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  • 1. Adding and Subtracting Real Numbers;1.3 Properties of Real Numbers 1. Add integers. 2. Add rational numbers. 3. Find the additive inverse of a number. 4. Subtract rational numbers. Copyright © 2011 Pearson Education, Inc.
  • 2. Objective 1 Add integers. Copyright © 2011 Pearson Education, Inc. Slide 1- 2
  • 3. Parts of an addition statement: The numbers added are called addends and the answer is called a sum. 2+3=5 Addends Sum Copyright © 2011 Pearson Education, Inc. Slide 1- 3
  • 4. Adding Numbers with the Same SignTo add two numbers that have the same sign, addtheir absolute values and keep the same sign. Copyright © 2011 Pearson Education, Inc. Slide 1- 4
  • 5. Example 2Add.a. 27 + 12 b. –16 + (– 22)Solutiona. 27 + 12 = 39b. –16 + (–22) = –38 Copyright © 2011 Pearson Education, Inc. Slide 1- 5
  • 6. Adding Numbers with Different SignsTo add two numbers that have different signs,subtract the smaller absolute value from thegreater absolute value and keep the sign of thenumber with the greater absolute value. Copyright © 2011 Pearson Education, Inc. Slide 1- 6
  • 7. Example 3Add.a. 35 + (–17) b. –29 + 7Solutiona. 35 + (–17) = 18b. –29 + 7 = –22 Copyright © 2011 Pearson Education, Inc. Slide 1- 7
  • 8. Objective 2 Add rational numbers. Copyright © 2011 Pearson Education, Inc. Slide 1- 8
  • 9. Adding Fractions with the Same DenominatorTo add fractions with the same denominator, addthe numerators and keep the same denominator;then simplify. Copyright © 2011 Pearson Education, Inc. Slide 1- 9
  • 10. Example 4A 2 4 4  5 + − +− ÷ 9 9 12  12  4  5a 2 4 2+4 6 b. − +  − ÷ + = = 12  12  9 9 9 9 −4 + ( −5 ) 9 2 g3 2 = =− = = 12 12 3 g3 3 3 g3 3Replace 6 and 9 with their prime =− =−factorizations, divide out the 3 g2 g 2 4common factor, 3, then multiply Simplify to lowest terms by dividingthe remaining factors. out the common factor, 3. Copyright © 2011 Pearson Education, Inc. Slide 1- 10
  • 11. Adding FractionsTo add fractions with different denominators:1. Write each fraction as an equivalent fraction with the LCD.2. Add the numerators and keep the LCD.3. Simplify. Copyright © 2011 Pearson Education, Inc. Slide 1- 11
  • 12. Example 5a 1 1Add: + 3 4Solution 1 1 + Write equivalent fractions with 12 in the denominator. 3 4 Add numerators and keep the common denominator. Because the addends have the same sign, we add and keep the same sign. Copyright © 2011 Pearson Education, Inc. Slide 1- 12
  • 13. Example 5b 5 3Add: − + 6 4 Write equivalent fractions with 12 in the denominator. Add numerators and keep the common denominator. Because the addends have different signs, we subtract and keep the sign of the number with the greater absolute value. Copyright © 2011 Pearson Education, Inc. Slide 1- 13
  • 14. Example 5c 7 9 Add: − + 8 30Solution 7 9 7 ( 15 ) 9(4) − + =− + Write equivalent fractions 8 30 8 ( 15 ) 30(4) with 120 in the denominator. 105 36 =− + Add numerators and keep the common denominator. 120 120 −105 + 36 Reduce to lowest terms. = 120 −69 3 ×23 23 = =− =− 120 2 ×2 ×2 × × 3 5 40 Copyright © 2011 Pearson Education, Inc. Slide 1- 14
  • 15. Example 6Bank account : Copyright © 2011 Pearson Education, Inc. Slide 1- 15
  • 16. Objective 4 Subtract rational numbers. Copyright © 2011 Pearson Education, Inc. Slide 1- 16
  • 17. Parts of a subtraction statement: 8–5=3 Difference Minuend Subtrahend Copyright © 2011 Pearson Education, Inc. Slide 1- 17
  • 18. Rewriting Subtraction To write a subtraction statement as an equivalent addition statement, change the operation symbol from a minus sign to a plus sign, and change the subtrahend to its additive inverse. Copyright © 2011 Pearson Education, Inc. Slide 1- 18
