(a)
(b)
(c)
(d)
(e)
(
)
Page 1 of 2
Section 1.8
Simplifying Expressions
Example 1
Simplify each expression.
(a)
(b) –
A term is a number, a variable, or a product or quotient of numbers and variables raised to
powers. The numerical coefficient of a term is the number in front of the variables.
Identify the numerical coefficient of each term:
Term
Coefficient
It is important to be able to distinguish between terms (separated by a
factors (multiplied):
or ) and
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Terms with exactly the same variables that have the same exponents, but possibly different
coefficients, are called
.
Determine whether the terms are like terms:
and
and
and
and
Using the distributive property (“in the reverse direction”) is called combining like terms.
For example,
Dist. Property
(in reverse)
Example 2
Combine like terms.
(a)
(b)
(c)
Example 3
Simplify each expression.
(a)
(b)
(c)
Section 2.1
Solving Linear Equations Part I: Addition Property of Equality
What is a Linear Equation?
Definitions
A linear equation in one variable can be written in the form
A, B and C , with A  0 .
The solution, or root, of an equation in x is ____________________________________
_______________________________________________________________________.
Equivalent equations are equations that ______________________________________.
Tool #1: Addition Property of Equality
If A, B, and C are real numbers, then the equations A  B and A  C  B  C are
______________________
Example 1: Solve each equation.
a) x  8  9
c)
1
4
x7   x
5
5
b)
y  6  10
d)
x  21.5  13.4
Solving Linear Equations
Example 2: Solve each equation and check your solution
a) 4 x  3x  6  6 x  10  3
b) 8x  3  7 x  1  6
c)  53x  3  1  16 x   0
e)  51  2 x   43  x   73  x   0
d)
6
3 4 1
1
x   x
7
4 5 7
6
Section 2.2
Linear Equations Part II: The Multiplication Property of Equality
Tool #2: Multiplication Property of Equality
If A, B, and C are real numbers, then the equations
are equivalent equations
Example 2: Solve each equation and check your solution
a) 7 x  10
b) 5x  70
c) .9 x  18
d)
1
x  3
5
e) 
5
4
x
6
9
f)  y  
1
2
Ex 2 Solve each equation and check your solution
a) 9 x  2 x  121
b) 11x  5x  6 x  168
c)  5x  4 x  8x  0
d) 10 x  6 x  3x  4
Page 1 of 4
2.3 More on Solving Linear Equations
Class Notes
Steps for solving a linear equation (An equation where the variable
power of 1):
only appears to a
Step 1: Simplify each side separately.
Step 2: Isolate the variable term on one side.
Step 3: Isolate the variable.
Then check your answer by plugging in your solution into the original equation!
Example 1
Solve.
(a)
(c)
(b)
(
)
(d)
(
(
)
)
(
)
Page 2 of 4
If fractions appear in an equation, CLEAR THE FRACTIONS by multiplying both
sides of the equation (or every term!) by the least common denominator.
If decimals appear in an equation, CLEAR THE DECIMALS by multiplying both sides
of the equation (or every term!) by either 10, 100, 1000, etc…
Example 2
Solve.
( )
( )
( )
(
)
(
(
)
)
(
)
Page 3 of 4
There are 3 types of linear equations:
1)
An equation with exactly one solution is called a ________________________________
____________________________.
2)
An equation for which every real number is a solution is called an __________________.
3)
An equation that has no solution is called a ____________________________________.
Example 3
Solve each equation. Then state whether the equation is a conditional equation, an
identity, or a contradiction.
(a)
(
(b)
(
(c)
(
)
(
)
)
)
(
)
(
)
Page 4 of 4
Algebra in Everyday Life
1. A football player gained y yards on a punt return. On the next return, he gained 4 yd.
What expression represents the number of yards he gained altogether?
2. A hockey player scored 42 goals in one season. He scored n goals in one game. What
expression represents the number of goals he scored in the rest of the games?
3. Chandler is b years old. What expression represents his age 3 yr ago? 5 yr from now?
4. Jean has y dimes. Express the value of the dimes in cents.
5. A clerk has v dollars, all in $20 bills. What expression represents the number of $20
bills the clerk has?
6. A concert ticket costs c dollars for an adult and f dollars for a child. Find an expression
that represents the total cost for 4 adults and 6 children.
Page 1 of 4
Section 2.4
An Introduction to Applications of Linear Equations
Solving an Applied Problem
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Read the problem carefully until you understand what is given and what is to be found.
Assign a variable to represent the unknown value, using diagrams or tables as needed.
Write down what the variable represents. If necessary, express any other unknown
values in terms of the variable.
Write an equation using the variable expression(s).
Solve the equation.
State the answer. Does it seem reasonable?
Check the answer in the words of the original problem.
Example 1
If 5 is added to the product of 9 and a number, the result is 19 less than the number. Find the
number.
(Assign the variable . Write an equation and solve.)
Example 2
In the 2006 Winter Olympics in Torino, Italy, the United States won 6 more medals than Norway.
The two countries won a total of 44 medals. How many medals did each country win?
(Assign the variable . Write an equation and solve.)
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Example 3
The owner of Terry’s Coffeehouse found that the number of orders for croissants was 1/6 the
number of orders for muffins. If the total number for the two breakfast rolls was 56, how many
orders were placed for croissants?
(Assign the variable . Write an equation and solve.)
Example 4
At a meeting of the local computer user group, each member brought two nonmembers. If a
total of 27 people attended, how many were members and how many were nonmembers?
(Assign the variable . Write an equation and solve.)
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Example 5
A piece of pipe is 50 in. long. It is cut into three pieces. The longest piece is 10 in. more than the
middle-sized piece, and the shortest piece measures 5 in. less than the middle-sized piece. Find
the lengths of the three pieces.
(Assign the variable . Write an equation and solve.)
Definition: Two integers that differ by 1 are called consecutive integers.
Examples:
5 and 6
102 and 103
-3 and -2
If x represents an integer, then _________ represents the next larger consecutive integer.
Consecutive even integers and consecutive odd integers differ by 2.
Examples:
4 and 6 (consecutive even integers)
7 and 9 (consecutive odd integers)
If x represents an even integer, then _________ represents the next larger even consecutive integer.
If x represents an odd integer, then _________ represents the next larger odd consecutive integer.
Page 4 of 4
Example 6
Two back-to-back page numbers in this book have a sum of 569. What are the page numbers?
(Assign the variable . Write an equation and solve.)
Example 7
Find two consecutive even integers such that six times the lesser added to the greater gives a
sum of 86.
(Assign the variable . Write an equation and solve.)
Section 2.5: Formulas and Applications from Geometry
Example: Find the value of the remaining variable.
P = 2 L + 2W
P = 126, W = 25
L
W
Example: A farmer has 800 m of fencing material to enclose a rectangular field. The width of the field
is 50 m less than the length. Find the dimensions of the field.
Step 1
Read
Step 2
Assign a variable
Step 3
Write an equation
Step 4
Solve
Step 5
State the answer
Step 6
Check
Pg. 1 of 4
Section 2.5: Formulas and Applications from Geometry
Example: The longest side of a triangle is 1 in. longer than the medium side. The medium side is 5 in.
longer than the shortest side. If the perimeter is 32 in., what are the lengths of the three sides?
Step 1
Step 2
Read
Assign a variable
Step 3
Write an equation
Step 4
Solve
Step 5
State the answer
Step 6
Check
Example: The area of a triangle is 120 m 2 . The height is 24 m. Find the length of the base of the
triangle.
Step 1
Step 2
Read
Assign a variable
Step 3
Write an equation
Step 4
Solve
Step 5
State the answer
Step 6
Check
Pg. 2 of 4
Section 2.5: Formulas and Applications from Geometry
Geometry Notes:
Complementary Angles
Supplementary Angles
Vertical Angles
Example: Find the measure of each marked angle in the figures.
(a)
(6 x + 29)
(b)
( x + 11)
(4 x + 19)
(6 x − 5)
Pg. 3 of 4
Section 2.5: Formulas and Applications from Geometry
Example: Solve I = prt for t .
Example: Solve S = 2π rh + 2π r 2 for h .
Example: Solve A = p + prt for t .
Example: Solve A =
1
h (b + B ) for h .
2
Pg. 4 of 4
Pg. 1 of 6
Section 2.7
Further Applications of Linear Equations
Percent means “per hundred”. For example, 45% represents the ratio 45 to 100, or
45
= 0.45.
100
Mixture Problems:
Solution
Substance X
(Amount of Solution)
x
(% concentration of substance X)
=
Amount of substance X
Interest Problems:
(Principal)
x
(% Interest Rate)
=
Annual Interest
Example 1:
a) What is the amount of pure acid in 40 L of a 16% acid solution?
b) Find the annual interest if $5000 is invested at 4%.
Pg. 2 of 6
Example 2: A certain metal is 40% copper. How many kilograms of this metal must be
mixed with 80 kg of a metal that is 70% copper to get a metal that is 50% copper?
Step 1
Step 2
Read
Assign a variable
Amount of
Total Metal (kg)
40% copper metal
70% copper metal
Mixture
Step 3
Write an equation
Step 4
Solve
Step 5
State the answer
Step 6
Check
% Concentration
of Copper
Amount of Copper (kg)
Pg. 3 of 6
Example3: With income earned by selling a patent, an engineer invests some money at 5% and $3000
more than twice as much at 8%. The total annual income from the investments is $1710. Find the
amount invested at 5%.
Step 1
Step 2
Read
Assign a variable
Principal
5% investment
8% investment
Step 3
Write an equation
Step 4
Solve
Step 5
State the answer
Step 6
Check
Interest Rate
Annual Interest
Pg. 4 of 6
Money Denominations Problems:
=
(Number)
x
(Value of one item)
=
Total Value
Example 4: A man has $2.55 in quarters and nickels. He has 9 more nickels than quarters. How many
nickels and how many quarters does he have?
Step 1
Step 2
Read
Assign a variable
Number of
coins
Quarters
Nickels
Step 3
Write an equation
Step 4
Solve
Step 5
State the answer
Step 6
Check
Value of one
Coin
Total Value
Pg. 5 of 6
Distance-Rate-Time Problems:
=
65 mph
(Rate)
x
(Time)
=
Distance
Example 5: A new world record in the men’s 100-m dash was set in 2005 by Asafa Powell of
Jamaica, who ran it in 9.77 sec. What was his speed in meters per second?
Example 6: Two airplanes leave Boston at 12:00 noon and fly in opposite directions. If one flies at
410 mph and the other 120 mph faster, how long will it take them to be 3290 mi apart?
Step 1
Step 2
Read
Assign a variable
Rate
Faster Airplane
Slower Airplane
Step 3
Write an equation
Step 4
Solve
Step 5
Step 6
State the answer
Check
Time
Distance
Pg. 6 of 6
Example7: Two buses left the downtown terminal, traveling in opposite directions. One had an
average speed of 10 mph more than the other. Twelve minutes (1/5 hour) later, they were 12 mi apart.
What were their speeds?
Step 1
Step 2
Read
Assign a variable
Rate
Faster Bus
Slower Bus
Step 3
Write an equation
Step 4
Solve
Step 5
State the answer
Step 6
Check
Time
Distance
Section 2.8
Solving Linear Inequalities
Definition
A linear inequality in one variable can be written in the form
where A, B, and C are real numbers, with A  0 .
Multiplication Property of Inequality
For any real numbers A, B, and C, with C  0 ,
1. If C is positive, then the inequalities A  B and
have exactly the same solutions.
2. If C is negative, then the inequalities A  B and
have exactly the same solutions.
Ex 1 Write each inequality in the interval notation and graph the interval
a) r  11
b)  3  x  5
Solving a Linear Inequality
1. Simplify each side separately.
2. Isolate the variable terms on one side.
3. Isolate the variable (make sure you reverse the inequality symbol only when multiply
or divide both sides of an inequality by a negative number)
Ex 2 Solve each inequality. Write the solution set in interval notation and graph it.
7
a) 5x  2  4 x  5
b)  x  14
8
c)  4 x  12  9 x  x  9  x
d)
7
 x  4   4  x  5
9
3
Definition
A compound inequality or a three-part inequality says that one number is between two
other numbers.
For example, using -7, 10, and 16: ________________________
Or using 4, x 1, and 9: _______________________________ (Now Solve!)
Ex 3 Solve the following compound inequalities. Write the solution set in interval notation and
graph it.
a)  1  1  5 y  16
b)  12 
3
x  2  4
7
Page 1 of 3
3.1 Linear Equations in Two Variables
A linear equation in two variables is an equation that can be written in the form:
where
and
are real numbers and
and
are not both 0.
Example 1
Write the following equations in the form
. Then identify
and .
(a)
(b)
(c)
A solution of a linear equation is a pair of numbers,
and , that make the equation true.
We write a solution of a linear equation as an ordered pair
A linear equation in two variables has infinitely many solutions.
Example 2
List a few solutions of the linear equation
.
(Note that there are infinitely many solutions!)
.
Page 2 of 3
Example 3
Decide whether the ordered pair
is a solution of the equation
Example 4
Complete each ordered pair for the equation
(a)
.
.
(b)
In order to graph the solutions to linear equations in two variables, we need a number line for
each value and .
The horizontal number line used to represent the value of a solution is called the - axis.
