Objectives: 1. To identify properties of addition and multiplication.

1. Properties of Numbers Lesson 2-1 Objectives: 1. To identify properties of addition and multiplication. 2. To use properties to solve problems.

2. Properties of Numbers Lesson 2-1 New Terms: Commutative Properties – changing the order of the values you are adding or multiplying does not change the sum or product. Associative Properties – Changing the grouping of the values you are adding or multiplying does not change the sum or product. Additive Identity – when you add a number to 0, the sum equals the original number. Multiplicative Identity – when you multiply a number to 1, the product equals the original number.

3. Properties of Numbers Lesson 2-1 Additional Examples Carlos spent $42 on his golf game. He then bought a bottle of water for $2 and a chef’s salad for $8. What was the total cost for his golf game and meal? You can use the Associative Property of Addition to find the total cost in two different ways. 42 + (2 + 8) = 42 + 10 = 52 Add 2 and 8 first. (42 + 2) + 8 = 44 + 8 = 52 Add 42 and 2 first. Carlos’s total cost was $52.

4. Properties of Numbers Lesson 2-1 Additional Examples Name each property shown. a. 17 + x + 3 = 17 + 3 + x Commutative Property of Addition b. (36  2)10 = 36(2  10) Associative Property of Multiplication c.km = km • 1 Identity Property of Multiplication d.  (103 + 26) + 4 = 103 + (26 + 4) Associative Property of Addition

5. Properties of Numbers Lesson 2-1 Additional Examples Use mental math to simplify (48 + 7) + 2. (48 + 7) + 2 = (7 + 48) + 2 Use the Commutative Property of Addition. = 7 + (48 + 2) Use the Associative Property of Addition. = 7 + 50 Add within parentheses. = 57 Add.

6. Properties of Numbers Lesson 2-1 Additional Examples Suppose you buy school supplies costing $.45, $.65, and $1.55. Use mental math to find the cost of these supplies. 0.45 + 0.65 + 1.55 = 0.65 + 0.45 + 1.55 Use the Commutative Property of Addition. = 0.65 + (0.45 + 1.55) Use the Associative Property of Addition. = 0.65 + 2.00Add within parentheses. = 2.65 Add. The cost of the school supplies is $2.65.

7. Properties of Numbers Lesson 2-1 Additional Examples Use mental math to simplify (20 • 13) • 5. (20 • 13) • 5 = (13 • 20) • 5 Use the Commutative Property of Multiplication. = 13 • (20 • 5) Use the Associative Property of Multiplication. = 13 • 100Multiply within parentheses. = 1,300 Multiply.

8. The Distributive Property Lesson 2-2 Objectives: 1. To use the Distributive Property with numerical expressions 2. To use the Distributive Property with algebraic expressions

9. The Distributive Property Lesson 2-2 New Terms: Distributive Property – to multiply a sum or difference, multiply each number within the parentheses by the number outside the parantheses. Tips: remember when multiplying by a negative number, the rules for integers still apply.

10. The Distributive Property Lesson 2-2 Additional Examples Use the Distributive Property to find 15(110) mentally. 15(110) = 15(100 + 10) Write 110 as (100 + 10). = 15 • 100 + 15 • 10Use the Distributive Property. = 1,500 + 150 Multiply. = 1,650 Add.

11. The Distributive Property Lesson 2-2 Additional Examples Ms. Thomas gave 5 pencils to each of her 37 students. What is the total number of pencils she gave to the students? (37)5 = (40 – 3)5Write 37 as (40 – 3). = 40 • 5 – 3 • 5Use the Distributive Property. = 200 – 15 Multiply. = 185 Subtract. Ms. Thomas gave the students 185 pencils.

12. The Distributive Property Lesson 2-2 Additional Examples Simplify 11(23) + 11(7). 11(23) + 11(7) = 11(23 + 7) Use the Distributive Property. = 11(30) Add within parentheses. = 330 Multiply.

13. –9(2 – 8y) = –9(2) – (–9)(8y) Use the Distributive Property. (5m + 6)11 = (5m)11 + (6)11Use the Distributive Property. The Distributive Property Lesson 2-2 Additional Examples Multiply. a. –9(2 – 8y) = –18 – (–72y)Multiply. = –18 + 72ySimplify. b.  (5m + 6)11 = 55m + 66 Multiply.

14. Simplifying Variable Expressions Lesson 2-3 • Objectives: • To identify parts of a variable expression • 2. To simplify expressions

15. Simplifying Variable Expressions Lesson 2-3 New Terms: Term – a number or the product of a number and variable(s) Constant – a term that has no variable Like Terms – terms that have identical variables Coefficient – a number that multiples a variable Deductive Reasoning – the process of reasoning logically from given facts to a conclusion. As you use properties, rules, and definitions to justify the steps in a problem, you are using deductive reasoning. Tips: some variable terms have an unwritten coefficient of 1, important to remember when adding like terms.

