Equations and Inequalities

Equations and Inequalities Algebra 2 Chapter 1 George Hardy

Do Now – Vocab Quiz • Close books and notebooks! On Looseleaf Paper: • 1) Define: coefficient • 2) Define: Like terms • 3) Define: rational numbers • 4) The class has been strategically developed to be a challenge, and to promote ______________ , ACCOUNTABILITY, and ________________. Fill in one of the blanks. • 5) True or False: I accept late work.

Do Now – Vocab Quiz • Close books and notebooks! On Looseleaf Paper: • 1) Define: variables • 2) Define: Like terms • 3) Define: irrational numbers • 4) The class has been strategically developed to be a challenge, and to promote ______________ , ACCOUNTABILITY, and ________________. Fill in one of the blanks. • 5) The quiz category is worth __% of your grade

Discussion Preparation • In your 5-subject notebook, answer the following question. Support it with specific and accurate facts. Does .999… = 1? Why or Why Not? • What do these two numbers represent?

Let’s Get Started • CHAPTER 1 is a REVIEW • By the end of next week, we should be on Chapter 2 • Try your best to keep up • Ask QUESTIONS when necessary! • We’re in this together!

Numbers Subsets • Whole Numbers • 0, 1, 2, 3, 4, … • Integers • Positive and negative whole numbers • -∞, …, -4, -3, -2, -1, 0, 1, 2, 3, …, ∞ • Rational Numbers • A number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator • Integers and fractions: 1, -3, -50, -⅞, ½

Number Subsets • Irrational Numbers • A real number that CANNOT be written as a fraction of two integers • Decimal that never ends nor forever repeats • EX. π, √2 ≈ 1.41421356 • Real Numbers • Includes all of the measuring numbers, including decimals (Every RATIONAL and IRRATIONAL is a REAL, but every REAL is NOT a RATIONAL) • Repeating decimals or terminating decimals • EX. 123.456 or 1/3

Quiz • √5 • .333333 • 1 • -56 • 3/9 • -7/8 • √100 • e= 2.7182818 Irrational Rational, Real Real, Integer, Whole, Rational Real, Integer, Rational Real, Rational Real, Rational Real, Rational, Whole, Integer Irrational

Multiplication Properties of Real Numbers Addition Commutative A + B = B + A Associative (A + B) + C = A + (B + C) Identity A + 0 = A 0 + A = A Inverse A + (-A) = 0 Commutative AB = BA Associative (AB)C = A(BC) Identity A * 1 = A 1 * A = A Inverse A * 1/A = 1, when a ≠ 0

Identify the Properties • (3 + 9) + 8 = 3 + (9 + 8) • Associative Property of Addition • 14 * 1 = 14 • Identity Property of Multiplication • 2(b + c) = 2b + 2c • Distributive Property

NOTE! • The opposite, or additive inverse of any number a is –a. The reciprocal, or multiplicative inverse, of any nonzero number a is 1/a. • Subtraction is defined as adding the opposite, and division is defined as multiplying by the reciprocal.

TRY THIS • You are exchanging $400 for Mexican pesos. The exchange rate is 13.05 pesos per dollar and the bank charges a 1% fee to make the exchange • How much money should you take to the bank if you do not want to use part of the $400 to pay the exchange fee? • How much will you receive in pesos?

ANSWER Section A One percent = .01 400 * .01 = 4 400 + 4 = 404 You want to bring 404 dollars Section B 400 dollars * 13.05 pesos 1 dollar = 5,220 pesos

Converting When you return from Mexico, you have 425 pesos left. How much can you get in US dollars? Remember, there is a one percent processing fee 425 * .01 = 4.25 Next, subtract the $4.25 from $425 = 420. 75

Now… Convert 420.75 pesos * 1 dollar 13.05 pesos = 32.24 dollars

Question • When converting units with unit analysis/dimensional analysis, how do you choose whether to use a particular conversion factor or its reciprocal? • Choose the one that lets you “cancel” units, until the desired unit remains

Critical Thinking In Nordstrom, I found a sign that read “20% off the sale price.” • Sperry’s that originally $80 had a sale price that was 15% less than the original price. What was the final price of the shoes, according to the sign? • Would the final price of the shoes be the same if 35% had been taken off the original price? If not, which price would be lower?

Challenge • At the register, another shopper was buying a pair of shoes that had been subject to the same two discounts. The final price of the shoes was $81.60. What was the original price?

Try These • Agree or Disagree, and support with an example: (a – b) – c = a – (b – c) • Agree or Disagree, and support with an example: (a * b) * c = a * (b * c)

Order of Operations • Which is correct? • PEDMAS • PEMDAS • PEDMSA • PEMDSA • All are correct! • Division or Multiplication • Addition or Subtraction • Just remember to go left to right!

Simplify

Summary • What does it mean to evaluate an expression? • What symbols act as grouping symbols in the order of operations?

