1. Algebra II Review of Algebra

2. Real Numbers and Number Operations (1.1) Real numbers can be pictured as points on the real number line. {.. -2 -1 0 1 2..} Negative # Positive # Enclosing the numbers in braces {} indicates that they for a set. The dots indicates that the list continues.

3. Real Numbers Within the set of real numbers there are subsets. Natural Numbers (counting numbers) Whole Numbers (includes zero) Integers (negative and positive whole numbers, including zero) Rational Numbers (any number that can be written as a ratio of two integers) Irrational Numbers (any number that is not rational such as  )

4. Ordering Numbers The real number line can be used to order numbers as well as inequality symbols. 2 is less than 5 (verbal statement) 2 < 5 (inequality) 2 is to the left of 5 (number line interpretation) 5 > 2 also gives the same information

5. Opposites Two points on a number line that are the same distance from the origin but are on opposite sides of the origin (zero) are opposites. If a is positive, then the opposite is –a. The opposite of –a is a. The opposite of 0 is 0.

6. Properties of Real Numbers Let a, b and c be any real numbers. PropertyAdditionMultiplication Closure a + b is realab is real Commutative a + b = b + aab = ba Associative (a+b)+c = a+(b+c)(ab)c = a(bc) Identity a+0=a, 0+a=aa·1=a, 1·a=a Inverse a+(-a)=0

7. Subtraction and Division Are division and subtraction associative or commutative? Does (a-b)-c=a-(b-c)? Does (a-b)=(b-a)? Does (a  b)  c=a  (b  c)? Does (a  b)=(b  a)

8. Properties Relating Addition and Multiplication Left Distributive Property a(b+c)=ab+ac Right Distributive Property (a+b)c=ac+bc The reciprocal of a is 1/a, provided a is not zero. Zero has no reciprocal. Subtraction is defined by adding the opposite. Division is defined as multiplying by the reciprocal.

9. Operations with Real Numbers The sum of -12 and 14 is -12 + 14 = 2 The difference of 5 and -13 is 5 – (-13) = 18 The product of -3 and -11 is (-3)(-11) = 33 The quotient of -42 and 14 is -42 = -3 14

10. Practice Is zero a positive number, a negative number, or neither? Is zero an integer? Give examples of real numbers that are natural numbers, whole numbers, integers, rational numbers and irrational numbers. Which of the following is false? Explain. A) Every whole number is an integer. B) Every integer is a rational number. C) No integer is an irrational number. D) Every rational number is an integer.

11. Practice Find the unite of measure for the product. Explain your reasoning.

12. Homework 1.1 Pages 6-7: 12-14 all; 17-24 all; 26-32 even; 33 and 34.

13. Algebraic Expressions and Models (1.2) An algebraic expression is a collection of numbers, variables, operations, and grouping symbols. When the variables in an expression are replaced with numbers, you are evaluating the expression. Evaluate 2x – 3y when x = 2 and y = 5 The resulting number is the value of the expression.

14. Algebraic Expressions In the sum 4x + 3, 4x and 3 are called terms In the product 4x, 4 and x are called factors. In the quotient 3x, 3x is called the 8numerator and 8 is the denominator.

15. Order of Operations 1) Do operations that occur within grouping symbols using Rules 2-4. 2) Evaluate powers. 3) Do multiplication and division from left to right. 4) Do addition and subtraction from left to right.

16. Algebraic Expressions as Mathematical Models For 1980 through 1990, the number, B (in millions), of children’s books that were sold in the United States can be modeled by B = 18.7t + 119 where t = 0 represents 1980. Use the model to approximate the number of children’s books sold in 1985. Solution: If t = 0 represents 1980, then t = 5 represents 1985. B = 18.7(5) +119 B = 212.5 Therefore, 212.5 million children’s books were sold in 1985.

17. Practice Give an example of an algebraic expression for which the value of the expression is positive regardless of the value of the variable. Describe the difference between the terms of an expression and the factors of an expression. Identify the terms of 6x³ - 17x + 5. Identify the factors of 10xy. Write a verbal description of this expression (x³+1)². Explain how the Order of Operations is used to evaluate 3 - 8²  4+1.

18. Homework 1.2 Pages 12 and 13: 7-43 odd; 49-52 all

19. Solving Linear Equations (1.3) A linear equation is an equation in which x is to the power of one. To solve a linear equation is to fine the number (solution) that causes the equation to be true. 3x – 10 = 8 has a solution of 6, because 3(6)-10=8. Two equations are equivalent if they have the same solutions.

