Transcript Elementary Algebra Exam 3 Material Formulas, Proportions, Linear Inequalities

Elementary Algebra
Exam 3 Material
Formulas, Proportions, Linear
Inequalities
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 a Formula for One Variable
Given Values of Other Variables
• If you know the values of all variables in a
formula, except for one:
– Make substitutions for the variables whose
values are known
– The resulting equation has only one variable
– If the equation is linear for that variable, solve
as other linear equations
Example of Solving a Formula
for One Variable Given Others
• Given the formula: P  2L  2W
and P  40 , W  5 , solve for the
remaining variable:
   2L  2 
40  2L  25 Equation is linear for L
40  2L  10
30  2L
15  L
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 W :
A  LW
Since A and L are assumed to be constants
this is similar to : 3  5W
How would you solve this for W?
Divide both sides by 5
How would you solve the real formula?
Divide both sides by L :
A
W
L
Example
• Solve the formula for L :
P  2W  2L
Since P and W are assumed to be constants
this is similar to : 7  4  2 L
How would you solve this for L?
Subtract 4 and then divide by 2 on both sides
How would you solve the real formula?
Subtract 2W on both sides : P - 2W  2L
P - 2W
Divide both sides by 2 :
L
2
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
Solving Application Problems
Involving Geometric Figures
• If an application problem describes a geometric
figure (rectangle, triangle, circle, etc.) it often
helps, as part of the first step, to begin by
drawing a picture and looking up formulas that
pertain to that figure (these are usually found on
an inside cover of your book)
• Continue with other steps already discussed (list
of unknowns, name most basic unknown, name
other unknowns, etc.)
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
Problems Involving Straight
Angles
• As previously discussed, a “straight angle”
is an angle whose measure is 180o
• When two angles add to form a straight
angle, the sum of their measures is 180o
A
B
• A + B is a straight angle so:
A  B  180
Example of Problem Involving
Straight Angles
• Given that the two angles in the following
diagram have the measures shown with
variable expressions, find the exact value
of the measure of each angle:
2x 15
2x  15  x  180
3x  15  180
3x  165
x  550
x
2x  15  255  15  125
0
Problems Involving Vertical
Angles
• When two lines intersect, four angles are
formed, angles opposite each other are
called “vertical angles”
• Pairs of vertical angles always have equal
measures
B
C
A
D
• A and C are “vertical” so:
• B and D are “vertical” so:
AC
BD
Example of Problem Involving
Vertical Angles
• Given the variable expression measures of
the angles shown in the following diagram,
find the actual measure of each marked
angle
x
2x  30
2x  30  x
2x  x  30
x  30
Both angles have a measure of 30
0
Homework Problems
• Section:
• Page:
• Problems:
2.5
138
Odd: 3 – 45, 57 – 85
• MyMathLab Section 2.5 for practice
• MyMathLab Homework Quiz 2.5 is due for
a grade on the date of our next class
meeting
Ratios
• A ratio is a comparison of two numbers
using a quotient
• There are three common ways of showing
a ratio:
a to b
a:b
a
b
• The last way is most common in algebra
Ratios Involving
Same Type of Measurement
• When ratios involve two quantities that measure the
same type of thing (both measure time, both measure
length, both measure volume, etc.), always convert both
to the same unit, then reduce to lowest terms
• Example: What is the ratio of 12 hours to 2 days?
2 days is the same as how many hours? 48
12 hr
12 hr
1


2 day
48 hr
4
• In this case the answer has no units
Ratios Involving
Different Types of Measurement
• When ratios involve two quantities that measure different
things (one measures cost and the other measures
distance, one measures distance and the other
measures time, etc.), it is not necessary to make any unit
conversions, but you do need to reduce to lowest terms
• Example: What is the ratio of 69 miles to 3 gallons?
69 miles
 23 miles gal
3 gal
• In this case the answer has units
Proportions
• A proportion is an equation that says that
two ratios are equal
• An example of a proportion is:
6 2

9 3
• We read this as 6 is to 9 as 2 is to 3
Terminology of Proportions
• In general a proportion looks like:
a c

b d
• a, b, c, and d are called “terms”
• a and d are called “extremes”
• b and c are called “means”
Characteristics of Proportions
• For every proportion:
a c

b d
• the product of the “extremes” always
equals the product of the “means” ad  bc
• sometimes this last fact is stated as:
“the cross products are equal”
6 2