16. Simplifying Variable Expressions Lesson 2-3 Additional Examples Name the coefficients, the like terms, and the constants in 7x + y – 2x – 7. Coefficients: 7, 1, –2 Like terms: 7x, –2x Constant: –7

17. Simplifying Variable Expressions Lesson 2-3 Additional Examples Simplify 2b + b – 4. 2b + b – 4 = 2b + 1b – 4 Use the Identity Property of Multiplication. = (2 + 1)b – 4 Use the Distributive Property. = 3b – 4 Simplify.

18. Simplifying Variable Expressions Lesson 2-3 Additional Examples Simplify (7 – 3x)5 + 20x. (7 – 3x)5 + 20x = 35 – 15x + 20xUse the Distributive Property. = 35 + (–15x + 20x) Use the Associative Property of Addition. = 35 + (–15 + 20)xUse the Distributive Property to combine like terms. = 35 + 5x Simplify.

19. Variables and Equations Lesson 2-4 • Objectives: • To classify types of equations. • 2. To check equations using substitution

20. Variables and Equations Lesson 2-4 New Terms: Equation – is a mathematical sentence with an equal sign Open Sentence – an equation with one or more variables Solution to an Equation – a value to a variable that make the equation “true” Tips: ≠ means not equal The verb “is” indicates an equal sign

21. = / false, because 13 1 Variables and Equations Lesson 2-4 Additional Examples State whether each equation is true, false, or an open sentence. Explain. a. 3(b – 8) = 12 open sentence, because there is a variable b. 7 – (–6) = 1 c. –9 + 5 = – 4 true, because – 4 = – 4

22. the number the opposite of forty-two Words six times the number added to is –42 x 6x is added to x + = –42 6x Equation Variables and Equations Lesson 2-4 Additional Examples Write an equation for Six times a number added to the number is the opposite of forty-two. State whether the equation is true, false, or an open sentence. Explain. The equation is an open sentence, because there is a variable.

23. = / 120 + 45 0 75 Substitute 45 for x. 165 75 Variables and Equations Lesson 2-4 Additional Examples Is 45 a solution of the equation 120 + x = 75? 120 + x = 75 No, 45 is not a solution of the equation.

24. = / weight of apples weight of jam weight of cheese Words plus plus is 20 lb j = weight of jam. Let Equation 9 + 5 + j = 20 14 + 7 20 Substitute 7 for the variable. 21 20 Variables and Equations Lesson 2-4 Additional Examples A gift pack must hold 20 lb of food. Apples weigh 9 lb and cheese weighs 5 lb. Can the jar of jam that completes the package weigh 7 lb? 9 + 5 + j = 20 14 + j = 20 Add. No, the jar of jam cannot weigh 7 lb.

25. Solving Equations by Adding or Subtracting Lesson 2-5 Objectives: 1. To solve one-step equations using subtraction 2. To solve one-step equations using addition

26. Solving Equations by Adding or Subtracting Lesson 2-5 New Terms: Inverse Operations – used to get the variable alone Tips: The goal to solving any equation is to “isolate” the variable using inverse operations. You should always simplify both sides of an equation before isolating the variable. Remember to add or subtract BOTH sides by the same number

27. y + 5 = 13 Subtract 5 y + 5 – 5 = 13 – 5from each side. – 5 = – 5 Check:y + 5 = 13 8 + 5 13 Replace y with 8. 13 = 13 Solving Equations by Adding or Subtracting Lesson 2-5 Additional Examples Solve y + 5 = 13. Method 1: Method 2: y + 5 = 13 y = 8 Simplify. y = 8

28. Words target number is 250 plus number to buy Let x = number to buy. 327 = 250 + x Equation Solving Equations by Adding or Subtracting Lesson 2-5 Additional Examples Larissa wants to increase the number of books in her collection to 327 books. She has 250 books now. Find the number of books she needs to buy.

29. Check:  Is the answer reasonable? 250 plus the number of books bought should be a total collection of 327. 250 + 77 = 327 Solving Equations by Adding or Subtracting Lesson 2-5 Additional Examples (continued) 327 = 250 + x 327 = x + 250 Use the Commutative Property of Addition. 327 – 250 = x + 250 – 250Subtract 250 from each side. 77 = xSimplify. Larissa needs to buy 77 more books.

30. Solving Equations by Adding or Subtracting Lesson 2-5 Additional Examples Solve c – 23 = – 40. c – 23 = – 40 c – 23 + 23 = – 40 + 23Add 23 to each side. c = –17 Simplify.

31. Words cost of CD player was $115 less than cost of DVD player t Let = cost of the DVD player. Equation 78 = t – 115 Solving Equations by Adding or Subtracting Lesson 2-5 Additional Examples Marcy’s CD player cost $115 less than her DVD player. Her CD player cost $78. How much did her DVD player cost? 78 = t – 115 Write an equation. 78 + 115 = t – 115 + 115 Add 115 to each side. 193 = tSimplify. Marcy’s DVD player cost $193.

32. Solving Equations by Multiplying or Dividing Lesson 2-6 • Objectives: • To solve one-step equations using division • 2. To solve one-step equations using multiplication

33. Solving Equations by Multiplying or Dividing Lesson 2-6 Tips: The division property of equality suggests you can divide each side of an equation by the same nonzero number. Divisors are restricted to nonzero values because division by zero is undefined. Remember to multiply or divide BOTH sides by the same number.