The vertical number line used to represent the value of a solution is called the - axis.
When both axes are represented together, we call this the rectangular (or Cartesian)
coordinate system. The numbers in the ordered pairs are called the coordinates of the point.
Page 3 of 3
Example 5
Complete the table of values for the equation. Then write the results as ordered pairs and plot
the solutions.
0
0
3
Example 6
Complete the table of values for the equation. Then write the results as ordered pairs and plot
the solutions.
pg. 1
Math 251 Beginning Algebra
3.2 Graphing Linear Equations in Two Variables
The graph of any linear equation in two variables, Ax + By = C , is a straight line.
Example 1
Graph 5 x + 2 y = −10 .
x
y
Example 2
Graph y = 23 x − 2 .
x
y
The x-intercept (x,0) is where a graph crosses the x-axis.
To find the x-intercept, let y = 0 and solve for x.
The y-intercept (0, y ) is where a graph crosses the y-axis.
To find the y-intercept, let x = 0 and solve for y.
Example 3
Find the intercepts for 5 x + 2 y = 10 .
Then draw the graph.
Math 251 Beginning Algebra
3.2 Graphing Linear Equations in Two Variables
pg. 2
Example 4
Graph 4 x − 2 y = 0 .
x
y
The graph of a linear equation of the form Ax + By = 0 passes through the origin (0,0) .
Example 5
Graph y = −5 .
x
Example 6
Graph x − 2 = 0 .
y
x
y
The graph of y = k, where k is a real number, is a horizontal line
with y-intercept (0, k ) and no x-intercept.
The graph of x = h, where h is a real number, is a vertical line
with x-intercept (h,0) and no y-intercept.
pg. 1
Math 251 Beginning Algebra
3.3 The Slope of a Line
Slope, m =
vertical change in y
Rise
=
horizontal change in x Run
Q (3,4)
P (−1,2)
P (0,−1)
Q (4,−5)
Example 2
Find the slope of the line.
Example 1
Find the slope of the line.
( −4,−2)
(2,−1)
( −3,−5)
(3,−6)
pg. 2
Math 251 Beginning Algebra
3.3 The Slope of a Line
x 2 − x1
( x2 , y 2 )
y 2 − y1
( x1 , y1 )
Slope Formula
The slope of the line through the points ( x1 , y1 ) and ( x 2 , y 2 ) is
m=
y 2 − y1
x 2 − x1
1) Find the slope given two points:
Example 1
Find the slope of the line through (6,−8) and (−2,4) .
Example 2
Find the slope of the line through (−2,3) and (−1,0) .
pg. 3
Math 251 Beginning Algebra
3.3 The Slope of a Line
Example 3
Find the slope of the line through (2,5) and (−1,5) .
Example 4
Find the slope of the line through (3,1) and (3,−4) .
Positive Slope
Line slants UPWARD
from left to right
Negative Slope
Line slants DOWNWARD
from left to right
Slope = 0
Line is HORIZONTAL
Slope is undefined
Line is VERTICAL
pg. 4
Math 251 Beginning Algebra
3.3 The Slope of a Line
2) Find the slope from the equation of a line:
Example 5
Find the slope of the line y = −2 x + 3 .
How to Find the Slope of a Line from its Equation
Method #1
Step 1 Pick any two points on the line
Step 2 Use the slope formula.
Example 6
Find the slope of the line y = 9 +
2
x.
3
Example 7
Find the slope of the line 3 x + 2 y = 9 .
Method #2 (FASTEST method!)
Step 1 Solve the equation for y.
Step 2 The slope is given by the coefficient of x.
pg. 5
Math 251 Beginning Algebra
3.3 The Slope of a Line
Parallel Lines
Same slope
Perpendicular Lines
Slopes have a product of -1
ie) negative reciprocals
Example 8
Decide whether the pair of lines is parallel, perpendicular, or neither.
(a)
(c)
3x − y = 4
x + 3y = 9
5 x − 3 y = −2
3 x − 5 y = −8
(b)
− 4x + 3 = 4
− 8x + 6 y = 0
Page 1 of 6
Section 3.4
Equations of a Line
Recall from the previous section, if given the equation of a line,
we may find the slope of the line as follows:
1)
2)
Example 1
Given the following equation of a line,
(a) Solve for .
(b) What is the slope of the line?
(c) Find the -intercept of the line.
Slope-Intercept Form of the Equation of a Line
Page 2 of 6
Example 2
Identify the slope and -intercept of each line.
(a)
Slope: _________
-int: _________
(b)
Slope: _________
-int: _________
(c)
Slope: _________
-int: _________
(d)
Slope: _________
-int: _________
Example 3
Find an equation of the line with slope
and -intercept
.
Page 3 of 6
Example 4
Graph the line with -intercept
and slope
.
Example 5
Graph the line passing through the
point
with slope
.
Example 6
Graph the line passing through
the point
with
.
Example 7
Graph the line passing through the
point
with undefined slope.
Example 8
Graph the equation of the line by using the slope and -intercept.
Page 4 of 6
Example 9
Write an equation in slope-intercept form of the line passing through the point
slope
.
with
In this last example, we were given a point on the line, and the slope of the line.
Since we didn’t know the -intercept, we could not immediately write the equation of the line
in slope-intercept form. We had to do a little bit of work to first find the -intercept.
Wouldn’t it be nice if there was a form of the equation of a line that didn’t
require that we know the -intercept?
Let’s derive another form of the equation of a line!
Suppose you are given that the point
is on a line with slope
.
Page 5 of 6
Point-Slope Form of the Equation of a Line
Example 10
Write an equation for the line passing through the point
Give the final answer in slope-intercept form.
Example 11
Write an equation for the line passing through the point
with slope
.
with undefined slope.
Page 6 of 6
Example 12
Write an equation for the line passing through the points
Give the final answer in slope-intercept form.
and
.
Example 13
Write an equation for the line passing through the points
and
.
Example 14
Write an equation of the line passing through the point
and perpendicular to
.
Section 3.5 Graphing Linear Inequalities in
Two Variables
Definition
An inequality that can be written as Ax  By  C or Ax  By  C , where A, B, and C are real
numbers and A and B are not both 0, is a linear inequality in two variables.
Steps in graphing a linear inequality
1. Graph the line.
a) Draw the line solid if  or  appears.
b) Draw the line dashed if > or < appears.
2. Test points from each interval in the inequality.
a) If the test point leads to a true statement, then the interval containing the point is part
of the solution. Shade that region.
b) If the test point leads to a false statement, then the interval containing the point is not
part of the solution. Shade the region not containing the point.
Example: Graph each linear inequality.
a) x  y  3
b) 3x  4 y  14
Example: Graph each linear inequality.
a) y  3x  1
b) y  2 x
Math 251: Beginning Algebra
Section 3.6 Notes
Introduction to Functions
When the elements in one set are linked to elements in a second set, we call this a relation.
Animal
Dog
Cat
Duck
Lion
Rabbit
Set of Inputs=Domain
Life Expectancy
(years)
11
10
7
Set of Outputs=Range
If x is an element in the domain and y is an element in the range, and if a relation exists between x and
y, then we say that y depends on x, and we write x → y . We can also represent this relation as a set of
ordered pairs ( x, y ) , where x represents the input and y represents the output:
{(Dog,11), (Cat,11), (Duck,10), (Lion,10), (Rabbit,7)}
x
y
Example 1:
Set A
Set B
1
5
8
10
12
3
Domain
4
6
0
Range
Represent this relation as a set of ordered pairs ( x, y ) , where x represents the input and y represents the
output:
Identify the domain and range of this relation:
Any set of ordered pairs is a relation.
Math 251: Beginning Algebra
Section 3.6 Notes
Example 2:
Identify the domain and range of the relation: {(2,4), (2, -3), (1, 5)}
A function is a special relation.
It is a set of ordered pairs in which each input corresponds to exactly one output.
Example 3: Determine whether each relation is a function.
(a)
Animal
Dog
Cat
Duck
Lion
Rabbit
(b)
Life Expectancy
(years)
11
10
7
Domain
Range
1
5
8
10
12
3
(c) {(-2,8), (-1,1), (0,0), (1,1), (2,8)}
(d) {(5,2), (5,1), (3,4)}
4
6
0
Math 251: Beginning Algebra
Section 3.6 Notes
Most useful functions have an infinite number of ordered pairs and are usually defined with equations
that tell how y depends on x.
Everyday Examples ☺
1. The distance d a car moving at 45 mph travels is a function of the time t:
d = 45 t
2. The cost y in dollars charged by an express mail company is a function of the weight x in pounds
determined by the equation:
y = 1.5( x − 1) + 9
One way to determine if a relation is a function is to look at the graph of the equation!
y
y
x
x2 + y2 = 1
y = x2
y
x
y = x+2
x
Math 251: Beginning Algebra
Section 3.6 Notes
Vertical Line Test
Intersects
in one point
Intersects in more
than one point
Passes the test
Function
Fails the test
Not a Function
If a vertical line intersects a graph in more than one point,
then the graph is not the graph of a function.
Example 4: Determine whether each relation is a function.
(a)
(c)
(b)
(b)
(d) x = 5
Math 251: Beginning Algebra
Section 3.6 Notes
Function Notation
x
f
f(x)=y
The letters _____, _____, and _____ are commonly used to name functions.
For example, since the equation y = 4 x − 3 describes a function, we may use function notation:
f ( x) = 4 x − 3
Say: “f of x”
For functions, the notations y and f (x) can be used interchangeably.
For the function defined by f ( x) = 4 x − 3 , if x = 2 , then f (2) = _____________________.
The statement f (2) = 5 says that the value of y is _______ when x is _______.
Say: “f of 2 equals 5”
The statement f (2) = 5 also indicates that the point (
,
) lies on the graph of f.
Math 251: Beginning Algebra
Example 5: For each function f, find f (2) , f (−1) , and f (0) .
(a) f ( x) = x − 1
f ( 2) =
f (−1) =
f ( 0) =
(b) f ( x) = x 2 − x + 1
f ( 2) =
f (−1) =
f ( 0) =
Section 3.6 Notes
Math 251: Beginning Algebra
Section 4.1 Notes
Solving Systems of Linear Equations by Graphing
Recall, an equation in two variables is linear provided it can be written in the form:
Ax + By = C
A system of linear equations is a grouping of two or more linear equations.
Definition:
Examples:
(a)
(b)
A solution of a system of linear equations is an ordered pair that makes both equations true at the
same time. A solution of an equation is said to satisfy the equation.
Example 1: Decide whether the ordered pair (4,−1) is a solution of each system.
(a)
(b)
5 x + 6 y = 14
2x + 5 y = 3
− x − y = −3
x + y = −3
Math 251: Beginning Algebra
Section 4.1 Notes
A system of two equations containing two variables represents a pair of lines.
The points of intersection are the solutions of the system.
Hence, we can look at their graphs to solve the system! Their graphs can appear in one of 3 ways:
Intersect at exactly one point
Parallel
They are the same line
One solution
No solution
Infinite number of solutions
Example 2: Solve the system by graphing.
2x − y = 4
4x + y = 2
Math 251: Beginning Algebra
Section 4.1 Notes
Example 3: Solve the system by graphing.
x + 2y = 4
2 x + 4 y = 12
Example 4: Solve the system by graphing.
2x − y = 4
4x = 2 y + 8
Math 251: Beginning Algebra
Section 4.2 Notes
Solving Systems of Linear Equations by Substitution
Solving a system by graphing is very difficult, especially without graph paper! Thus, we prefer
algebraic methods for solving systems of linear equations. There are two such algebraic methods:
1. Substitution
2. Elimination
We look at the method of substitution in this section.
Example 1: Solve the system by the substitution method.
2 x + 7 y = −12
x = −2 y
Step 1
Solve one equation for either variable.
Step 2
Substitute for that variable in the other equation.
Step 3
Solve the equation from Step 2.
Step 4
Substitute the result from Step 3 into the equation from Step 1 to find the value
of the other variable.
Step 5
Check the solution in both of the original equations.
Math 251: Beginning Algebra
Section 4.2 Notes
Solving a Linear System of Substitution
Step 1
Step 2
Step 3
Step 4
Step 5
Solve one equation for either variable.
Substitute for that variable in the other equation.
Solve the equation from Step 2.
Substitute the result from Step 3 into the equation from Step 1 to find the
value of the other variable.
Check the solution in both of the original equations.
Example 2: Solve the system by the substitution method.
x + 1 = −4 y
2 x − 5 y = 11
Example 3: Solve the system by the substitution method.
y = 8x + 4
16 x − 2 y = 8
Math 251: Beginning Algebra
Section 4.2 Notes
Example 4: Solve the system by the substitution method.
x + 3 y = −7
4 x + 12 y = −28
Example 5: Solve the system by the substitution method.
1
1
1
x+ y=−
2
3
3
1
x + 2 y = −2
2
(Clear fractions first!!)
Math 251: Beginning Algebra
Section 4.3 Notes
Solving Systems of Linear Equations by Elimination
Solving a system by graphing is very difficult, especially without graph paper! Thus, we prefer
algebraic methods for solving systems of linear equations. There are two such algebraic methods:
1. Substitution
2. Elimination
We look at the method of elimination in this section.
Example 1: Solve the system by the elimination method.
3x − y = 7
2x + y = 3
Step 1
Write both equations in standard form, Ax + By = C
Step 2
Transform the equations as needed so that the coefficients of one pair of
variable terms are opposites. Multiply one or both equations by appropriate numbers so
that the sum of the coefficients of either the x- or y- terms is 0.
Step 3
Add the new equations to eliminate a variable.
Step 4
Solve the equation from Step 3 for the remaining variable.
Step 5
Substitute the result from Step 4 into either of the original equations, and solve
for the other variable.
Step 6
Check the solution in both of the original equations. Then write the solution set.
Math 251: Beginning Algebra
Section 4.3 Notes