9 3
and
63  29
Solving Proportions When One
Term is Unknown
• When a proportion is stated or implied by a problem, but
one term is unknown:
– use a variable to represent the unknown term
– set the cross products equal to each other
– solve the resulting equation
• Example:
If it cost $15.20 for 5 gallons of gas, how much would it
cost for 7 gallons of gas?
• We can think of this as the proportion: $15.20 is to 5
gallons as x (dollars) is to 7 gallons.
15.20 x

106.40  5x
21.28  x
5
7
106.40
$
x
15.20 7  5  x
7
gal
is
21.28
5
Geometry Applications of
Proportions
• Under certain conditions, two triangles are
said to be “similar triangles”
• When two triangles are similar, certain
proportions are always true
• On the slides that follow, we will discuss
these concepts and practical applications
Similar Triangles
• Triangles that have exactly the same
shape, but not necessarily the same
size are similar triangles
A
B
D
C
E
F
Conditions for Similar Triangles
• Corresponding angles must have the
same measure.
A  D, B  E , C  F
• Corresponding side lengths must be
proportional. (That is, their ratios must be
equal.) AB BC AC
DE

EF

DF
A
B
D
C
E
F
Example: Finding Side Lengths
on
Similar
Triangles
Write a proportion involving correspond ing sides with one unknown :
• Triangles ABC and
DEF are similar. Find
the lengths of the
unknown sides in
triangle DEF.
D
A
112
35
64
F
24
32
C
112
33
48
B
32 64

16
x
32 x  1024
x  32
• To find side FE:
32
16
• To find side DE:
E
32 48

16 x
32 x  768
x  24
Example: Application of Similar
Triangles
• A lighthouse casts a
• Since the two
shadow 64 m long. At
triangles are similar,
the same time, the
corresponding sides
shadow cast by a
are proportional:
mailbox 3 m high is 4
3 x
m long. Find the

