# Equations and Inequalities Algebra 2 Chapter 1 George

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Equations and Inequalities Algebra 2 Chapter 1 George Hardy

DO NOW: Real Numbers Use the number subsets to fill in the chart below

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

Quiz √ 5 . 333333 1 -56 3/9 -7/8 √ 100 e= 2. 7182818 Irrational Rational, Real, Integer, Whole, Rational Real, Integer, Rational Real, Rational, Whole, Integer Irrational

Properties of Real Numbers Addition Multiplication Commutative A+B=B+A Associative (A + B) + C = A + (B + C) Identity A+0=A 0+A=A Inverse A + (-A) = 0 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) = 2 b + 2 c 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.

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?

DO NOW 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!

Insert Parenthesis. . …To make the statements true

Evaluate the Expressions

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. 12 x describes the average neurotic level for people x years old. Evaluate the expression for x = 80. Describe what it means in practical terms.

Checkpoint What does it mean to evaluate an expression? What symbols act as grouping symbols in the order of operations? Why is 3(x + 7) – 4 x not simplified? What must be done to simplify the expression?

Homework Chapter 1. 1 #16 – 50 even, and 1. 2 #16 – 52 even Read 1. 3 and 1. 4 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.

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 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(2 x + 1) – 29 = 3(2 x – 5)

College Tuition Costs The model T = 974 x + 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. 2 x + 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) – 2 x = -2(x + 1) 4 x 2 – 2(x 2 – 3 x) + 6 x + 8

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

College Tuition Costs The model T = 974 x + 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. 2 x + 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 -3 < 2 x + 1 ≤ 3 OR 2 x – 3 < 7 0 r 35 – 4 x ≤ 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 3 n + 1 > 10 and ½n – 1> 3

Classwork Chapter 1. 6 #4 – 11 ALL

Homework Quiz: D Hour Close text books, show ALL work, box answers Chapter 1. 2 #36, 44, 46, 56 Chapter 1. 3 #30, 36, 42

Homework Quiz: E Hour Close text books, show ALL work, box answers Chapter 1. 1 #54, 62 Chapter 1. 2 #36, 56 Chapter 1. 3 #32, 44, 46

Homework Quiz: F Hour Close text books, show ALL work, box answers Chapter 1. 1 #60 Chapter 1. 2 #46, 50, 52 Chapter 1. 3 #30, 34, 42

Homework Quiz G Hour Close text books, show ALL work, box answers Chapter 1. 1 #46, 54 Chapter 1. 2 #46, 56 Chapter 1. 3 #32, 44, 46

Equations Examples Solve the following equations 6 a – 4[2 – 3(4 a – 3)] = -17 ½ - 3(x + 1) = 8

DO NOW Solve the equations

Recall The state sales tax in Michigan is 0. 06 (or 6%). If your total bill at the music store included $1. 32 in tax, how much did the merchandise cost?

Video Arcade You have $4. 25 to spend at a video arcade. Some games cost $0. 75 to play and other games cost $0. 50 to play. You decide to play 2 games that cost $0. 75. Write and solve an inequality to find the possible number of $0. 50 video games you can play.

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

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

DO NOW: ACT Practice When x = 3 and y = 5, by how much does the value of 3 x 2 – 2 y exceed the value of 2 x 2 – 3 y ? How many irrational numbers are there between 1 and 6 ?

Critical Thinking Determine whether the statement makes sense or does not make sense. Explain your reasoning. If you tell me that three times a number is less than two times that number, it’s obvious that no number satisfies this condition, and there is no need to write and solve an inequality.

Intro to 1. 4 Solve for y:

Why is it important… …to rewrite equations?

Rewriting Equations and Formulas The formula for the perimeter of a rectangle is P = 2 l + 2 w. Solve for l.

Rewriting Equations and Formulas V = ⅓πr 2 h, solve for r

Volume The silo is a cylinder with a half sphere on top. The silo can hold 576π cubic feet of grain. The radius of the sphere is 6 ft. Given the volume of a cylinder is V=πr 2 h and the volume of a sphere is V=4/3πr 3, write a formula for the volume of the silo Solve the formula for h; then, determine the height of the silo.

Geometry Challenge The formula for the area of a circle is A=πr 2. The formula for the circumference of a circle is C=2πr. Write the formula for the area of a circle in terms of its circumference.

Classwork Chapter 1. 4 Practice Worksheet Copy the Table of Common Formulas

Homework Chapter 1. 4 #12 – 32 even. 33 – 34 all, 37 – 39 all. Read Chapter 1. 7 and Take Notes

Do Now Complete the Mixed Review on Page 47 #61 - 69

DO NOW: Homework Quiz D-Hour Chapter 1. 4 #12, 16, 20, 22, 26, 28, 30

Critical Thinking Write an inequality whose solutions are all real numbers. Show why the solutions are all real numbers

Lesson Opener What is the definition of absolute value?

Discussion What is the value of x? Explain the two statements.

Absolute Value Equations Remember, that x can have two possible values- a positive and a negative value! Therefore: |ax + b| = c, where c > 0 (positive) is equivalent to the compound statement ax + b = c or ax + b = -c

Example 1

Example 2

Example 3

Critical Thinking

Try These

Absolute Value Inequalities The inequality |ax + b| < c, where c > 0, means that ax + b is between -c and c. This is equivalent to –c < ax + b < can be replaced with ≤ The inequality |ax + b| > c, where c > 0, means that ax + b is beyond -c and c. This is equivalent to ax + b > c or ax + b < -c. > can be replaced with ≥

A Way to Remember! GREAT(OR) Than Less TH(AND)

Example 1

Example 2

Try These

Classwork Chapter 1. 7 4#4 -15 all

Homework Chapter 1. 7 #18 – 58 even Read chapter 1. 5 and take notes

DO NOW: Homework Quiz F-Hour Chapter 1. 4 #24, 26, 28 Chapter 1. 6 #32, 36 Chapter 1. 7 #32, 38

DO NOW: Homework Quiz G-Hour Chapter 1. 4 #24, 26, 28 Chapter 1. 7 #32, 38

DO NOW: Homework Quiz D-Hour Chapter 1. 4 #12, 16, 20, 22, 26, 28, 30

DO NOW: Homework Quiz E-Hour Chapter 1. 4 #12, 16, 22, 24, 26, 28, 30

DO NOW/RECALL Solve the inequality 1) |7 x + 5| < 23 2) |-½x + 3| ≥ -3 3) |x + 7| < 0

Writing Abs Value Inequalities Physicians consider an adult’s normal body temperature to be within 1°F and 98. 6°F. Write an absolute value inequality that describes the range of normal body temperatures.

Intro to Chapter 1. 5 Problem Solving Using Algebraic Models Strategies Group Work/Discussion Presentation/Review

Strategies for Word Problems Read the problem carefully! State the problem in your own words and state what the problem is asking. Let x (or any variable) represent one of the unknowns Write an equation that models verbal condition Solve the equation Check the solution Draw diagrams and Patterns

Homework Problem Solving Worksheet Complete the Chapter 1 Review

DO NOW Take out Chapter 1. 4 worksheet AND the problem solving strategies worksheet! Pass them to the front Find your seat (in alphabetical order)

- Algebra Equations and Inequalities Algebra Equations and Inequalities
- Equations with Inequalities Solving Equations with Inequalities is
- Equations and Inequalities College Algebra Radical Equations Extraneous
- EQUATIONS INEQUALITIES Properties WHAT ARE EQUATIONS Equations are
- Solving Inequalities Compound Inequalities and Absolute Value Inequalities