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Properties of Numbers

Properties of Numbers. PRE-ALGEBRA LESSON 2-1. At the Red Valley Sports Camp, 15 kids went horseback riding, 14 played tennis, 23 went hiking, and the rest of the campers stayed in their cabins. If 83 kids were in the camp, how many stayed indoors?. 31. 2-1. Properties of Numbers.

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Properties of Numbers

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  1. Properties of Numbers PRE-ALGEBRA LESSON 2-1 At the Red Valley Sports Camp, 15 kids went horseback riding, 14 played tennis, 23 went hiking, and the rest of the campers stayed in their cabins. If 83 kids were in the camp, how many stayed indoors? 31 2-1

  2. Properties of Numbers PRE-ALGEBRA LESSON 2-1 (For help, go to Lesson 1-5.) Simplify. 1. –18 + (–7) 2. 32 – (–3) 3. (–13) + 6 4. 2 – 48 Check Skills You’ll Need 2-1

  3. Properties of Numbers PRE-ALGEBRA LESSON 2-1 Solutions 1. –18 + (–7) = –25 2. 32 – (–3) = 32 + 3 = 35 32 – (–3) = 35 3. (–13) + 6 = –7 4. 2 – 48 = 2 + (–48) = –46 2 – 48 = –46 2-1

  4. Properties of Numbers PRE-ALGEBRA LESSON 2-1 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. Quick Check 2-1

  5. Properties of Numbers PRE-ALGEBRA LESSON 2-1 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 Quick Check 2-1

  6. Properties of Numbers PRE-ALGEBRA LESSON 2-1 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. Quick Check 2-1

  7. Properties of Numbers PRE-ALGEBRA LESSON 2-1 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. Quick Check 2-1

  8. Properties of Numbers PRE-ALGEBRA LESSON 2-1 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. Quick Check 2-1

  9. Properties of Numbers PRE-ALGEBRA LESSON 2-1 Name each property shown. 1.d 1 = d 2. 3 + (2 + 5) = (3 + 2) + 5 3.fg = gf 4. 32 + 45 + 102 = 45 + 32 + 102 Identity Property of Multiplication Associative Property of Addition Commutative Property of Multiplication Commutative Property of Addition 2-1

  10. The Distributive Property PRE-ALGEBRA LESSON 2-2 Evaluate each expression. a. 24 – 6 ÷ 2 • 4 + 36 b. 6 • (3 – 1) + 18 c. 10 + 6 • 5 – 4 ÷ 2 38 48 30 2-2

  11. The Distributive Property PRE-ALGEBRA LESSON 2-2 (For help, go to Lesson 1-2.) Simplify each expression. 1. 3 • 7 – 9 2. (9 – 5)6 3. 8 + 2 • 6 4. 2(6 – 3) 5. 4 • 5 – 4 • 3 6. 3 • 2 – 1 • 2 Check Skills You’ll Need 2-2

  12. Solutions 1. 3 • 7 – 9 2. (9 – 5) • 6 21 – 9 4 • 6 12 24 3. 8 + 2 • 6 4. 2 • (6 – 3) 8 + 12 2 • 3 20 6 5. 4 • 5 – 4 • 3 6. 3 • 2 – 1 • 2 20 – 12 6 – 2 8 4 The Distributive Property PRE-ALGEBRA LESSON 2-2 2-2

  13. The Distributive Property PRE-ALGEBRA LESSON 2-2 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. Quick Check 2-2

  14. The Distributive Property PRE-ALGEBRA LESSON 2-2 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. Quick Check 2-2

  15. The Distributive Property PRE-ALGEBRA LESSON 2-2 Simplify 11(23) + 11(7). 11(23) + 11(7) = 11(23 + 7) Use the Distributive Property. = 11(30) Add within parentheses. = 330 Multiply. Quick Check 2-2

  16. Model four groups of 3x – 4. Group like tiles. The Distributive Property PRE-ALGEBRA LESSON 2-2 Use algebra tiles to multiply 4(3x – 4). So 4(3x – 4) = 12x – 16. Quick Check 2-2