Solving a Linear System by Elimination
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Write both equations in standard form, Ax + By = C
Transform the equations as needed so that the coefficients of one pair of
variable terms are opposites. Multiply one or both equations by appropriate
numbers so that the sum of the coefficients of either the x- or y- terms is 0.
Add the new equations to eliminate a variable.
Solve the equation from Step 3 for the remaining variable.
Substitute the result from Step 4 into either of the original equations, and
solve for the other variable.
Check the solution in both of the original equations. Then write the solution
set.
Example 2: Solve the system by the elimination method.
x −2 = −y
2 x = y + 10
Math 251: Beginning Algebra
Section 4.3 Notes
Example 3: Solve the system by the elimination method.
4 x − 5 y = −18
3 x + 2 y = −2
Math 251: Beginning Algebra
Section 4.3 Notes
Example 4: Solve the system by the elimination method.
3y = 8 + 4x
6x = 9 − 2 y
Math 251: Beginning Algebra
Section 4.3 Notes
Example 5: Solve the system by the elimination method.
3 x + y = −7
6x + 2 y = 5
Example 6: Solve the system by the elimination method.
2x + 5y = 1
−4 x − 10 y = −2
Solving Systems of Equations Algebraically
There are two methods we can use to solve a system of equations using algebra (rather
than graphing). The two methods are the substitution method and the additionelimination method.
Substitution Method
The idea of the substitution method is to substitute one equation into the other to acquire
an equation with one variable only.
Example 1: Solve the system using substitution:
x− y =3
y = 5− x
Solution: Since the second equation is already solved for y, we can substitute the second
equation in for y in the first equation. The first equation will look like the following:
x − (5 − x ) = 3
x−5+ x = 3
2x − 5 = 3
2x = 8
x=4
Once you know x, you can go back to the original equations and find y. y = 5 − 4 = 1 .
The intersection point of the two lines is ( 4, 1) .
Example 2: Solve the system using substitution:
3x + y = 4
y = −3 x − 2
Solution: Again, the second equation had already been solved for y; therefore, we can
directly substitute into the first equation.
3 x + ( −3 x − 2 ) = 4
3x − 3 x − 2 = 4
−2 ≠ 4
This statement is never true; therefore, there is no solution to this system (remember this
means the lines are parallel.)
Example 3: Solve the system using substitution:
2 x − y = −2
y = 2x + 2
Solution: Again, we will substitute the second equation into the first equation.
2 x − ( 2 x + 2 ) = −2
2 x − 2 x − 2 = −2
−2 = −2
This statement is always true; therefore, the answer is an infinite number of solutions
(remember this means the lines are the same.)
Addition-Elimination Method
The addition-elimination method allows you to add the two equations together; thereby,
eliminating one of the variables. The idea is to get the coefficients of either the x- or the
y-values to be opposing numbers.
Example 1: Solve the system:
x+ y =3
x− y =7
Solution: Notice the coefficients to y are opposing values (1 and -1). If we add the two
equations together, we will get the following:
x+ y =3
+
x− y =7
2 x = 10
x=5
We can substitute into either one of the original equations to find y. 5 + y = 3 , y = −2 .
The lines intersect at the point ( 5, − 2 ) .
Example 2: Solve the system:
x + 2 y = 11
12 x − 6 y = 12
Solution: The coefficients of the y have opposing signs, but they also need to be the
same number. We can multiply all of equation 1 by 3 and add the two equations.
3 x + 6 y = 33
12 x − 6 y = 12
15 x = 45
x=3
Substituting back into the first equation, we can find y. 3 + 2 y = 11 , 2 y = 8 , y = 4 . The
lines intersect at the point ( 3, 4 ) .
Practice Problems
Solve the following systems using either method:
1.
4.
7.
3 x − 7 y = 22
5x − y = 2
3x − 4 y = 7
x − 3y = 2
3x − 2 y = 4
4 x + 5 y = −10
2.
5.
8.
2 x + y = −5
y = 2x + 3
2x − 5y = 9
4 x − 10 y = 18
6x − 5 y = 4
y = 4x + 2
3.
6.
9.
4 x − 3 y = −5
8 x = 6 y − 10
15 x + 6 y = 13
12 x − 10 y = 3
2 x + 3 y = 73
x = 5y + 4
Math 251: Beginning Algebra
Section 4.4 Notes
Solving an Applied Problem with Two Variables
Step 1
Read the problem carefully until you understand what is given and what is to be found.
Step 2
Assign variables to represent the unknown values, using diagrams or tables as needed.
Write down what each variable represents.
Step 3
Write two equations using both variables.
Step 4
Solve the system of two equations.
Step 5
State the answer. Does it seem reasonable?
Step 6
Check the answer in the words of the original problem.
Example 1: Two top-grossing Disney movies in 2002 were Lilo and Stitch and The Santa Clause 2.
Together they grossed $284.2 million. The Santa Clause 2 grossed $7.4 million less than Lilo and
Stitch. How much did each movie gross?
Step 1
Step 2
Read
Assign variables
Step 3
Write two equations
Step 4
Solve
Step 5
Step 6
State the answer
Check
Math 251: Beginning Algebra
Section 4.4 Notes
Example 2: In 1997-1998, the average movie ticket (to the nearest U.S. dollar) cost $10 in Geneva
and $8 in Paris. If a group of 36 people from these two cities paid $298 for tickets to see The Rookie,
how many people from each city were there?
Step 1
Step 2
Read
Assign variables
Step 3
Write two equations
# of Tickets
Step 4
Solve
Step 5
Step 6
State the answer
Check
Price per ticket
Total value
Math 251: Beginning Algebra
Section 4.4 Notes
Example 3: How many liters of a 25% alcohol solution must be mixed with a 12% solution to get
13 L of a 15% solution?
Step 1
Step 2
Read
Assign variables
Step 3
Write two equations
25%
12%
+
% Concentration
of alcohol
Step 4
Solve
Step 5
Step 6
State the answer
Check
15%
=
Liters of
Solution
Liters of pure alcohol
Math 251: Beginning Algebra
Section 4.4 Notes
Water/Air Current Problems:
x = speed of the boat in still water (speedometer reading)
y = speed of the current
r = true rate of the boat
x
x
y
y
Downstream (with current)
r=x+y
Upstream (against current)
r=x–y
Example 4: In one hour, Abby can row 2 mi against the current or 10 mi with the current. Find the
speed of the current and Abby’s speed in still water.
Step 1
Step 2
Read
Assign variables
Step 3
Write two equations
Rate
Step 4
Solve
Step 5
Step 6
State the answer
Check
Time
Distance
Applications of Linear Systems
There are two types of applications we will look at in this section: mixture problems and
motion problems. We will be using the same set-up as we saw in chapter two; however,
we now have the flexibility to use two variables rather than one.
Mixture Problems
There are two types of mixture problems: those involving quantities and costs and those
involving percents. The following two formulas will be used to make the tables:
Quantity x Cost per Product = Total Value
Amount of solution x % of concentrate = Amount of pure concentrate
Example 1: Candy worth $1.05 per pound was mixed with candy worth $1.35 per pound
to produce a mixture worth $1.17 per pound. How many pounds of each kind of candy
were used to make 30 pounds of the mixture?
Solution: Let x represent the number of pounds of the $1.05 candy and let y represent
the number of pounds of the $1.35 candy.
$1.05
$1.35
Total Mixture
Quantity (lbs)
x
y
30
x
Cost per lb
1.05
1.35
1.17
=
Total Value
1.05x
1.35y
1.17(30)
x + y = 30
1.05 x + 1.35 y = 1.17 ( 30 )
You can use any method to solve the system, below I used the substitution method.
The two equations will be:
If x + y = 30 , then x = 30 − y . This can be substituted into the second equation:
1.05 ( 30 − y ) + 1.35 y = 35.1
31.5 − 1.05 y + 1.35 y = 35.1
31.5 + .30 y = 35.1
.30 y = 3.6
y = 12
x = 30 − 12 = 18
We will mix 18 lbs of the $1.05 candy with 12 lbs of the $1.35 candy.
Example 2: A 3% sulfuric acid solution was mixed with an 18% sulfuric acid solution to
produce an 8% sulfuric acid solution. How much of each solution was used to produce
15 L of the 8% solution?
Solution: Let x represent the number of liters of the 3% solution and let y represent the
number of liters of the 18% solution.
3% solution
18% solution
8% solution
The two equations are:
Liters of solution
x
y
15
x
% of sulfuric acid =
.03
.18
.08
Sulfuric acid
.03x
.18(y)
.08 (15 )
x + y = 15
.03 x + .18 y = .08 (15 )
Again, I will use the substitution method. If x + y = 15 , then x = 15 − y and we can
substitute into the second equation.
.03 (15 − y ) + .18 y = .08 (15)
.45 − .03 y + .18 y = 1.2
.15 y = .75
y =5
x = 15 − 5 = 10
We will mix 10 liters of 3% solution with 5 liters of 18% solution.
Motion Problems
Remember motion problems are set up using the formula: Rate x Time = Distance.
A plane’s speed will depend on the rate of the wind and if the plane is flying with the
wind or against the wind. A boat’s speed will depend on the rate of the current and if the
boat is going with the current or against the current. The set up we will use for these
types of motion problems are the following:
Let x represent the rate of the plane or boat
Let y represent the rate of the wind or current
The total rate in our table will be x + y if the plane/boat is going with the wind/current
and x − y if the plane/boat is going against the wind/current. (Notice the rate of the
plane/boat is always first – the assumption is that the plane/boat is faster than the
wind/current.)
Example 3: In Grape Creek, Jim can row 30 km downstream in 3 hours or he can row
18 km upstream in the same amount of time. Find the rate at which Jim can row and the
rate of the current.
Solution: Let x represent the rate at which Jim can row and let y represent the rate of the
current.
Rate
x
Time
=
Distance
x
+
y
with current
3
30
x− y
3
18
against current
The two equations are:
( x + y ) 3 = 30
( x − y ) 3 = 18
use distributive property
3 x + 3 y = 30
3 x − 3 y = 18
I will use the addition-elimination method, by adding the two equations together to
eliminate the y variable.
3 x + 3 y = 30
3 x − 3 y = 18
6 x = 48
x =8
Substitute into one of the original equations: 3 ( 8) + 3 y = 30 to obtain y = 2 .
The rate at which Jim rows is 8 km/hr and the current is 2 km/hr.
Practice Problems
1. Starbucks would like to make a new blend of coffee that is lower in caffeine. The
company decides to mix decaf coffee worth $3.75 per lb with their Verona coffee
worth $4.35 per lb to produce a blend worth $4.11 per lb. How much of each kind of
coffee was used to produce 40 lbs of blended coffee?
2. A 25% salt solution was mixed with an 18% salt solution to produce a 20% salt
solution. How much of each solution was used to produce 28 L of the 20% solution?
3. Joe flew his small airplane 900 km in 5 hours flying with the wind. He flew 700 km
against the wind in 7 hours. Find the rate of the plane and the rate of the wind.
4. A jet liner flying east with the wind traveled 3000 km in 5 hours. The return trip,
flying against the wind, took 6 hours 15 minutes. Find the rate of the jet and the rate
of the wind.
5. How much pure tin must be mixed with a 4% tin alloy to produce 32 kg of a 16% tin
alloy?
6. The cost of 5 avocados and 6 tomatoes is $7.15. Three avocados and 8 tomatoes cost
$6.27. Find the cost of each avocado and each tomato.
7. The cost of 6 adult’s tickets and 12 children’s tickets is $210. The cost of 4 adult’s
tickets and 8 children’s tickets is $140. Find the cost of each adult’s ticket and each
child’s ticket.
4.5 Solving Systems of Linear Inequalities
To Solve a System of Linear Inequalities:
a)
b)
Step 1: Graph the first inequality.
Graph the boundary line (Dashed or Solid?)
Test a point not on the line to shade the correct region.
a)
b)
Step 2: Graph the second inequality.
Graph the boundary line (Dashed or Solid?)
Test a point not on the line to shade the correct region.
Step 3: The solution is the INTERSECTION of the two regions.
Example 1
Graph the solution set of the system:
3x  y  6
x  2y  8
Step 1: Graph
Step 2: Graph
Step 3: The solution is the intersection of the two regions. Make this region stand out by shading it
even darker!
Example 2
Graph the solution set of the system:
Example 3
Graph the solution set of the system:
x  2y  0
3x  4 y  12
x2
x 1  0
Section 5.1 The Product Rule and Power Rules for Exponents
Definition: Recall, for 52 , 5 is the __________ and 2 is the _______________.
Ex. 1 Write 2  2  2 in exponential form and evaluate.
Ex. 2 Evaluate. Name the base and the exponent
(a)
26
(b)
26
(c)
( 2)6
Some more examples of this:
By definition of exponents:
43  _____,(4)3  _____,  (4)3  _____
24  23  (2  2  2  2)(2  2  2)  27
Product Rule for Exponents: a m  a n  a mn
(Keep the same base; add the exponents)
Ex. 3 Use the product rule for exponents to find each product if possible.
(a) ( 7)5 ( 7)3
(b) ( 4 p5 )(3 p8 )
(c) m  m4
(d) z 2 z 5 z 6
(e) 42  35
(f) 64  62
Note the difference between adding and multiplying:
8x2  2 x2 
(8 x 2 )(2 x 2 ) 
By definition of exponents:
(42 )3  (42 )(42 )(42 ) 
Power Rule (a) for Exponents: (a m )n  a mn
(Raise a power to a power by multiplying exponents)
Ex. 4 Simplify.
(a) (62 )5
(b) ( z 4 )9
Note the difference between product and power rules:
4 2  43  4 2  3 
(42 )3  42(3) 
By definition of exponents:
(3x)4  (3x)(3x)(3x)(3x) 
Power Rule (b) for Exponents: (ab)m  a mbm
(Raise a product to a power by raising each factor to the power)
Ex. 5 Simplify.
(a) (3a 2b4 )5
(b) 4( xy )2
(c) ( 3m2 )3
Note: you must raise (-1)
to a power of 3.
(d) 3(2 x 3 )2
Note the power rule does not apply to a sum:
(4 x)2  42 x2 , but (4  x)2  42  x2
3
By definition of exponents:
3
 2   2  2  2  2