4 64
height of the
Unknowns :
lighthouse.
4 x  192
Height of LH x
3
4
x
64
x  48
• The lighthouse is
48 m high.
Homework Problems
• Section:
• Page:
• Problems:
2.6
146
Odd: 3 – 69
• MyMathLab Section 2.6 for practice
• MyMathLab Homework Quiz 2.6 is due for
a grade on the date of our next class
meeting
Section 2.7 Will be Omitted
• Material in this section is very important,
but will not be covered until college
algebra
• We now skip to the final section for this
chapter
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”
• Examples:
3  8
5  4 1
1 9 5
1  7
29  7
4  11  2
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
Using Parenthesis and Bracket
in Graphing
• A parenthesis pointing left, ) , is used to
mean “less than this number”
• A parenthesis pointing right, ( , is used to
mean “greater than this number”
• A bracket pointing left, ] , is used to mean
“less than or equal to this number”
• A bracket pointing right, [ , is used to mean
“greater than or equal to this number”
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
Addition and Inequalities
• Consider following true inequalities:
6  4
2  10
 8  4
• Are the inequalities true with the same
inequality symbol after 3 is added on both
sides?
3  7
5  13
 5  1
• Yes, adding the same number on both
sides preserves the truthfulness
Subtraction and Inequalities
• Consider following true inequalities:
6  4
2  10
 8  4
• Are the inequalities true with the same
inequality symbol after 5 is subtracted on
both sides?
 11  1
3  5
13  9
• Yes, subtracting the same number on both
sides preserves the truthfulness
Multiplication and Inequalities
• Consider following true inequalities:
6  4
2  10
 8  4
• Are the inequalities true with the same
inequality symbol after positive 3 is
multiplied on both sides?
18  12
6  30
 24  12
• Yes, multiplying by a positive number on
both sides preserves the truthfulness
Multiplication and Inequalities
• Consider following true inequalities:
 8  4
6  4
2  10
• Are the inequalities true with the same inequality
symbol after negative 3 is multiplied on both
sides?
24  12
18  12
 6  30
• No, multiplying by a negative number on both
sides requires that the inequality symbol be
reversed to preserve the truthfulness
24  12
18  12
 6  30
Division and Inequalities
• Consider following true inequalities:
6  4
2  10
 8  4
• Are the inequalities true with the same
inequality symbol after both sides are
divided by positive 2?
3  2
1 5
 4  2
• Yes, dividing by a positive number on
both sides preserves the truthfulness
Division and Inequalities
• Consider following true inequalities:
 8  4
6  4
2  10
• Are the inequalities true with the same inequality
symbol after both sides are divided by negative
2?
42
3  2
1  5
• No, dividing by a negative number on both
sides requires that the inequality symbol be
reversed to preserve the truthfulness
42
3  2
1  5
Summary of Math Operations
on Inequalities
• Adding or subtracting the same value on
both sides maintains the sense of an
inequality
• Multiplying or dividing by the same
positive number on both sides maintains
the sense of the inequality
• Multiplying or dividing by the same
negative number on both sides reverses
the sense of the inequality
Principles of Inequalities
• When an inequality has the same
expression added or subtracted on both
sides of the inequality symbol, the
inequality symbol direction remains the
same and the new inequality has the
same solutions as the original
• Example of equivalent inequalities:
3 has been added on both sides
x 3  7
x  10
and numbers less than or equal to
10 are solutions to both inequaliti es
Principles of Inequalities
• When an inequality has the same positive
number multiplied or divided on both
sides of the inequality symbol, the
inequality symbol direction remains the
same and the new inequality has the
same solutions as the original
• Example of equivalent inequalities:
both sides have been divided by positive 4
4x  12
x3
and numbers greater th an 3 are solutions to
both inequaliti es
Principles of Inequalities
• When an inequality has the same
negative number multiplied or divided
on both sides of the inequality symbol, the
inequality symbol direction reverses, but
the new inequality has the same solutions
as the original
• Example of equivalent inequalities:
both sides have been multiplied by negative 3
1
 x  2 and numbers less than or equal to - 6 are solutions to
3
x  6 both inequaliti es
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
4x  5  13
4x  5  5  13  5
4x  8
4
8
x
4
4
x2
(
0
2
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
Example of Solving Linear
Inequality
3
x  2 x  3
5
3
x  2x  6
5
3 
5 x   52 x  6 
5 
3x  10x  30
]
4
2
7
3x 10x  10x 10x  30
 7 x  30
 7 x 30

7 7
30
2
x
 4
7
7
0
Application Problems Involving
Inequalities
• Word problems using the phrases similar
to these will translate to inequalities:
– the result is less than
– the result is greater than or equal to
– the answer is at least
– the answer is at most
Phrases that Translate to
Inequality Symbols
English Phrase
• the result is less than
• the result is greater
than or equal to
• the answer is at least
• the answer is at most
Inequality Symbol




Example
• Susan has scores of 72, 84, and 78 on her
first three exams. What score must she
make on the last exam to insure that her
average is at least 80?
• What is unknown? Score on last exam  x
• How do you calculate average for four
scores? Add four scores and divide by 4
• What inequality symbol means “at least”? 
72  84  78  x
• Inequality:
4
 80
Example Continued
72  84  78  x
 80
4
234  x
 80
4
 234  x 
4
  480 
 4 
234  x  320
234  234  x  320  234
x  86
Susan must make
at least 86 on her
last exam to have
an average of 80.
Example
• When 6 is added to twice a number, the
result is at most four less than the sum of
three times the number and 5. Find all
such numbers.
• What is unknown? the number  x
• What inequality symbol means “at most”? 
• Inequality: 2 x  6  3x  5  4
Example Continued
2x  6  3x  5  4
x5
2x  6  3x  5  4
2x  6  3x 1
2x  3x  6  3x  3x 1
 x  6 1
 x  6  6  1 6
 x  5
1 x   1 5
Any number greater
than or equal to 5 will
give the desired result.
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
Homework Problems
• Section:
• Page:
• Problems:
2.8
174
Odd: 3 – 25, 29 – 71,
77 – 83
• MyMathLab Section 2.8 for practice
• MyMathLab Homework Quiz 2.8 is due for
a grade on the date of our next class
meeting