Formulas
• A “formula” is an equation containing more than one variable
• Familiar Examples:LWA
2WL2P
Rectangle) a of (Area
Rectangle) a of (Perimeter
cba P triangle)a of (Perimeter
bh2
1A Triangle) a of (Area
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:
and , , solve for the remaining variable:
2WL2P
40P 5W
2L2 52L240
10L240 L230
L15
Lfor linear isEquation
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 LWA :W
W53 :similar to is this
constants be toassumed are L andA Since
for W? thissolveyou wouldHow
5by sidesboth Divideformula? real thesolveyou wouldHow
:Lby sidesboth DivideW
A
L
Example
• Solve the formula for 2L2WP :L
L247 :similar to is this
constants be toassumed are W and P Since
L?for thissolveyou wouldHow
sidesboth on 2by divide then and 4Subtract formula? real thesolveyou wouldHow
:sidesboth on 2W Subtract
L2
2W-P:2by sidesboth Divide
2L2W-P
Example
• Solve the formula for
ABA3
2
2
1
B:
ABA3
2
2
1
2
1
ABA
3
26
2
1
2
16
ABA 433
AABAA 34333
AB 3
33
3 AB
3
AB
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:
• What is the rectangle formula that applies for this problem?
width times3 than less inches 4 isLength
habout widt know Nothing inches 32 isPerimeter
WLP 22
Geometric Example Continued
• List of unknowns:– Length of rectangle:– Width of rectangle:
• What other information is given that hasn’t been used?
• Use perimeter formula with given perimeter and algebra names for unknowns:
width times3 than less inches 4 isLength
unknown basicmost theis This
43 xx
inches 32 isPerimeter
WLP 22
xx 243232
Geometric Example Continued
• Solve the equation:
• What is the answer to the problem?The length of the rectangle is:
xx 243232 xx 28632
8832 xx840 x5
45343x 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 is a straight angle so:
A B
180BA
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:
152x x
180x152 x180153 x1653 x
055x 012515552152x
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
• A and C are “vertical” so:• B and D are “vertical” so:
AD
CB
CA DB
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
302x x
x302 x30x2 x
30x030 of measure a have anglesBoth
Homework Problems
• Section: 2.5
• Page: 138
• Problems: 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:
• The last way is most common in algebra
b toa b:ab
a
Ratios InvolvingSame 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?
• In this case the answer has no units
hours?many how as same theis days 2 48
day 2
hr 12
hr 48
hr 12
4
1
Ratios InvolvingDifferent 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?
• In this case the answer has units
gal 3
miles 69galmiles23
Proportions
• A proportion is an equation that says that two ratios are equal
• An example of a proportion is:
• We read this as 6 is to 9 as 2 is to 3
3
2
9
6
Terminology of Proportions
• In general a proportion looks like:
• a, b, c, and d are called “terms”
• a and d are called “extremes”
• b and c are called “means”
d
c
b
a
Characteristics of Proportions
• For every proportion:
• the product of the “extremes” always equals the product of the “means”
• sometimes this last fact is stated as:
“the cross products are equal”
d
c
b
a
bcad
3
2
9
6 and 9236
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.
75
15.20 x
x 5720.15
x540.106
x5
40.106
x28.21
21.28is gal 7 $
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 C
D
E F
Conditions for Similar Triangles
• Corresponding angles must have the same measure.
• Corresponding side lengths must be proportional. (That is, their ratios must be equal.)
A
B C
D
E F
FCEBDA , ,
DF
AC
EF
BC
DE
AB
Example: Finding Side Lengths on Similar Triangles
• Triangles ABC and DEF are similar. Find the lengths of the unknown sides in triangle DEF.
• To find side DE:
• To find side FE:A
C B
F E
D
35
112 33
112
32
48
64
16
64
32 1024
3
32
6
2
1
xx
x
48
32
3
768
2
6
2
1
4
xx
x
:unknown one with sides ingcorrespond involving proportion a Write
32
24
Example: Application of Similar Triangles
• A lighthouse casts a shadow 64 m long. At the same time, the shadow cast by a mailbox 3 m high is 4 m long. Find the height of the lighthouse.
• Since the two triangles are similar, corresponding sides are proportional:
• The lighthouse is 48 m high.