  • 19. Example 9aSubtracta. –17 – (–5)SolutionWrite the subtraction as an equivalent addition. –17 – (–5)Change the operation Change the subtrahend from minus to plus. to its additive inverse. = –17 + 5 = –12 Copyright © 2011 Pearson Education, Inc. Slide 1- 19
  • 20. Example 9b 3 1Subtract: − − 8 4Solution 3 1 3 1 − − =− − 8 4 8 4 3  1 = − +− ÷ 8  4 3  1(2)  = − +− ÷ Write equivalent fractions with the common denominator, 8. 8  4(2)  3  2 5 = − +− ÷ = − 8  8 8 Copyright © 2011 Pearson Education, Inc. Slide 1- 20
  • 21. Multiplying and Dividing Real Numbers;1.4 Properties of Real Numbers 1. Multiply integers. 2. Multiply more than two numbers. 3. Multiply rational numbers. 4. Find the multiplicative inverse of a number. 5. Divide rational numbers. Copyright © 2011 Pearson Education, Inc.
  • 22. Objective 1 Multiply integers. Copyright © 2011 Pearson Education, Inc. Slide 1- 22
  • 23. In a multiplication statement, factors aremultiplied to equal a product. 2 g 3 = 6 Factors Product Copyright © 2011 Pearson Education, Inc. Slide 1- 23
  • 24. Multiplying Two Numbers with Different SignsWhen multiplying two numbers that have differentsigns, the product is negative. Copyright © 2011 Pearson Education, Inc. Slide 1- 24
  • 25. Example 2Multiply.a. 7(–4) b. (–15)3Solutiona. 7(–4) = –28 Warning: Make sure you see the difference between 7(–4), which indicates multiplication, and 7 – 4,b. (–15)3 = –45 which indicates subtraction. Copyright © 2011 Pearson Education, Inc. Slide 1- 25
  • 26. Multiplying Two Numbers with the Same SignWhen multiplying two numbers that have the samesign, the product is positive. Copyright © 2011 Pearson Education, Inc. Slide 1- 26
  • 27. Example 3Multiply.a. –5(–9) b. (–6)(–8)Solutiona. –5(–9) = 45b. (–6)(–8) = 48 Copyright © 2011 Pearson Education, Inc. Slide 1- 27
  • 28. Objective 2 Multiply more than two numbers. Copyright © 2011 Pearson Education, Inc. Slide 1- 28
  • 29. Multiplying with Negative FactorsThe product of an even number of negative factorsis positive, whereas the product of an odd numberof negative factors is negative. Copyright © 2011 Pearson Education, Inc. Slide 1- 29
  • 30. Objective 3 Multiply rational numbers. Copyright © 2011 Pearson Education, Inc. Slide 1- 30
  • 31. Example 5a 3 4Multiply − g  ÷. 5 9Solution 3 4 3  2 g2  − g  ÷= − g  ÷ Divide out the common factor, 3. 5 9 5  3 g3  4 Because we are multiplying two =− numbers that have different signs, 15 the product is negative. Copyright © 2011 Pearson Education, Inc. Slide 1- 31
  • 32. Objective 5 Divide rational numbers. Copyright © 2011 Pearson Education, Inc. Slide 1- 32
  • 33. Parts of a division statement: 8 ÷ 2 = 4 Dividend Quotient Divisor Copyright © 2011 Pearson Education, Inc. Slide 1- 33
  • 34. Dividing Signed Numbers When dividing two numbers that have the same sign, the quotient is positive. When dividing two numbers that have different signs, the quotient is negative. Copyright © 2011 Pearson Education, Inc. Slide 1- 34
  • 35. Example 8Divide.a. 56 ÷ (−8) b. −72 ÷ ( −6 )Solutiona. 56 ÷ (−8) = −7 b.−72 ÷ ( −6 ) = 12 Copyright © 2011 Pearson Education, Inc. Slide 1- 35
  • 36. Division Involving 0 0 ÷ n = 0 when n ≠ 0. n ÷ 0 is undefined when n ≠ 0. 0 ÷ 0 is indeterminate. Copyright © 2011 Pearson Education, Inc. Slide 1- 36
  • 37. Example 9 3 4Divide − ÷ . 10 5Solution 3 4 3 5 − ÷ =− g Write an equivalent multiplication. 10 5 10 4 3 5 Divide out the common factor, 5. =− g 5 g2 2 g2 3 Because we are dividing two numbers =− that have different signs, the result is 8 negative. Copyright © 2011 Pearson Education, Inc. Slide 1- 37
  • 38. Exponents, Roots, and Order of1.5 Operations 1. Evaluate numbers in exponential form. 2. Evaluate square roots. 3. Use the order-of-operations agreement to simplify numerical expressions. 4. Find the mean of a set of data. Copyright © 2011 Pearson Education, Inc.