20. Using Equivalent Equations to Solve Linear Equations (Transformations) 1) Add the same number to both sides 2) Subtract the same number from both sides. 3) Multiply both sides by the same nonzero number. 4) Divide both sides by the same nonzero number. 5) Simplify one or both sides. 6) Interchange the two sides.

21. Equations 8x + 15 = -4x + 51 15(1 – x) = -3 (-x – 2) 3(x – 2) + 6 = 4(2 – x) (4/5)x – 6 = 14 5(3x – 18) = 15 -4(6x – 9) 4(x – 3) = -2(9 – 8x) Is -4 the only solution of 2x + 9 = 1?

22. Mathematical Models using Linear Equations The weekly earnings of a baker at the Bakery Corp is $7 per hour and time and half for anything over 40 hours per week. How many hours would the baker work in a week to earn $332.50. (Hourly wage)(40 hours)+1.5(Hourly wage)(over time hours) Mathematical model: 7(40) + 1.5(7)(t – 40) = 332.50 t = ?

23. Practice Check whether the number is a solution of the equation. 3x -11 = 1 (4) 6x + 1 = 7 (1) Solve the equation. Check your solution. -4(3 + x) + 5 = 4(x + 3) -(x – 1) + 10 = -3(x – 2)

24. Word Problems You have two summer jobs. In the first job, you work 40 hours a week and earn $6.25 an hour. The second one, you earn $5.50 an hour and can work as many hours as you want. If you want to earn $316 a week, how many hours must you work at your second job? The bill for the repair of your car was $415. The cost for parts was $265. The cost for labor was $25 per hour. How many hours did the repair take?

25. Homework 1.3 Pages 21-22: 1-29 odd;43-48 all.

26. Problem Solving Strategies (1.4) Guess a reasonable solution and check. Make a table using the data from the problem and look for a pattern. Use the pattern to complete the table and solve the problem. Draw a diagram that shows the facts. Create an algebraic model.

27. Algebraic Model Write a verbal model. Assign labels. Write an algebraic model. Solve the algebraic model. Answer the question.

28. Word Problems To study life in arctic waters, scientists worked in a submerged Sub-Igloo station in Resolute Bay in northern Canada. The water pressure at the floor of the station was 2184 pounds per square foot. Water weights 62.4 pounds per cubic foot. The water pressure at a depth, d (in feet), is 62.4(d) pounds per square foot. How deep was the station floor? Verbal model: (water pressure)=(weight of water)(depth of station) Labels: Water pressure = 2184 (Pounds per square foot) Weigh of water = 62.4 (Pounds per cubit foot) Depth of water = x (Feet) Equation: 2184 = 62.4x x = ? Let’s look at units.

29. Construction Problem In 1862, two railroad companies were given the rights to build a railroad connecting Omaha, Nebraska, with Sacramento, CA. The Central Pacific Company began building eastward from CA in 1863. The Union Pacific started westward 24 months later. The Central Pacific Company averaged 8.75 miles of track per month. The Union Pacific Company averaged 20 miles of track per month. The two companies met in Promontory, Utah, when the 1,590 miles of track were completed. How long did it take to complete the railroad? How many miles of track did each company build? Verbal Model: (Total miles)=(miles/month)(#months)+(miles/month)(#months) Labels: Total miles of track = 1,590 (miles) Central Pacific rate = 8.75 (miles/month) Central Pacific time = t (months) Union Pacific rate = 20 (miles/month) Union Pacific time = t – 24 months.

30. Equation and Solution Equation: 1,590 = 8.75t + 20(t – 24); t = ? How many miles of track did Central Pacific build? How many miles of track did Union Pacific build? Why did Central Pacific build less track in a longer period of time?

31. Business Application You have started a small business making bracelets. The cost of each bracelet is $2.50. Your pieces sell at a local consignment shop for $15.00 and you receive 50% of the selling price. Each bracelet takes about 2 hours to complete. If you spend 16 hours a week working on the bracelets, how many weeks will you work to earn a profit of $400? Verbal Model: (Total profit)=(Profit per bracelet)(bracelets/week)(#weeks) Labels: Total Profit = $400 (Dollars) Profit/bracelet = $5.00 (Dollars/bracelet) Bracelets/week = 8 (Bracelets/week) Number of weeks = x (weeks) Equation: 400 = 5(8)(x) x = ?

32. Practice Your sister is selling Girl Scout cookies for $2.60 a box. Your family bought 6 boxes. How many boxes must she sell to collect $130?

