1. Exam 3 Material Formulas, Proportions, Linear Inequalities
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 hasn’t 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 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?
In this case the answer has no units

20. 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?
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: A C B F E D
35
112
33 32 48 64 16

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 m high. 64 4 3 x

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:
< “is less than”
“is less than or equal to”
> “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:

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 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

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 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

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. Solving Three 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 Solving Three Part Linear Inequalities

61. Homework Problems Section: 2.8 Page: 174
Problems: Odd: 3 – 25, 29 – 71, – 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