  • 39. Objective 1 Evaluate numbers in exponential form. Copyright © 2011 Pearson Education, Inc. Slide 1- 39
  • 40. Evaluating an Exponential Form To evaluate an exponential form raised to a natural number exponent, write the base as a factor the number of times indicated by the exponent; then multiply. Copyright © 2011 Pearson Education, Inc. Slide 1- 40
  • 41. Example 1aEvaluate. (–9)2SolutionThe exponent 2 indicates we have two factors of –9.Because we multiply two negative numbers, the resultis positive. (–9)2 = (–9)(–9) = 81 Copyright © 2011 Pearson Education, Inc. Slide 1- 41
  • 42. Example 1b 3  3Evaluate. − ÷  5SolutionThe exponent 3 means we must multiply the base byitself three times. 3  3  3  3   3   − ÷ =  − ÷ − ÷ − ÷  5  5  5   5  27 =− 125 Copyright © 2011 Pearson Education, Inc. Slide 1- 42
  • 43. Evaluating Exponential Forms with Negative BasesIf the base of an exponential form is a negative number and the exponent is even, then the product is positive.If the base is a negative number and the exponent is odd, then the product is negative. Copyright © 2011 Pearson Education, Inc. Slide 1- 43
  • 44. Example 2Evaluate.a. (−3) 4 b. −34 c. (−2)3 d. −23 Copyright © 2011 Pearson Education, Inc. Slide 1- 44
  • 45. Objective 2 Evaluate square roots. Copyright © 2011 Pearson Education, Inc. Slide 1- 45
  • 46. Roots are inverses of exponents. Morespecifically, a square root is the inverse of asquare, so a square root of a given number is anumber that, when squared, equals the givennumber.Square RootsEvery positive number has two square roots, a positive root and a negative root.Negative numbers have no real-number square roots. Copyright © 2011 Pearson Education, Inc. Slide 1- 46
  • 47. Example 3Find all square roots of the given number.Solutiona. 49Answer ± 7b. −81Answer No real-number square roots exist. Copyright © 2011 Pearson Education, Inc. Slide 1- 47
  • 48. The symbol, , called the radical, is used toindicate finding only the positive (or principal)square root of a given number. The given number orexpression inside the radical is called the radicand. Radical Principal Square Root 25 = 5 Radicand Copyright © 2011 Pearson Education, Inc. Slide 1- 48
  • 49. Example 4Evaluate the square root.a. 169 b. 64 c. 0.64 d. −25 81Solution Copyright © 2011 Pearson Education, Inc. Slide 1- 49
  • 50. Objective 3 Use the order-of-operations agreement to simplify numerical expressions. Copyright © 2011 Pearson Education, Inc. Slide 1- 50
  • 51. Order-of- Operations AgreementPerform operations in the following order:1. Within grouping symbols: parentheses ( ), brackets [ ], braces { }, above/below fraction bars, absolute value | |, and radicals .2. Exponents/Roots from left to right, in order as they occur.3. Multiplication/Division from left to right, in order as they occur.4. Addition/Subtraction from left to right, in order as they occur. Copyright © 2011 Pearson Education, Inc. Slide 1- 51
  • 52. Example 5aSimplify. −26 + 15 ÷ (−5) ×2Solution −26 + 15 ÷ (−5) ×2 = −26 + (−3) ×2 Divide 15 ÷ (−5) = –3 = −26 + (−6) Multiply (–3) ⋅ 2 = –6 = −32 Add –26 + (–6) = –32 Copyright © 2011 Pearson Education, Inc. Slide 1- 52