  17. –9(2 – 8y) = –9(2) – (–9)(8y) Use the Distributive Property. (5m + 6)11 = (5m)11 + (6)11Use the Distributive Property. The Distributive Property PRE-ALGEBRA LESSON 2-2 Multiply. a. –9(2 – 8y) = –18 – (–72y)Multiply. = –18 + 72ySimplify. b.  (5m + 6)11 = 55m + 66 Multiply. Quick Check 2-2

  18. The Distributive Property PRE-ALGEBRA LESSON 2-2 Use the Distributive Property to simplify. 1. 15(203) 2. 7(180) 3. 6(26) – 6(16) 4. (8a – 4)9 5. 2(14 + y) 3,045 1,260 60 72a – 36 28 + 2y 2-2

  19. 1 3 2 7 Simplifying Variable Expressions PRE-ALGEBRA LESSON 2-3 Use numbers to write the following word expression: eight and one-third minus five and two-sevenths. 8 – 5 2-3

  20. Simplifying Variable Expressions PRE-ALGEBRA LESSON 2-3 (For help, go to Lesson 2-2.) Simplify each expression. 1. 5(b + 4) 2. –3(2x + 5) 3. 4(–8 – 3q) 4. –6(2b – 7) Check Skills You’ll Need 2-3

  21. Simplifying Variable Expressions PRE-ALGEBRA LESSON 2-3 Solutions 1. 5(b + 4) = 5(b) + 5(4) 2. –3(2x + 5) = –3(2x) + (–3)(5) = 5b + 20 = –6x – 15 3. 4(–8 – 3q) = 4(–8) + 4(–3q) 4. –6(2b – 7) = –6(2b) + (–6)(–7) = –32 – 12q = –12b + 42 2-3

  22. Simplifying Variable Expressions PRE-ALGEBRA LESSON 2-3 Name the coefficients, the like terms, and the constants in 7x + y – 2x – 7. Coefficients: 7, 1, –2 Like terms: 7x, –2x Constant: –7 Quick Check 2-3

  23. 9 +  4f + 3  + 2f 6f  + 12 Simplifying Variable Expressions PRE-ALGEBRA LESSON 2-3 Simplify 9 + 4f + 3 + 2f. Quick Check 2-3

  24. Simplifying Variable Expressions PRE-ALGEBRA LESSON 2-3 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. Quick Check 2-3

  25. Simplifying Variable Expressions PRE-ALGEBRA LESSON 2-3 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. Quick Check 2-3

  26. Simplifying Variable Expressions PRE-ALGEBRA LESSON 2-3 Name coefficients, like terms, and constants. 1. 4f – 2f + 3 2.z + 2y – 14 Simplify each expression. 3. 3(a + c – 1) – 2c4. 4(4v) – 4(v – 9) 4, –2; 4f, –2f; 3 1, 2; none; –14 12v + 36 3a + c – 3 2-3

  27. c 5 Variables and Equations PRE-ALGEBRA LESSON 2-4 Evaluate 10 – for c = 10. 8 2-4

  28. Variables and Equations PRE-ALGEBRA LESSON 2-4 (For help, go to Lesson 1-1.) Write a variable expression for each phrase. 1. the sum of x and 46 2. four less than g 3.t decreased by five 4. the quotient of z and 26 Check Skills You’ll Need 2-4

  29. z 26 Variables and Equations PRE-ALGEBRA LESSON 2-4 Solutions 1. the sum of x and 46 2. four less than g x plus 46 g minus 4 x + 46 g – 4 3.t decreased by five 4. the quotient of z and 26 t minus 5 z divided by 26 t – 5 2-4

  30. = / false, because 13 1 Variables and Equations PRE-ALGEBRA LESSON 2-4 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 Quick Check 2-4

  31. 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 PRE-ALGEBRA LESSON 2-4 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. Quick Check 2-4

  32. = / 120 + 45 0 75 Substitute 45 for x. 165 75 Variables and Equations PRE-ALGEBRA LESSON 2-4 Is 45 a solution of the equation 120 + x = 75? 120 + x = 75 No, 45 is not a solution of the equation. Quick Check 2-4

  33. = / 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 PRE-ALGEBRA LESSON 2-4 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. Quick Check No, the jar of jam cannot weigh 7 lb. 2-4