33. Homework 1.4 Pages 28-29: 9-16 all; 18.

34. Literal Equations and Formulas (1.5) A literal equation is an equation that has more than one variable. 5x + 2y = 7 is a literal equation and can be solved (rearranged) for either variable. x = 7 – 2yy = 7 -5x 5 2 We use the same process to rearrange a literal equation as to solve an equation.

35. Practice Solve for h: V =  r²h Solve for r: C = 2  r Solve for P: A = P + Prt Solve for L: S = L – rL Surface area of a cylinder: S = 2  rh+2  r². Solve for h, then find h given S = 105 in² and r = 3 inches.

36. Homework 1.5 Page 35 5-23 all Literal equations worksheet

37. N-spire Activity Simple inequalities.

38. Solving Linear Inequalities (1.6) A solution of an inequality in one variable is any value of the variable that makes the inequality true. Most inequalities have infinitely many solutions. The graph of the linear inequality in one variable is on the real number line. The graph of x > 2, will have an open dot and will arrow to the right. The graph of x ≥ 2, will have a closed dot and will arrow to the right. The graph of x < 2 will have an open dot and will arrow to the left. The graph of x ≤ 2 will have a closed dot and will arrow to the left.

39. More Solving The equivalent equation process for solving a linear inequality is the same as solving a linear equation with one major exception. When multiplying or dividing by a negative number, the direction of the sign must change. There are two types of linear inequalities, simple and compound. A compound inequality is two simple inequalities joined by an “or” or an “and” statement.

40. Simple Inequalities Solve and graph the following: 3y – 8 < 10 3x – 2 ≤ 5x – 3 6t + 7 > 11 -x + 5 < 3x + 1

41. Compound Inequalities Solve and graph. (An “and” statement, the graph will be bounded). Notice the direction of both signs are the same. -4 ≤ 2t – 6 ≤ 12 Solve and graph. (An “or” statement, the graph will not be bounded.) Notice there are two separate statements. Signs go in opposite directions. Cannot be written as one statement. 4x + 3 7

42. Practice Each group create 3 “and” inequalities and 3 “or” inequalities to be solve by your classmates.

43. Word Problem The range of a human’s voice frequency, h (in cycles per second), is about 85≤h≤1100. The relationship between a human’s voice frequency and a bat’s voice frequency, b, is h = 85 + (203/22,000)(b – 10,000) Find the range of the bat’s voice frequency.

44. Critical Thinking Write an inequality that corresponds to the phrase. All real numbers less than or equal to -5. All real numbers less than 7 and greater than or equal to -2. True or False. Multiplying both sides of an inequality by a nonzero number requires you to switch the direction of the sign.

45. Homework 1.6 Pages 41-42: 7-37 odd; 43-45 all

46. Absolute Value Equations and Inequalities (1.7) The absolute value of a number, x, is the distance the number is from 0. To solve an absolute value equation, the expression inside the absolute value symbols can be either positive or negative.

47. Solve the Equation A linear absolute value equation will have two solutions.  2x - 3  = 7 2x – 3 = 7 or 2x – 3 = -7 2x = 10 2x = -4 X = 5 x = -2 Both solutions cause the equation to be true. Solve  3x + 5  = -10

48. Absolute Value Inequalities – Rewriting as Compound Inequalities The inequality  ax + b  <c means that ax+b is between –c and c. This is an and statement and is written and solved as one statement. (Inclusive) -c < ax + b < c The inequality  ax + b  > c means that ax+b is greater than c OR less than –c. This is not an inclusive statement and must be written as two statements. ax + b > c OR ax + b < -c

49. More Absolute Value Inequalities  x  < 5 means that -5 < x < 5 and looks like this on the number line. -5 -4 -3 -2 -1 0 1 2 3 4 5 The distance from -5 to 5 is ten. What happens when we change the inequality to  x + 2  < 5? Does it change the distance? Solution set?

50. Critical Thinking The graph of the inequality  x - 3  < 2 can be described as all real numbers that are within 2 units of 3. Give a similar description of the graph of  x - 4  < 1. The graph of the inequality  y - 1  > 3 can by described as all real numbers that are greater than 3 units from 1 or less than 3 units from 1. Give a similar description of the graph of  y + 2  > 4.

51. Solve and Graph  2x - 3  < 5  3x + 4  ≥ 10  d + 1  ≤ 6 4 +  2x - 5  > 15

52. Homework 1.7 Pages 47-48; 5-33 odd; 36; 39 & 40.