     
3
 3   3  3  3  3
m
am
a
Power Rule (c) for Exponents:    m
b
b
(Raise a quotient to a power by raising both numerator and denominator to
the power.)
Ex. 6 Simplify.
 3
(a)  
x
3
( x  0)
1
(b)  
 3
5
SUMMARY
Rules for Exponents:
a m  a n  a mn
( a m ) n  a mn
( ab) m  a mb m
m
am
a

 
bm
b
Ex. 7 Simplify
4
2
1
(a)   (2 x ) 2
 5
 5k 3 
(b) 

 3 
(c) ( 3xy 2 )3 ( x 2 y )4
(d) (4 x)2 (4 x)3
Ex. 8 Find an expression that represents the area of the rectangle.
(Note: Area and Volume formulas are on the inside back cover)
4x 2
8x 4
5.2 Integer Exponents and the Quotient Rule
24 
20 
23 
2 1 
22 
2 2 
21 
Zero Exponent: For any nonzero real number a, a 0  1 .
Ex. 1
Evaluate:
(a) 70
(c) 70
(b) ( 7)0
Negative Exponents: For any nonzero real number a, and any integer n, a  n 
Ex. 2 Evaluate the following
(a) 4 2
 3
(c)  
5
(e)
1
(b)  
4
2
1
m 3
3
(d) 21  51
1
.
an
1
23 23
Consider: 4 