  34. Variables and Equations PRE-ALGEBRA LESSON 2-4 1. Is b + 3b = 15 true, false, or an open sentence? 2. Write an equation for A number added to seven times the number is the opposite of thirty-two. Is the equation true, false, or an open sentence? 3. Is 25 a solution of the equation 175 – x = 150? 4. Can you stack eight books that are each 3 in. thick inside a box that is 18 in. tall? open sentence 7x + x = –32; open sentence yes no 2-4

  35. 2 3 1 3 Solving Equations by Adding or Subtracting PRE-ALGEBRA LESSON 2-5 Lauralee pitched 12 innings. Maureen pitched 3 fewer innings. How many innings did Maureen pitch? 8 2-5

  36. Solving Equations by Adding or Subtracting PRE-ALGEBRA LESSON 2-5 (For help, go to Lesson 2-1.) Simplify each expression. 1. 3 + 4 – 4 2. 7 + 9 – 7 3. 8 – 2 + 2 4. 6 + 2 – 2 Check Skills You’ll Need 2-5

  37. Solving Equations by Adding or Subtracting PRE-ALGEBRA LESSON 2-5 Solutions 1. 3 + 4 – 4 2. 7 + 9 – 7 = 3 + (4 – 4) = 9 + (7 – 7) = 3 + 0 = 9 + 0 = 3 = 9 3. 8 – 2 + 2 4. 6 + 2 – 2 = 8 + (2 – 2) = 6 + (2 – 2) = 8 + 0 = 6 + 0 = 8 = 6 2-5

  38. 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 PRE-ALGEBRA LESSON 2-5 Solve y + 5 = 13. Method 1 Method 2 y + 5 = 13 y = 8 Simplify. y = 8 Quick Check 2-5

  39. Words target number is 250 plus number to buy Let x = number to buy. 327 = 250 + x Equation Solving Equations by Adding or Subtracting PRE-ALGEBRA LESSON 2-5 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. 2-5

  40. 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 PRE-ALGEBRA LESSON 2-5 (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. Quick Check 2-5

  41. Solving Equations by Adding or Subtracting PRE-ALGEBRA LESSON 2-5 Solve c – 23 = – 40. c – 23 = – 40 c – 23 + 23 = – 40 + 23Add 23 to each side. c = –17 Simplify. Quick Check 2-5

  42. 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 PRE-ALGEBRA LESSON 2-5 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 78 + 115 = t – 115 + 115 Add 115 to each side. 193 = tSimplify. Marcy’s DVD player cost $193. Quick Check 2-5

  43. Solving Equations by Adding or Subtracting PRE-ALGEBRA LESSON 2-5 Solve each equation. 1.y + 8 = 12 2. 7 + f – 21 = –20 3. 67 = g – (–36) 4. Ricky rides his bike 12 miles every day. He stops after 7 miles to rest. How much farther does he have to ride? 4 –6 31 5 mi 2-5

  44. 0.5952 1.6 10.5126 2.1 Solving Equations by Multiplying or Dividing PRE-ALGEBRA LESSON 2-6 Divide. Round each quotient to the nearest hundredth. a. b. 0.372, which rounds to 0.37 5.006, which rounds to 5.01 2-6

  45. Simplify each quotient. 1.2. 3.4. 18 18 –7 7 21 –21 –13 –13 Solving Equations by Multiplying or Dividing PRE-ALGEBRA LESSON 2-6 (For help, go to Lesson 1-9.) Check Skills You’ll Need 2-6

  46. Solutions 1. = 1 2. = –1 3. = –1 4. = 1 18 18 –7 7 21 –21 –13 –13 Solving Equations by Multiplying or Dividing PRE-ALGEBRA LESSON 2-6 2-6

  47. 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 PRE-ALGEBRA LESSON 2-6 288 pens are boxed by the dozen. How many boxes are needed? 2-6

  48. Divide each side by 12. = 288 12 12b 12 Solving Equations by Multiplying or Dividing PRE-ALGEBRA LESSON 2-6 (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. Quick Check 2-6

  49. 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 PRE-ALGEBRA LESSON 2-6 Solve –2v = –24. –2v = –24 v = 12 Simplify. Quick Check 2-6

  50. = – 5 8 = 8(–5) Multiply each side by 8. x 8 x 8 x 8 Solving Equations by Multiplying or Dividing PRE-ALGEBRA LESSON 2-6 Solve = – 5. x = – 40 Simplify. Quick Check 2-6

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