3
4 644 192
48
x
x
x
64
4
3
x
:UnknownsLH ofHeight x
Homework Problems
• Section: 2.6
• Page: 146
• Problems: 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:
83
145
591
71
792
2114
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:All numbers smaller than 5Solutions are often shown in graph form:
5x
0 5
)
thanlessmean tosparenthesi of use Notice
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
• Graph solutions to:
• Graph solutions to:
• Graph solutions to:
• Graph solutions to:
0
0
0
0
2x
2x
2x
2x
2
2
2
2
]
)
[
(
Addition and Inequalities
• Consider following true inequalities:
• Are the inequalities true with the same inequality symbol after 3 is added on both sides?
• Yes, adding the same number on both sides preserves the truthfulness
48 46 102
15 73 135
Subtraction and Inequalities
• Consider following true inequalities:
• Are the inequalities true with the same inequality symbol after 5 is subtracted on both sides?
• Yes, subtracting the same number on both sides preserves the truthfulness
48 46 102
913 111 53
Multiplication and Inequalities
• Consider following true inequalities:
• Are the inequalities true with the same inequality symbol after positive 3 is multiplied on both sides?
• Yes, multiplying by a positive number on both sides preserves the truthfulness
48 46 102
1224 1218 306
Multiplication and Inequalities
• Consider following true inequalities:
• Are the inequalities true with the same inequality symbol after negative 3 is multiplied on both sides?
• No, multiplying by a negative number on both sides requires that the inequality symbol be reversed to preserve the truthfulness
48 46 102
1224 1218 306
1224 1218 306
Division and Inequalities
• Consider following true inequalities:
• Are the inequalities true with the same inequality symbol after both sides are divided by positive 2?
• Yes, dividing by a positive number on both sides preserves the truthfulness
48 46 102
24 23 51
Division and Inequalities
• Consider following true inequalities:
• Are the inequalities true with the same inequality symbol after both sides are divided by negative 2?
• No, dividing by a negative number on both sides requires that the inequality symbol be reversed to preserve the truthfulness
48 46 102
24 23 51
24 23 51
Summary of Math Operationson 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:73 x
10xesinequalitiboth tosolutions are 10
toequalor than less numbers and
sidesboth on addedbeen has 3
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:124 x
3xesinequalitiboth
tosolutions are 3an greater th numbers and
4 positiveby dividedbeen have sidesboth
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:
23
1 x
6x esinequalitiboth
tosolutions are 6- toequalor than less numbers and
3 negativeby multipliedbeen have sidesboth
Linear Inequalities
• A linear inequality looks like a linear equation except the = has been replaced by:
• Examples:
• Our goal is to learn to solve linear inequalities
or , , ,
137 xx
325
3 xx1354 x
xxx 382
1627.
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
07
24
325
3 xx
625
3 xx
6255
35
xx
30103 xx
301010103 xxxx
307 x
7
30
7
7
x
7
30x
7
24
]
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 toInequality 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?• How do you calculate average for four
scores?• What inequality symbol means “at least”?• Inequality:
examlast on Score x
4by divide and scoresfour Add
804
788472
x
Example Continued
804
788472
x
804
234
x
8044
2344
x
320234 x
234320234234 x
86x
80. of averagean
have toexamlast
heron 86least at
makemust Susan
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?
• What inequality symbol means “at most”?
• Inequality:
number the x
45362 xx
Example Continued
result. desired thegive
will5 toequalor than
greaternumber Any
45362 xx
45362 xx
1362 xx
133632 xxxx
16 x6166 x
5 x 511 x
5x
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:
• Not Legitimate:
.
142
13 x
142
13 x
142
13 x
Sense Same Havet Don' Symbols Inequality
1- NOT is 3-
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:
• “Graphical notation” – same as with two part inequalities:
• “Interval notation” – same as with two part inequalities:
32 x
( ]2 3
]3,2(
SolvingThree Part Linear Inequalities
• Solved exactly like two part linear inequalities except that solution is achieved when variable is isolated in the middle
Example of SolvingThree Part Linear Inequalities
142
13 x
122
13 x
246 x
22 x SolutionNotation Standard
2 2
[ ) SolutionNotation Graphical
SolutionNotation Interval)2,2[