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:

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:

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

Example Solve the formula for

Example Solve the formula for

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?

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:

Geometric Example Continued Solve the equation: What is the answer to the problem? The length of the rectangle is:

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:

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:

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:

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

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

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

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

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

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”

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”

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.

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

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

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

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. 64 4 3 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:

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:

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:

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

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

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

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

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

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

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:

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:

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:

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

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

Example of Solving Linear Inequality

Example of Solving Linear Inequality

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? How do you calculate average for four scores? What inequality symbol means “at least”? Inequality:

Example Continued

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:

Example Continued

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: .

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:

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

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