  • 53. Example 5cSimplify. ( −3) + 5 6 − ( 2 + 1)  − 49 2  Solution Calculate within the innermost parenthesis. Evaluate the exponential form, brackets, and square root. Multiply 5(3). Add 9 + 15. Subtract 24 – 7. Copyright © 2011 Pearson Education, Inc. Slide 1- 53
  • 54. Example 7a 8(−5) − 23Simplify. 4(8) − 8Solution Evaluate the exponential form in the numerator and multiply in the denominator. Multiply in the numerator and subtract in the denominator. Subtract in the numerator. Divide. Copyright © 2011 Pearson Education, Inc. Slide 1- 54
  • 55. Example 7b 9(4) + 12Simplify. 43 + (8)(−8)Solution Because the denominator or divisor is 0, the answer is undefined. Copyright © 2011 Pearson Education, Inc. Slide 1- 55
  • 56. 1.6 Translating Word Phrases to Expressions 1. Translate word phrases to expressions. Copyright © 2011 Pearson Education, Inc.
  • 57. Objective 1 Translating word phrases to ExpressionsLook at the pg Copyright © 2011 Pearson Education, Inc. Slide 1- 57
  • 58. The key words sum, difference, product, and quotientindicate the answer for their respective operations.sum of x and 3 difference of x and 3 x+3 x–3 product of x and 3 quotient of x and 3 x⋅3 x÷3 Copyright © 2011 Pearson Education, Inc. Slide 1- 58
  • 59. Example 1Translate to an algebraic expression.a. five more than two times a numberTranslation: 5 + 2n or 2n + 5b. seven less than the cube of a numberTranslation: n3 – 7c. the sum of h raised to the fourth power and twelveTranslation: h4 + 12 Copyright © 2011 Pearson Education, Inc. Slide 1- 59
  • 60. Example 2Translate to an algebraic expression.a. seven times the sum of a and bTranslation: 7(a + b)b. the product of a and b divided by the sum of w2 and 4 abTranslation: ab ÷ (w + 4) or 2 2 w +4 Copyright © 2011 Pearson Education, Inc. Slide 1- 60
  • 61. 1.7 Evaluating and Rewriting Expressions 1. Evaluate an expression. 2. Determine all values that cause an expression to be undefined. 3. Rewrite an expression using the distributive property. 4. Rewrite an expression by combining like terms. Copyright © 2011 Pearson Education, Inc.
  • 62. Objective 1 Evaluate an expression. Copyright © 2011 Pearson Education, Inc. Slide 1- 62
  • 63. Evaluating an Algebraic ExpressionTo evaluate an algebraic expression:1. Replace the variables with their corresponding given values.2. Calculate the numerical expression using the order of operations. Copyright © 2011 Pearson Education, Inc. Slide 1- 63
  • 64. Example 1aEvaluate 3w – 4(a – 6) when w = 5 and a = 7.Solution 3w – 4(a − 6) 3(5) – 4(7 – 6) Replace w with 5 and a with 7. = 3(5) – 4(1) Simplify inside the parentheses first. = 15 – 4 Multiply. = 11 Subtract. Copyright © 2011 Pearson Education, Inc. Slide 1- 64
  • 65. Objective 2 Determine all values that cause an expression to be undefined. Copyright © 2011 Pearson Education, Inc. Slide 1- 65
  • 66. Examples: Copyright © 2011 Pearson Education, Inc. Slide 1- 66
  • 67. The Distributive Property of Multiplication overAddition a(b + c) = ab + acThis property gives us an alternative to the order ofoperations.2(5 + 6) = 2(11) 2(5 + 6) = 2⋅5 + 2⋅6 = 22 = 10 + 12 = 22 Copyright © 2011 Pearson Education, Inc. Slide 1- 67