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: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



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:




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


• 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


• 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


• 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


• 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


• 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


• 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


• 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

1354 x

513554 x

84 x

4

8

4

4x

2x

0 2

(


04(

137 xx

13373 xxxx

172 x

71772 x

82 x

2

8

2

2

x

4x


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[

Homework Problems

• Section: 2.8

• Page: 174

• Problems: Odd: 3 – 25, 29 – 71, 77 – 83



elementary algebra exam 3 material formulas, proportions, linear inequalities

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