1
3
34
Changing from Positive to Negative Exponents: For any nonzero numbers a and b
and any integers m and n,
a  m bn
a
 m and  
n
b
a
b
m
m
b
  .
a
Ex. 3
Simplify by writing with positive exponents. Assume that all variables represent nonzero
real numbers.
52
4h 5
(a) 3
(b)
m 2 k
3
 x2 
(c)  3 
 2y 
3
52  31
(d)
7  2 3
45 4  4  4  4  4


43
444
Consider:
42
44


5
4
44444
Quotient Rule
For any nonzero number a and any integers m and n,
am
a
 a mn and  
n
a
b
Keep the same base; subtract the exponents.
m
m
b
  .
a
Ex. 4
Simplify by writing with positive exponents. Assume that all variables represent nonzero
real numbers.
47
45
(a) 5
(b) 7
4
4
(c)
x 6
x 12
(d)
84 m 9n 3
85 m10n 2
Summary of All Rules for Exponents:
m
1. a  a  a
m
n
mn
2. a 0  1
3. a
n
1
 n
a
am
a
7.    m
b
b
a  m bn
8.  n  m
b
a
a
9.  
b
m
b
 
a
m
am
 a mn
n
a
5. (a m )n  a mn
6. (ab)m  a mbm
4.
Ex. 5
Simplify. Assume that all variables represent nonzero real numbers
(b) (4 x 2 )(4 x)2
(34 ) 2
(a)
33
 5y2 
(c) 

 6 
2
39  ( x 2 y ) 2
(d)
33  x 4 y
Rules for Exponents
The following is a list of the rules for exponents:
1. x m i x n = x m+ n
4.
xm
= x m− n
n
x
x
6.  
 y
−m
2.
5. x
m
ym
 y
=  = m
x
x
m
(x )
−m
n
= xmi n
3.
( xy )
m
m
x
xm
4.   = m
y
 y
1
= m
x
x−m y n
7. − n = m
y
x
8. x 0 = 1
Example 1: Simplify (the answer should contain only positive exponents):
Solution: We will use rule #4.
= xm y m
x7
x3
x7
= x 7 −3 = x 4
3
x
Example 2: Simplify (the answer should contain only positive exponents):
Solution: We will use both rules #4 and #5.
48 x 3
43 x 7
48 x 3
45
8 − 3 3− 7
5 −4
=
4
x
=
4
x
=
43 x 7
x4
12m5 n3
Example 3: Simplify (the answer should contain only positive exponents):
36m 2 n12
Solution: Again, we will use both rules #4 and #5.
12m5 n3 12 3 −9 1 m3
m3
=
m
n
=
=
36m 2 n12 36
3 n9
3n9
Example 4: Simplify (the answer should contain only positive exponents):
Solution: We will use rules #2, #3, & #4.
1 16 9 a16b9
a b =
3
3
20 a 5b3
5
( )
60 ( a b )
3 2 3
20 a 5
5
3 5
( ) (b )
=
60 ( a ) ( b )
3 3
2 3
=
20 a 5b3
5
( )
60 ( a b )
3 2 3
20a 25b15
=
60a 9b6
Practice Problems
Simplify (the answers should contain only positive exponents):
4
1.
( 2x) ( 2x)
3.
5 9
7
2
2 3
( 2a b ) ( 3a b )
3
2
3
4
( )(7 y )
5. −2 x y 5 x
 x −7 y 8 
7.  −2 −4 
x y 
4
4
8
7
4 3
9
7 2
2.
( 3x y )
4.
(2x y )
(6x y )
 16a 3b14 
6. 
7 9 
 4a b 
8
3 2
−6
8.
5
8
( 2 xy ) ( x y )
( 4x y )
7
9 2
9. 5−1 + 7 −1
10. 3−2 − 12−1
 3a 4b 2   8a12b 2 
11.  13 6   2 5 
 4 a b   9a b 
 5 x 7 y 3   75 x −3 y 4 
12.  2 9  
8 −3 
 3 x y   18 x y 
−2
Pg. 1 of 2
5.3 Scientific Notation
Use a scientific calculator to compute the following: 5,000,000 × 6,000,000 = ___________________
Very large and very small numbers often occur in the sciences. For such numbers, we use
scientific notation simply because it’s easier to work with AND most calculators cannot display
enough digits to give the answer in decimal form!
The form for scientific notation is:
a  10n where 1  a  10 and n is an integer.
Example 1: Determine whether or not each number is written in scientific notation.
Circle the numbers written in scientific notation.
1.3  102
2  103
0.4  102
53.2  106
21  103
5.99 1046
Observe the following:
1
2.14 × 10−1 = 2.14 × 10 = 0.214
2.14 × 101 = 2.14 × 10 = 21.4
1
2.14 × 102 = 2.14 × 100 = 214
1
2.14 × 103 = 2.14 × 1000 = 2,140
2.14 × 10−2 = 2.14 × 100 = 0.0214
2.14 × 10−3 = 2.14 × 1000 = 0.00214
Do you notice a pattern??
How to write a number in decimal notation (without exponents):
For numbers greater than 10, the exponent n is positive and equal to the number of places the
decimal point in the number a moves to the right.
scientific notation
decimal notation (without exponents)
3.45 × 104 = 3. 4 5 0 0 . = 34,500
For numbers less than 1, the exponent n is negative and equal to the number of places the
decimal point in the number a moves to the left.
scientific notation
decimal notation (without exponents)
3.45 × 10−4 = . 0 0 0 3. 45 = .000345
Example 2: Write in decimal notation:
1. 7.4  108
_____________________
4. 2  102
_____________________
2. 3.54  106 _____________________
5. 1.333  104 _____________________
3. 2.5  102 _____________________
6. 8  101
_____________________
Pg. 2 of 2
How to write a number in scientific notation:
First write 𝑎 (where 1 ≤ 𝑎 < 10) by placing the decimal after the first nonzero digit.
For numbers greater than 10, the exponent n on the base 10 should be positive and equal to the
number of places the decimal point in 𝑎 would need to move to the right to yield the number
without exponents.
decimal notation (without exponents)
scientific notation
34,500 = 3. 4 5 0 0 . = 3.45 × 104
For numbers less than 1, the exponent n on the base 10 should be negative and equal to the
number of places the decimal point in 𝑎 would need to move to the left to yield the number
without exponents.
decimal notation (without exponents)
scientific notation
0.000345 = . 0 0 0 3. 45 = 3.45 × 10−4
Example 3: Write each number in scientific notation:
1. 0.00035
_____________________
4. 280,000
_____________________
2. 358
_____________________
5. 0.125
_____________________
3. 0.0000056 _____________________
6. 43,000,000 _____________________
Example 4: Perform the indicated operations. Write each answer (a) in scientific notation
and (b) without exponents.
1.
2.
4 × 107 (3 × 10−3 )
3×10 9
6×10 5
Page 1 of 5
5.4 Adding and Subtracting Polynomials
Name the coefficient of each term in the expression: 2 x3  5x 2  x  4
Simplify by adding like terms: x 2  3xy 2  5x 2  10 xy 2
A polynomial in x is a term or the sum of a finite number of terms of the form ax n .
Examples:
NOT polynomials:
The degree of a term is the sum of the exponents on the variables:
Examples:
2x 7
degree:_______________
5
degree:_______________
3x 2 y 4
degree:_______________
The degree of a polynomial is the greatest degree of any nonzero term of the polynomial.
Examples:
1  4 x 2  3x 5
degree:_______________
4x  8
degree:_______________
3x 3 y 5  4 xy 6  20
degree:_______________
Page 2 of 5
A polynomial with only one term is called a ______________________.
Example:
A polynomial with exactly two terms is called a _______________________.
Example:
A polynomial with exactly three terms is called a _______________________.
Example:
Example 1
For each polynomial, first simplify, if possible, and write it in descending powers of the variable.
Then give the degree of the resulting polynomial and tell whether it is a monomial, binomial,
trinomial, or none of these.
(a) 6 p5  4 p3  8 p5  10 p 2
(b)
4 6 1 6
r  r
5
5
Example 2
Evaluate the polynomial 2 y 3  8 y  6 at y  1 .
Page 3 of 5
To add two polynomials, add like terms.
Example 3
Add.
(2 x 4  6 x 2  7)  ( 3x 4  5x 2  2)
Method 1 (Vertically):
Method 2 (Horizontally):
Example 4
Subtract.
(2 x 4  6 x 2  7)  ( 3x 4  5x 2  2)
Method 1 (Vertically):
Method 2 (Horizontally):
Page 4 of 5
Example 5
Subtract:
(5x3 y  3x 2 y 2  4 xy )  (7 x 3 y  x 2 y 2  6 xy )
Example 6
Find a polynomial that represents the perimeter of the rectangle:
Example 7
Graph the equation by completing the table of values.
0
1
2
Page 5 of 5
Math 251
5.5 Notes
pg. 1
5.5 Multiplying Polynomials
To find the product of polynomials, we use the distributive property:
a (b + c) = ab + ac
Example 1: Find the product.
(a)
2 x 4 (3 x 2 + 2 x − 5)
(b)
( m3 − 2m + 1) ⋅ (2m 2 + 4m + 3)
To multiply two polynomials, multiply each term of the second polynomial by each term of the
first polynomial and add the products.
(c)
Multiply Vertically:
3x 2 + 4 x − 5
x+4
Math 251
5.5 Notes
Find the product of 5 x 3 − 10 x 2 + 20 and
(d)
(e)
1 2 2
x +
5
5
Use the rectangle method to find the product (4 x + 3)( x + 2)
This approach can be extended to polynomials with any number of terms!
pg. 2
Math 251
5.5 Notes
pg. 3
Let’s look at the last example part (e). We could have also used the F.O.I.L. method of multiplying
two binomials:
Outer
First
F.
O.
I.
L.
( ax + b)( cx + d ) = acx 2 + adx + bcx + bd
Inner
Last
Example 2: Use the FOIL method to multiply.
(a)
(4 x + 3)( x + 2)
(b)
( m + 6) 2
(c)
( −4 y + x )(2 y − 3x )
(d)
3 x 3 ( x − 2)(2 x + 1)
Compare this answer to your answer from the last example part (e).
Math 251
5.6 Notes
5.6 Special Products
Multiply:
( x + y )2 =
( x − y )2 =
Square of a Binomial
( x + y )2 = x 2 + 2 xy + y 2
( x − y )2 = x 2 − 2 xy + y 2
Example 1:
(a)
(2 x + 1) 2
(b)
(5r − 6 s )2
(c)
1

