Elementary Algebra

1
Elementary Algebra

• Exam 3 Material
• Formulas, Proportions, Linear Inequalities

2
Formulas

• A formula is an equation containing more than
  one variable
• Familiar Examples

3
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

4
Example of Solving a Formula for One Variable
Given Others

• Given the formula
• and , , solve for the
  remaining variable

5
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)

6
Example

• Solve the formula for

7
Example

• Solve the formula for

8
Example

• Solve the formula for

9
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.)

10
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?

11
Geometric Example Continued

• List of unknowns
• Length of rectangle
• Width of rectangle
• What other information is given that hasnt been
  used?
• Use perimeter formula with given perimeter and
  algebra names for unknowns

12
Geometric Example Continued

• Solve the equation
• What is the answer to the problem?
• The length of the rectangle is

13
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

14
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

15
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

16
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

17
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

18
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

19
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

20
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

21
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

22
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

23
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

24
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.

25
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

26
Similar Triangles

• Triangles that have exactly the same shape, but
  not necessarily the same size are similar
  triangles

27
Conditions for Similar Triangles

• Corresponding angles must have the same measure.
• Corresponding side lengths must be proportional.
  (That is, their ratios must be equal.)

28
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

29
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.

30
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

31
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

32
Inequalities

• An inequality is a comparison between
  expressions involving these symbols
• lt is less than
• is less than or equal to
• gt is greater than
• is greater than or equal to
• Examples

33
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 5
• Solutions are often shown in graph form

34
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

35
Graphing Solutions to Inequalities

• Graph solutions to
• Graph solutions to
• Graph solutions to
• Graph solutions to

36
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

37
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

38
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

39
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

40
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

41
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

42
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

43
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

44
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

45
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

46
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

47
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

48
Example of Solving Linear Inequality

49
Example of Solving Linear Inequality

50
Example of Solving Linear Inequality

51
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

52
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

53
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

54
Example Continued
55
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

56
Example Continued
57
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
• .

58
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

59
SolvingThree Part Linear Inequalities

• Solved exactly like two part linear inequalities
  except that solution is achieved when variable is
  isolated in the middle

60
Example of SolvingThree Part Linear Inequalities

61
Homework Problems

• Section 2.8
• Page 174
• Problems 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