34. number of pens times Words 12 number of boxes is b Let = number of boxes. 288 Equation = • b 12 Solving Equations by Multiplying or Dividing Lesson 2-6 Additional Examples 288 pens are boxed by the dozen. How many boxes are needed?

35. Divide each side by 12. = 288 12 12b 12 Solving Equations by Multiplying or Dividing Lesson 2-6 Additional Examples (continued) 288 = 12b 24 = bSimplify. 24 boxes are needed. Check: Is the answer reasonable? Twelve times the number of boxes is the number of pens. Since 12  24 = 288, the answer is reasonable.

36. Divide each side by –2. = Check: –2v = –24 –2 • (12) –24 Replace v with 12. –24 = –24 –24 –2 –2v –2 Solving Equations by Multiplying or Dividing Lesson 2-6 Additional Examples Solve –2v = –24. –2v = –24 v = 12 Simplify.

37. = – 5 8 = 8(–5) Multiply each side by 8. x 8 x 8 x 8 Solving Equations by Multiplying or Dividing Lesson 2-6 Additional Examples Solve = – 5. x = – 40 Simplify.

38. Problem Solving Strategy: Guess, Check, Revise Lesson 2-7 Objectives: 1.To solve a problem using the Guess, Check, Revise strategy

39. Problem Solving Strategy: Guess, Check, Revise Lesson 2-7 Additional Examples During the intermission of the play, the Theater Club sold cups of popcorn and soda. The club sold 79 cups of popcorn and 96 sodas for a total of $271. What was the selling price of a cup of popcorn? Of a soda? You can organize conjectures in a table. As a first conjecture, try both with a price of $1.

40. Popcorn Soda Price Price Total Price $1 $1 79(1) + 96(1) = 79 + 96 The total is too low. Increase = 175 the price of the popcorn only. Problem Solving Strategy: Guess, Check, Revise Lesson 2-7 Additional Examples (continued) $2 $1 79(2) + 96(1) = 158 + 96 The total is too low.= 254 Increase the price of the soda. $2 $2 79(2) + 96(2) = 158 + 192 The total is too high. = 350 Decrease the price of the popcorn. $1 $2 79(1) + 96(2) = 79 + 192 The total is correct. = 271 The popcorn price was $1, and the soda price was $2.

41. Inequalities and Their Graphs Lesson 2-8 • Objectives: • To graph inequalities • 2. To write inequalities

42. Inequalities and Their Graphs Lesson 2-8 New Terms: Inequality – a mathematical sentence that contains <,>,≤,≥, or ≠. Solution to an Inequality – any number that makes the inequality true. Tips: Know what the different signs mean, and we do not have gators or crocodiles in this class When reading the solution to an inequality, you should always start with the varible

43. > – An open dot shows that –2 is not a solution. Shade all the points to the right of –2. A closed dot shows that –5 is a solution. Shade all the points to the right of –5. Inequalities and Their Graphs Lesson 2-8 Additional Examples Graph the solutions of each inequality on a number line. a.x > –2   b.w –5 

44. < A closed dot shows that 4 is a solution. – Shade all the points to the left of 4. An open dot shows that 6 is not a solution. Shade all the points to the left of 6. Inequalities and Their Graphs Lesson 2-8 Additional Examples (continued) c.k 4   d.y < 6

45. > – x –3   Inequalities and Their Graphs Lesson 2-8 Additional Examples Write the inequality shown in each graph. a. b. x < 3

46. Inequalities and Their Graphs Lesson 2-8 Additional Examples Food can be labeled very low sodium only if it meets the requirement established by the federal government. Use the table to write an inequality for this requirement. Label Definition Sodium-free food Less than 5 mg per serving Very low sodium food At most 35 mg per serving Low-sodium food At most 140 mg per serving

47. < – a serving of very low sodium food Words has at most 35 mg sodium number of milligrams of sodium in a serving of very low sodium food. v Let = v 35 Inequality Inequalities and Their Graphs Lesson 2-8 Additional Examples (continued)

48. Solving One-Step Inequalities by Adding or Subtracting Lesson 2-9 Objectives: 1. To solve one-step inequalities using subtraction 2. To solve one-step inequalities using addition

49. Solving One-Step Inequalities by Adding or Subtracting Lesson 2-9 Tips: Just like the equality properties, you must add or subtract the same number from each side When rewriting an inequality in reverse order, you must pay attention to the direction of the inequality symbol

50. < – > > > > – – – – –16 y – 14 –16 y – 14 –16 + 14y – 14 + 14Add 14 to each side. –2 y or y –2 Simplify. Solving One-Step Inequalities by Adding or Subtracting Lesson 2-9 Additional Examples Solve each inequality. Graph the solutions. a. 4 + s < 12 4 + s < 12 4 + s– 4 < 12 – 4Subtract 4 from each side. s < 8 Simplify. b.