 3k − 
2

(d)
x (2 x + 7) 2
2
pg. 1
Math 251
Multiply:
5.6 Notes
( x + y )( x − y ) =
Product of the Sum and Difference of Two Terms
( x + y )( x − y ) = x 2 − y 2
Example 2:
(a)
(3 + y )(3 − y )
(b)
(10m + 7)(10m − 7)
(c)
1 
1

 3r −   3r + 
2 
2

(d)
x (6 x + 5)(6 x − 5)
Example 3 (Finding Greater Powers of Binomials):
(a)
(4 x + 1)3
(b)
(3k − 2)4
pg. 2
Page 1 of 4
Section 5.7
Dividing Polynomials
1. Division by a Monomial
Example 1
Divide
Example 2
Divide
Example 3
Divide
(
)
by
.
Page 2 of 4
2. Division by a Non-Monomial
Question: How should we divide
Answer:
Recall the long division process:
Example 4
Divide
Page 3 of 4
Example 5
Divide
Example 6
Divide
Page 4 of 4
Example 7
Divide
Example 8
Divide
by
.
Page 1 of 5
Section 6.1
The Greatest Common Factor; Factoring by Grouping
To factor means “to write a quantity as a product.” Factoring is the opposite of multiplying.
List all the positive factors of 12:
List all the positive factor of 18:
From the lists above, identify the greatest common factor (GCF) of 12 and 18: __________
Example 1
Find the greatest common factor (GCF) for each list of numbers.
(a) 20, 64
(b) 12, 18, 26, 32
(c) 12, 13, 14
Page 2 of 5
Fact: The greatest common factor (GCF) will be the product of every common prime
factor raised to the smallest exponent.
Example 2
Find the greatest common factor of 72 and 240 by first factoring the numbers into prime
factors.
(A factor tree is very helpful!)
Example 3
Find the greatest common factor for each list of terms.
(a)
(b)
(c)
,
,
Page 3 of 5
Example 4
Find the greatest common factor for each list of terms.
(a)
(b)
(
) (
(
)
)
(
)(
(
)
)
The process of applying the distribute property (reverse direction) to write a sum as a product
with the greatest common factor (GCF) as one of the factors is called factoring out the greatest
common factor (GCF).
Eg.
(
Example 5
Factor out the greatest common factor (GCF).
(a)
(b)
(c)
(d)
(e)
)
Page 4 of 5
Example 6
Factor out the greatest common factor.
(a)
(
)
(b)
(
)
(
(
)
)
When a polynomial has four terms, we can often factor by grouping.
Example 7
Factor by grouping.
(a)
(b)
Page 5 of 5
(c)
(d)
(e)
Page 1 of 3
Section 6.2
Factoring Trinomials
(
Observe:
)(
)
Here, we have factored the polynomial
as a product of two binomials.
Note the relationship between the coefficients of the original polynomial and those of the two
binomials.
Trinomials with a Leading Coefficient of 1
where
the product
and
the sum
If it is not possible to find such
and
.
)(
)
then the polynomial cannot be factored, and we say
the polynomial is __________________.
Example 1
Factor
(
Page 2 of 3
Example 2
Factor
.
Example 3
Factor
Example 4
Factor
.
.
Page 3 of 3
Example 5
Factor
Example 6
Factor
.
(Factor a trinomial with two variables)
The 1st step in EVERY factoring problem is to _______________________________________!!!
Example 7
Factor
Page 1 of 4
Section 6.3
More on Factoring Trinomials
Main Objective:
Factor the general polynomial
Techniques:
#1)
#2)
when the leading coefficient
By Grouping (a.k.a. AC Method)
By Using FOIL (a.k.a. Trial & Error Method)
Technique #1: By Grouping
Example 1
Factor by grouping method.
Example 2
Factor by grouping method.
is not 1.
Page 2 of 4
Example 3
Factor by grouping method.
Example 4
Factor by grouping method.
Technique #2: By Using FOIL
Example 5
Factor by using FOIL (trial & error).
Page 3 of 4
Example 6
Factor by using FOIL (trial & error).
Example 7
Factor.
Example 8
Factor.
Page 4 of 4
Example 9
Factor.
Example 10
Factor.
Example 11
Factor.
Page 1 of 4
Section 6.4
Special Factoring Techniques
Difference of Squares
(
Example 1
Factor completely.
(a)
(b)
(c)
(d)
(e)
)(
)
Page 2 of 4
(f)
(g)
Difference of Cubes
(
)(
)
)(
)
Sum of Cubes
(
Example 2
Factor completely.
(a)
(b)
Page 3 of 4
(c)
(d)
(e)
(f)
Perfect Square Trinomials
Example 3
Factor completely.
(a)
(
)
(
)
Page 4 of 4
(b)
(c)
(d)
(e)
Factoring Worksheet
Factor out the Greatest Common Factor (GCF):
1. 15a  25b
2. 7c3  28c2 d  35cd 3
3. 4a4b  16a 2b2  4ab
Factor by grouping:
4. 3x2  9 x  4 x  12
5. 2 x2  5x  2 x  5
6. 3x2  18x  7 x  42
7. m2  8mn  3mn  24n2
Factor completely:
8. x2  13x  36
9. x2  2 x  48
10. x2  12 x  45
11. x2  6 x  5
12. x2  5x  6
13. 4 x2  24 x  64
14. 2 x2  11x  15
15. 3x2  13x  14
16. 5x2  28x  15
17. 2 x2  3x  35
18. 2 x2  7 x  72
19. 12 x4  60 x3  27 x2
20. 36 x 2  49 y 2
21. 121  144y 2
22. 27 x3  125
23. 64  y 3
Factoring Worksheet Answers:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Did you remember to factor out the GCF first?
For problems #14-19, you should try to factor by trial & error first. I have shown the
method of grouping in the solutions since it is difficult to show the trial and error process.
14.
15.
16.
17.
18.
19.
[
[
]
remember to factor out the GCF first?
[
]
For problems #20-23, use the special formulas.
20. Difference of Squares:
21. Difference of Squares:
22. Sum of Cubes:
23. Difference of Cubes:
[
]
]
Did you
Page 1 of 4
6.5 Solving Quadratic Equations by Factoring
A quadratic equation is an equation that can be written in the form:
 Standard Form
where
and are real numbers and
.
How to recognize a quadratic equation: The highest power of
is ________.
Circle which of the following equations are quadratic?
Zero-Factor Property
If
Example 1
Solve the quadratic equation.
(a)
(b)
(c)
, then _________________________________.
Page 2 of 4
How to Solve a Quadratic Equation by ____________________.
1)
2)
3)
Example 2
Solve the quadratic equation.
(a)
(b)
(c)
Page 3 of 4
Example 3
Solve the quadratic equation.
(a)
(b)
(c)
The zero-factor property may be used to solve equations that are not quadratic, but that are
still factorable, as we will see in the next example…
Page 4 of 4
Example 4
Solve.
(a)
(b)
(c)
6.6 Applications of Quadratic Equations
Solving an Applied Problem
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Read the problem carefully until you understand what is given and what is to be found.
Assign a variable to represent the unknown value, using diagrams or tables as needed.
Write down what the variable represents. If necessary, express any other unknown
values in terms of the variable.
Write an equation using the variable expression(s).
Solve the equation.
State the answer. Does it seem reasonable?
Check the answer in the words of the original problem.
Example 1: Area of a parallelogram:
The area of this parallelogram is
sq. units. Find its base and height.
Example 2: Area of a rectangle:
A hardcover book is 3
longer than it is wide. The area of the cover is 54
length and width of the book.
Example 3: Volume of a rectangular box:
The volume of this rectangular box is 192 cu. units. Find its length and width.
4
x
x+2
Find the
Example 4: Area of a triangle:
The area of a triangle is 24
. The base of the triangle measures
height. Find the measures of the base and the height.
more than the
Two consecutive integers:
Two consecutive even or odd integers:
Example 5:
The product of the first and third of three consecutive odd integers is 16 more than the
middle integer. Find the integers.
Page 1 of 4
7.1 The Fundamental Property of Rational Expressions
Consider the following expressions:
Each of these is a ratio of two __________________________.
A rational expression is an expression of the form:
where
and
are __________________________ , with
Example 1
Find the value of the rational expression
when
(a)
(b)
(c)
_______.
Page 2 of 4
Note: A rational expression is undefined for any values of the variable that cause the
denominator to be zero. To find these values, we set the denominator equal to zero and solve.
Example 2
Find any values of the variable for which each rational expression is undefined. Write answers
with .
Page 3 of 4
A rational expression
(
) is in lowest terms if the numerator and denominator share no
common factors (except 1).
The Fundamental Property of Rational Expressions
If
(
) is a rational expression and if
represents any polynomial (
To write a rational expression in lowest terms:
1)
2)
Example 3
Write in lowest terms.
, then
Page 4 of 4
Example 4
Write in lowest terms.
Example 5
Write four equivalent forms for the rational expression.
Page 1 of 2
7.2 Multiplying and Dividing Rational Expressions
Multiplying rational expressions
Example 1
Multiply. Write your answer in lowest terms.
Page 2 of 2
Dividing rational expressions
Example 2
Divide. Write your answer in lowest terms.
Page 1 of 3
Section 7.3
Least Common Denominators
Recall how to add or subtract two rational numbers with a common denominator:
Recall how to add or subtract two rational numbers with different denominators:
Before two rational expressions with different denominators can be added or subtracted, both
rational expressions must be expressed in terms of a common denominator. This common
denominator is called the LCD (Least Common Denominator).
How to find the LCD of two (or more) rational expressions:
1) Factor each denominator into prime factors.
2) The LCD is the product of the LCM of the coefficients and each variable factor raised to
the greatest power that occurs in any one factorization.
Example 1
Find the LCD:
Page 2 of 3
Example 2
Find the LCD:
The next step after finding a common denominator is to rewrite each rational expression with
the new denominator.
Example 3
Rewrite each rational expression with the indicated denominator.
Page 3 of 3
Example 4
Rewrite each rational expression with the LCD as the denominator.
Page 1 of 4
Section 7.4
Adding and Subtracting Rational Expressions
To add/subtract rational expressions with the same denominator, add/subtract the numerators
and keep the same denominator.
Recall how to write a rational expression in lowest terms:
1) Factor the numerator and denominator
2) Cancel common factors
Example 1
Add/subtract. Write each answer in lowest terms.
Page 2 of 4
To add/subtract rational expressions with different denominators:
1)
2)
3)
4)
Rewrite the problem with the denominators written in factored form.
Identify the LCD, and rewrite each rational expression with the LCD as the denominator.
Add/subtract the numerators and write this result over the LCD.
Write the answer in lowest terms (Factor and cancel common factors).
Example 2
Add/subtract. Write each answer in lowest terms.
Page 3 of 4
Page 4 of 4
pg. 1
7.5 Complex Fractions
Method #1 to simplify a complex fraction:
1. Write the numerator as a single fraction.
2. Write the denominator as a single fraction.
3. Divide using multiplication by the reciprocal. Write in lowest terms.
Example 1
Simplify the complex fraction (Method #1):
Example 2
Simplify the complex fraction (Method #1):
pg. 2
Example 3
Simplify the complex fraction (Method #1):
pg. 3
Method #2 to simplify a complex fraction:
1. Find the LCD of all fractions within the complex fraction.
2. Multiply both the numerator and the denominator of the complex fraction by
this LCD using the distributive property as necessary. Write in lowest terms.
Example 4
Simplify the complex fraction (Method #2):
Example 5
Simplify the complex fraction (Method #2):
pg. 4
Example 6
Simplify the complex fraction (Method #2):
Page 1 of 4
Section 7.6
Solving Equations with Rational Expressions
1. When simplifying expressions, keep the LCD (denominator) throughout the simplification.
2. When solving an equation, multiply each side by the LCD to “clear the fractions”.
Example 1
Identify each of the following as an expression or an equation. Then simplify the expression or
solve the equation.
(a)
4
3
x x
7
5
Example 2
Solve.
(b)
2m  3 m
6
 