Homework • Chapter 1.1 #16 – 64 even, and 1.2 #16 – 60 even • Read 1.3 and 1.6 and take notes

Do Now: Geometry Connection • A triangle has a base of n + 10, and a height of n. Write and expression for the area of the figure. Then find the area when n = 40. • A square has a side of x + y. Write an expression for the area of the figure. Then evaluate the expression when x = 12 and y = 5.

Insert Parenthesis .. • …To make the statements true

Neurosis • A test measuring neurotic traits, such as anxiety and hostility indicate that people become less neurotic as they get older (Williams, 2006). • The algebraic expression 23 – 0.12x describes the average neurotic level for people x years old. Evaluate the expression for x = 80. Describe what it means in practical terms.

Challenge • At the register, another shopper was buying a pair of shoes that had been subject to the same two discounts. The final price of the shoes was $81.60. What was the original price?

Equations and Inequalities • An equation is a statement in which two expressions are equal • Inequalities have properties similar to those of equations; however, there are a few differences • What are some of the differences?

Dry Ice • Dry ice is solid carbon dioxide. Dry ice does not melt – it changes directly from a solid to a gas. Dry ice changes to a gas at -109.3° F. What is this temperature in degrees Celsius? • Use the formula

Stockbroker • A stockbroker earns a base salary $40,000 plus 5% of the total value of the stocks, mutual funds, and other investments that the stockbroker sells. Last year, the stockbroker earned $71,750. What was the total value of the investments the stockbroker sold?

Try These • How do we know our answers are correct?

Graphing Calculator • Solve the following equation without a graphing calculator • 4(2x + 1) – 29 = 3(2x – 5)

College Tuition Costs • The model • T = 974x + 15,410 represents the average cost of tuition and fees at private four-year colleges x years after 2000. Use the model to determine when tuition and fees will average $27,098.

Note: Types of Equations • An equation that is true for all real numbers for which both sides are defined is an IDENTITY. The solution set is all real numbers. • x + 3 = x + 2 + 1 • An equation that is not an identity, but that is true for at least one real number is a CONDITIONAL EQUATION. • 2x + 3 = 17 • An INCONSISTENT EQUATION is not true for even one real number • x = x + 7

Homework • Chapter 1.3 #16 – 48, 52 – 56 even, Chapter 1.6 #14 – 52 even. Try 54 in Fahrenheit • Read Chapter 1.4 and take notes!

Do Now • Solve the equations or simplify • The bill for the repair of your car was $390. The cost for parts was $215. The cost for labor was $35 per hour. How many hours did the repair work take? • –(x + 2) – 2x = -2(x + 1) • 4x2 – 2(x2 – 3x) + 6x + 8

Graphing Calculator • Solve the following equation without a graphing calculator • 4(2x + 1) – 29 = 3(2x – 5)

College Tuition Costs • The model • T = 974x + 15,410 represents the average cost of tuition and fees at private four-year colleges x years after 2000. Use the model to determine when tuition and fees will average $27,098.

Note: Types of Equations • An equation that is true for all real numbers for which both sides are defined is an IDENTITY. The solution set is all real numbers. • x + 3 = x + 2 + 1 • An equation that is not an identity, but that is true for at least one real number is a CONDITIONAL EQUATION. • 2x + 3 = 17 • An INCONSISTENT EQUATION is not true for even one real number • x = x + 7

Solving Linear Inequalities

Special Inequalities • Some inequalities have one solution, no solution, or all real solutions satisfy the set. • No solution – eliminate the variable, and have a false statement • All reals – eliminate the variable, and have a true statement.

Love • Write an inequality that expresses for which years in a relationship, intimacy is greater than commitment.

Applications

Car Rental • Acme Car Rental charges $4 a day plus $.15 per mile, whereas Interstate Car Rental charges $20 a day plus $.05 per mile. How many miles must be driven to make the daily cost of Acme Rental to be a better deal than Interstate?

Phone Call • You go online to find phone companies that have long distance rates. You’ve chosen a plan that has a monthly fee of $15 with a charge rate of 8 cents per minute for all long distance calls. Of course, there are other bills to pay; therefore, you don’t want to spend more than $35. • Write an inequality that represents the situation. • How many minutes of long distance calls can you make?

Grades • A professor announces that course grades will be computed by taking 40% of a student’s project score (0 – 100 points) and adding 60% of the student’s final exam score (0 – 100 points). If a student gets an 86 on the project, what scores can she get on the final exam to get a course grade of at least 90?

Compound Inequalities AND OR 2x – 3 < 7 0r 35 – 4x ≤ 3 • -3 < 2x + 1 ≤ 3

Car Repair • Parts for a repair Dodge Charger costs $175. The mechanic costs $34 per hour. If you receive an estimate for at least $226, and at most $294 for fixing the car. What is the time interval that the mechanic will be working on the car?

Connecting Ideas • Solve the compound inequality • 3n + 1 > 10 and ½n – 1> 3

Classwork • Chapter 1.6 #4 – 11 ALL

Equations and Inequalities