5
3
5
4
3
x  x 1
7
5
Page 2 of 4
When solving an equation with variables in the denominator, you must check your
answer for extraneous solutions which make the denominator equal 0.
Solving an Equation with Rational Expressions
1. Factor the denominators to find the LCD. Multiply each side of the equation by the
LCD to clear fractions.
2. Solve the equation
3. Check your solutions and reject any that cause a denominator to equal 0.
Example 3
Solve, and check your solutions.
1
2
2x

x 1 x 1
Example 4
Solve, and check your solutions.
2
3
 2
p  2p p  p
2
Page 3 of 4
Example 5
Solve, and check your solutions.
8r
3
3


2
4r  1 2r  1 2r  1
Example 6
Solve, and check your solutions.
1
1
2
 
2
x  2 5 5( x  4)
Page 4 of 4
Example 7
Solve, and check your solutions.
6
1
4

 2
5a  10 a  5 a  3a  10
Example 8
Solve each formula for the specified variable.
x
(a) z 
for y
x y
(b)
2 1 1
  for z
x y z
Rational Expressions vs. Equations
For each exercise, indicate “expression” if an expression is to be simplified or “equation”
if an equation is to be solved. Then simplify the expression or solve the equation.
2.
8
2
m5
1
4 x2

x2  x  2 2 x  2
4.
2
3
 2
k  4k k  16
5.
x4 x3

5
6
6.
3
4
9

 2
x  3 x  6 x  9 x  18
7.
3
1 7
 
t 1 t 2
8.
6 2

y 3y
1.
4
1

p  2 3p  6
3.
2
Answers:
1. Expression;
(
)
2. Equation; * +
3. Expression;
(
4. Expression;
(
5. Equation; *
6. Equation;
7. Equation; {
8. Expression;
)
)(
)
+
(
)
}
Pg. 1 of 1
7.7 Applications of Rational Expressions
Recall from Chapter 2 the distance formula:
Distance = Rate × Time
d = rt
Divide both sides by Rate or Time respectively to get two alternate forms of the distance
formula:
Example 1
(a) At the 2006 Winter Olympics, Joey Cheek of the United States won the 500-m speed
skating event for men in 69.76 sec. What was his rate?
(b) In 2004, the Indianapolis 500 race was only 450 mi. Buddy Rice won with a rate of
138.518 mph. What was his time?
Pg. 2 of 2
Rate of boat against current = Rate of boat with no current − Rate of current
Rate of boat with current = Rate of boat with no current + Rate of current
Example 2
A boat can go 10 mi against a current in the same time it can go 30 mi with the current.
The current flows at 4 mph. Find the speed of the boat with no current.
Assign a variable:
Let x =
Set up table:
Distance
÷
Rate
=
Time
Against current
(Upstream)
With current
(Downstream)
Write an equation:
(Hint: How do the times going against and with the current relate in this problem?)
Solve:
Pg. 3 of 3
Suppose that you can mow your lawn in 4 hours. Then your rate of work is:
1 job
1
= job per hour
4 hours 4
Rate of Work
If a job can be completed in t units of time, then the rate of work is
1
job per unit of time
t
Now, after 3 hours, the fractional part of the job done is:
1
4
3
⋅
Time
Worked
Rate of
Work
3
4
=
Fractional part
of job done
Since you can complete the job in 4 hours:
1
4
⋅
Rate of
Work
4
Time
Worked
1
=
Fractional part
of job done
When the job is completed, the fractional part of job done is ________.
In general:
Work Problems Formula
Rate of work
×
Time worked
=
Fractional part of job done
Pg. 4 of 4
Example 3
Al and Mario operate a small roofing company. Mario can roof an average house alone
in 9 hr. Al can roof a house alone in 8 hr. How long will it take them to do the job if they
work together?
Assign a variable:
Let x =
Al’s rate of work:
Mario’s rate of work:
Set up table:
Rate of
Work
Time Working
Together
Fractional Part of
Job Done when
Working Together
Al
Mario
Write an equation:
Hint:
Solve:
Fractional part
done by Al
+
Fractional part
done by Mario
=
1 whole
job
Page 1 of 5
Section 8.1
Evaluating Roots
“Square” of a number
If
,
If
If
,
,
then
_______.
then
_______.
then
_______.
In this chapter, we will consider the opposite process…
“Square root” of a number
If
,
then
__________________.
If
,
then
__________________.
, then
__________________.
If
The “radical sign” symbol √ represents the positive (or principal) square root.
Also, √ represents the negative square root.
So √
_______ , but
√
_______
Example 1
Find each square root.
(a) √
(b)
(d) √
(e) √
√
(c) √
(f) √
Page 2 of 5
Note: For nonnegative ,
√
√
Example 2
Find the square of each radical expression.
(a) √
(b) √
(c) √
A perfect square is any number whose positive square root is a rational number.
Circle the perfect squares:
If
If
is not a perfect square, then √ is irrational.
is negative, then √ is not a real number.
Example 3
Determine whether each number is rational, irrational, or not a real number. If a number is
rational, give its exact value. If a number is irrational, give a decimal approximation to the
nearest thousandth using a calculator.
(a) √
(b) √
(c) √
(d)
√
Page 3 of 5
c
a
Recall, the Pythagorean Theorem:
a2  b2  c2
b
If
, then the positive solution of the equation
is
Example 4
A boat is being pulled toward a dock with a rope attached at water level. When the
boat is 24 ft. from the dock, 30 ft. of rope is extended. What is the height of the dock
above the water?
30 ft
24 ft
************************************************************************
Finding the distance between two points:
Distance Formula
The distance between the points
and
is
Page 4 of 5
Example 5
Find the distance between
and
.
************************************************************************
The opposite of “cubing” a number is taking the “cube root”.
,
,
so
√
so
√
____________
____________
The opposite of finding the “fourth power” of a number is taking the “fourth root”
,
so
√
____________
In general, the th root of
If the index
is even, then
is written √ .
must be nonnegative to yield a real number root.
Example 6
Find each root.
(a) √
(b)
(d) √
(e) √
√
(c) √
(f)
√
Page 5 of 5
The following lists will come in handy when evaluating roots until you become more
familiar with them:
Perfect Squares
Perfect Cubes
Perfect Fourth Powers
Page 1 of 6
Section 8.2
Multiplying, Dividing, and Simplifying Radicals
Observe the following:
√
√
and
Product Rule
For nonnegative real numbers
√
√
and ,
√
and
√
WARNING: The rule does not apply to sums. In general, √
Example 1
Find each product. Assume that
(a) √
√
√
√
.
(b) √
√
(c) √
√
A square root radical is simplified when no perfect square factor remains under the radical sign.
We accomplish this by using the product rule in the form: √
√ √
Example 2
Simplify each radical.
(a) √
(Method #1: Identify the greatest perfect square factor)
(b)
√
(c) √
Page 2 of 6
Example 3
Simplify each radical.
(Method #2: Use a factor tree to write the prime factorization.)
(a) √
(b)
√
(c)
√
Example 4
Find each product and simplify.
(a) √
√
(b) √
√
Page 3 of 6
Quotient Rule
For nonnegative real numbers
√
and ,
and
,
√
√
Example 5
Simplify each radical.
(a)
48
3
(c)
5
36
(b)
4
49
Example 6
8 50
Simplify
4 5
Some problems require both the product and quotient rules.
Example 7
Simplify
3 7

8 2
Page 4 of 6
Radicals can involve variables. Simplifying such radicals can get a little tricky.
If
represents a nonnegative number , then √
If
represents a negative number , then √
For any real number ,
√
To avoid negative radicands, variables under radical signs will be assumed to be nonnegative in
this course. Therefore, absolute value bars are not necessary (in this course).
Example 8
Simplify each radical. Assume that all variables represent positive real numbers.
(a) √
(b) √
√
(d) √
(c)
(e) √
(f) √
Page 5 of 6
In general,
n
a n b 
Example 9
Simplify each radical.
(a)
3
108
(b)
4
160
(c)
4
16
625
n
and
n
a

b
Page 6 of 6
To simplify cube roots with variables, use the fact that for any real number a,
a3  a
This is true whether a is positive or negative.
3
Example 10
Simplify each radical.
(a)
3
(c)
3
z9
54t
5
(b)
3
8x 6
(d)
3
a15
64
Section 8.3
Adding and Subtracting Radicals
pg. 1
We can add or subtract “like radicals” ONLY.
Example 1
Add or subtract, as indicated.
(a) 8 5 + 2 5
(b) 3 11 − 12 11
(c)
7 + 10
(d) 2 3 + 23 3
Sometimes, one or more radical expression can be simplified first. Then it is possible to
add or subtract like radicals.
Example 2
Add or subtract, as indicated.
(a)
27 + 12
(b) 5 200 − 6 18
(c) 23 54 + 43 2
Section 8.3
Adding and Subtracting Radicals
pg. 2
Example 3
Simplify each radical expression. Assume that all variables represent nonnegative real
numbers.
(a)
7 ⋅ 21 + 2 27
(b)
6 ⋅ 3r + 8r
(c) y 72 − 18 y 2
(d)
3
81x 4 + 53 24 x 4
Section 8.4
Rationalizing the Denominator
pg. 1
It is sometimes easier to work with radical expressions if the denominators do not contain
any radicals.
For example, to eliminate the radical in the denominator of
number and denominator by
1
, we can multiply the
2
2:
1
=
2
This process of changing the denominator from a radical to a rational number is called
rationalizing the denominator.
Example 1
Rationalize each denominator:
(a)
18
24
(b)
16
8
A radical is considered to be in simplified form if the following three conditions are met:
1. The radicand contains no factor that is a perfect square (in dealing with square
roots), a perfect cube (in dealing with cube roots), and so on..
2. The radicand has no fractions.
3. No denominator contains a radical.
Example 2
Simplify
5
18
Section 8.4
Rationalizing the Denominator
Example 3
Simplify
1 5
⋅
2 6
Example 4
Simplify
5p
q
. Assume that p and q are positive real numbers.
Example 5
Simplify
5r 2 t 2
. Assume that r and t represent nonnegative real numbers.
7
pg. 2
Section 8.4
Rationalizing the Denominator
pg. 3
To rationalize a denominator with a cube root, we must try to “create” a perfect cube in
the denominator.
Example 6
Rationalize each denominator.
(a)
5
6
3
3
(b)
3
3
(c)
3
2
3
3
4x
(x ≠ 0)
Section 8.5
More Simplifying and Operations with Radicals
pg. 1
Simplifying Radical Expressions – Guidelines
1. Simplify radicals containing perfect squares, perfect cubes, etc...
Example: 49 = 7
2. Use the product rule for radicals to get a single radical.
Example: 3 ⋅ 7 = 21
3. Factor terms under the radical sign to obtain perfect squares, perfect cubes, etc...
Example: 3 16 = 3 8 ⋅ 2 = 3 8 ⋅ 3 2 = 2 3 2
4. Use the distributive property to combine like radicals.
Example: 3 2 + 4 2 = 7 2
5. Rationalize the denominator.
3
3
3 2
6
=
=
⋅
=
Example:
2
2
2
2 2
Example 1
Find each product and simplify.
a)
b)
2
(
(
8 + 20
)(
2 +5 3 ⋅
)
3−2 2
)
Section 8.5
c)
(
d)
(4
e)
(
More Simplifying and Operations with Radicals
2− 5
2 +5
y −4
)(
)
10 + 2
2
)(
y +4
)
)
pg. 2
Section 8.5
More Simplifying and Operations with Radicals
pg. 3
Definition
The conjugate of a + b is a − b
The conjugate of 2 + 6 is _______________. The conjugate of 5 − x is _______________.
Conjugates can be used to rationalize the denominators in more complicated quotients.
Example 2
Simplify by rationalizing each denominator.
a)
3
2+ 5
b)
5+2
2− 3
Section 8.6
Solving Equations with Radicals
To solve a radical equation of the form
pg. 1
M = N , square both sides to obtain M = N 2 .
However, you MUST CHECK all proposed solutions in the original equation when squaring
both sides of an equation since “false” extraneous solutions may show up occasionally.
Example 1
Solve 9 − x = 4 .
Example 2
Solve 3x + 9 = 2 x .
Example 3
Solve x = −4 .
Section 8.6
Solving Equations with Radicals
Solving a Radical Equation
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Example 4
Solve x = x 2 − 4 x − 16
Example 5
Solve 2 x − 1 = 10 x + 9
Isolate a radical
Square both sides.
Combine like terms.
Repeat Steps 1-3 if there is still a term with a radical
Solve the equation.
Check for extraneous solutions.
pg. 2
Section 8.6
Example 6
Solve 25 x − 6 = x
Example 7
Solve x + 1 − x − 4 = 1
Solving Equations with Radicals
pg. 3
Section 8.7
Using Rational Numbers as Exponents
Question: What is 51/3 equal to?
51/3  51/3  51/3 
By Laws of Exponents:
3
Also, observe:
5 3 5 3 5 
Answer: 51/3 =
If a is a nonnegative number and n is a positive integer, then
⁄
Example 1
Simplify.
a) 491/2
b) 321/5
Question: What is 163/4 equal to?
Answer: By the power rule, (a m )n  a mn :
163/4 
If a is a nonnegative number and m and n are integers with n  0 , then
⁄
Example 2
Evaluate.
a) 95/2
b) 85/3
c) 272/3
pg. 1
Section 8.7
Using Rational Numbers as Exponents
If a is a nonnegative number and m and n are integers with n  0 , then
⁄
Example 3
Evaluate.
a) 363/2
b) 813/4
Example 4
Simplify. Write each answer in exponential form with only positive exponents.
a) 71/3  72/3
b)
9 2/3
9 1/3
 27 
c)  
 8 
d)
5/3
31/2  32
35/2
pg. 2
Section 8.7
Using Rational Numbers as Exponents
pg. 3
Example 5
Simplify. Write each answer in exponential form with only positive exponents. Assume that all
variables represent positive numbers.
a) (a 2/3b1/3c2 )6
b)
r 2/3  r1/3
r 1
 a 2/3 
c)  1/4 
b 
3
Example 6
Simplify 4 122 by first writing it in exponential form.
Page 1 of 3
Section 9.1
Solving Quadratic Equations by the Square Root Property
Observe:
If
, then
___________________.
Square Root Property of Equations
If is a positive number and if
then
Example 1
Solve each equation by using the square root property. Simplify all radicals.
a)
b)
c)
How to Solve an Equation by the Square Root Property
1. __________________ the squared term.
2. Take ____________________________ of both sides, remembering the _________.
3. Solve for the variable.
Example 2
Solve each equation by using the square root property. Simplify all radicals.
a)
b)
Page 2 of 3
To solve an equation of the form: (
Apply the Square Root Property, using
)
√
as the base:
Example 3
Solve each equation by using the square root property. Simplify all radicals.
a)
(
)
b)
(
c)
(
)
d)
(
)
)
Page 3 of 3
Now let’s try a few application problems…
Example 4
The formula
is used to approximate the weight of a bass, in pounds, given its length and its girth ,
both measured in inches. Approximate the length of a bass weighing 2.80 lb and having girth
11 in.
Example 5
The area of a circle with radius is given by the formula
If a circle has area
, what is its radius?
Page 1 of 3
Section 9.2
Solving Quadratic Equations by Completing the Square
We have already seen numbers that are perfect squares:
is a perfect square since
But trinomials may also be perfect squares:
is a perfect square since
Example 1
Complete each trinomial so that it is a perfect square. Then factor the trinomial.
The first three problems have been done for you so you may observe the pattern.
9
1
⁄
(
)
Procedure for Completing the Square
Start
Add…
The Result
Factored Form
Page 2 of 3
How to Solve a Quadratic Equation by Completing the Square
1) Collect all terms on the left side of the equation. The constant term should be on the right side.
2) Divide both sides of the equation through by the leading coefficient to create a leading coefficient 1.
3) Complete the square by adding ( ) to both sides of the equation, where is the coefficient of .
4) Factor the perfect square trinomial on the left side of the equation. Simplify the right side.
5) Solve by the Square Root Property from the previous section.
Example 2: Solve each equation by completing the square.
a)
b)
Page 3 of 3
c)
d)
e)
Page 1 of 3
Section 9.3
Solving Quadratic Equations by the Quadratic Formula
We can solve any quadratic equation by completing the square, but the method is tedious. In
this section, we learn a formula which can be used to solve any general quadratic equation:
Example 1
Identify the values of , , and in the following quadratic equations.
a)
______
______
______
b)
______
______
______
c)
______
______
______
______
______
______
d)
(
)(
)
The Quadratic Formula
The solutions of the quadratic equation
are
√
Notice that the
is under both the – and the √
. When using
this formula:
1. First simplify the numerator
√
.
2. If possible, factor out the GCF from the numerator. Then you may divide
out the common factor.
Correct:
√
Wrong:
(
√ )
√
√
Factor out 2 in numerator.
Then we can cancel the 2’s.
√
We are NOT allowed to cancel these
2’s because 2 is not a factor of the
numerator.
Page 2 of 3
√
Example 2
Use the quadratic formula to solve each equation. Simplify all radicals, and write all answers in
lowest terms.
a)
b)
c)
Page 3 of 3
√
d)
e)
f)