The Negative Integers are negative numbers only: {…, -3, -2, -1}.

1. Integers In, algebra, it is necessary for us to be familiar with the various terms that are used to classify different types of Real Numbers: The Natural Numbers is the set of numbers we can use to count things: {1, 2, 3, 4, …}. It does not include zero or negative numbers.The three dots indicates that the list continues forever (there is no last element). The Whole Numbers are the nonnegative numbers, including zero: {0, 1, 2, 4, 5, …} The Positive Integers are the whole numbers greater than zero: {1, 2, 3, …}. The Negative Integers are negative numbers only: {…, -3, -2, -1}. Zero is neither positive nor negative. The Integers arenegative integers, zero, and positive integers: {…, -3, -2, -1, 0, 1, 2, 3, …} which is shown below on the number line. -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Next Slide

2. Inequalities Greater than: If a number is to the right of another number then it is greaterthan (>) the other number. Less than: If a number is to the left of another number, then it is lessthan (<) the other number. Note: The inequality symbol must always point toward the smaller number. Example 1a: i.) 6 > 4 6 is greater than 4 ii.) 8 < 12 8 is less than 12 iii.) –9 > –16 –9 is greater than –16. (–9 is to the right of –16.) We could also say that the inequality symbol opens up to the larger number. In part iii.) above, -9 is the larger number. Would you rather owe $9 or owe $16? You would probably rather owe the $9. Next Slide

3. Additive Inverse and the Absolute Value The Additive Inverse of a number is the opposite of the number Example 1b:NumberAdditive Inverse 5 5 a a 63 63 A negative sign in front of parenthesis means “Find opposite” Example 1c: (7) = 7 (7) = 7 (the opposite of –7 is +7) (b) = b (b) = b Absolute value: The distance between zero and a number on the number line. The symbol for absolute value is | |. Note: distance is a positive number. Example 1d: , , When negative sign precedes absolute value; Find absolute value; write inside parenthesis. Evaluate opposite. Examples 1e: Note: When evaluating absolute values, the sign of the answer will always be the sign in front of the parentheses. Next Slide

4. Your Turn Problem #1 Place the correct inequality symbol between the two numbers. • 32 12 • -18 -15 • 0 -7 • > • < • c. > Answers: Evaluate each of the following: • 13 • 7 • -8 • 45 • -(-13) • |-7| • -|-8| • |45|

5. Adding Integers Procedure: To add two numbers: If signs are like: Add the numbers as in arithmetic (add their absolute values), and then attach their common sign to the answer. If signs unlike: Subtract the numbers as in arithmetic (subtract their absolute values; you may have to mentally reverse the numbers so that the larger absolute value is first); attach the sign to the answer of the number with the larger absolute value. The following chart may be useful. 1st sign 2nd sign operation resulting sign Example 2. Add the following: + + addition + a) 7 + 12 – – – addition b) –7 + (–12) – keep sign of larger # (w/ larger abs. value). + subtraction c) –7 + 12 + subtraction – “same signs: add, keep the sign.” “different signs, subtract, keep the sign of the larger.” d) 7 + (–12) Your Turn Problem #2 Add the following: a) 12 + (–45) b) –20 + (–10) c) –14 + 18 d) 79 + 86 Answers: a) –33 b) –30 c) 4 d) 165 19 –19 5 –5

6. Procedure: To add more than two numbers: Method 1: Add the first two numbers; take the answer and add to the next number; take that answer and add to the next number, etc. Method 2: Add all the positives; add all the negatives; add the answer of the positives to the answer of the negatives. 10 + (–10) –5 + (–11) Answer: Answer: –19 0 Example 3. Add the following: a) –17 + 12 + (–11) b) 24 + (–14) + (–10) Your Turn Problem #3 Add the following: a) –16 + (–14) + (–11) a) –41 Answers: b) 25 + (–28) + (–3) b) –6

7. Subtracting Integers • Procedure: To subtract two integers: • Rewrite the first number. • Change the subtraction symbol to addition. • Change the sign of the second number. • Add using the rules for addition. Note: Make two marks, one to change subtraction to addition, the second mark is to change the sign of the second number. Example 4. Subtract the following: a) 15 – 24 b) –17 – 10 c) –8– (–12) d) –11– (–6) 15 + (–24) – 17 + (–10) –8 + (+12) – 11 + (+6) Answers: –9 –27 –5 4 Your Turn Problem #4 Subtract the following: a) 13 – (–18) b) –12– (–9) c) –14– 16 d) 28 – 50 Answers: a) 31 c) –30 b) –3 d) –22

8. When subtraction occurs several times in a problem, rewrite each subtraction as addition of the opposite. Then perform the addition. Example 5. Perform the following: a) 11 – 20 – 18 b) –13– 12 – (–10) –13 + (–12) + (+10) 11 + (–20) + (–18) –25 + (+10) –9 + (–18) Answers: –27 –15 Your Turn Problem #5 Perform the following: a) –18–13– (–10) b) 15– (–5) – (–7) a) –21 Answers: b) 27

9. Example 6. Mt. Korabi in Albania has an elevation of 3100 meters and Death Valley in the United States has an elevation of 86 meters. Find the difference in elevation between Mt. Korabi and Death Valley. Solution: Mt. Korabi = 3100 meters We want the difference in elevation. Difference means to subtract. Remember to keep the sign of the 86. Let D = the difference in elevation. D = 3100  (86) = 3100 + 86 Death Valley = 86 meters Your Turn Problem #6 Mrs. Lazzara has a checking account with Bank of America . The statement she received in May showed an account balance of $4875. In June, the account balance was $5 (overdrawn). Find the difference in statements. Answer: $4880

10. Multiplying Integers Procedure: To multiply two numbers: 1. Determine sign of answer: (+) (+) = + (+) () =  () (+) =  () () = + 2. Multiply numbers and write sign from step 1 in front of the product. Example 7. Multiply the following: a) (–7)(8) b) (–7)(–8) c) (7)(–8) Answers: Your Turn Problem #7 Multiply the following: a) (–12)(–10) b) (–18)(2) c) (15)(–4) Answers: a) 120 b) –36 c) –60 –56 –56 56

11. Multiplying by more than two factors follows same procedure as addition. Method 1: Find the product of the first two factors using procedures above; then multiply it by the third factor; then multiply its product by the fourth factor, etc. Method 2: Find the sum of the number of negative factors in the problem. If that sum is even, then the product to the problem will be + ; if that sum is odd, then the product to the problem will be  . Multiply the factors without regard to the signs. Attach the sign determined from counting the number of negatives to the product. Example 8. Multiply the following: a) (–2)(3)(–2)(–8) b) (–4)(–1)(–2)(–2)(3) (three negatives= –) (four negatives= +) Answers: –96 48 Your Turn Problem #8 Multiply the following: a) (–1)(–5)(–3)(–2) b) (–2)(3)(–2)(4)(–1) Answers: a) 30 b) –48

12. Dividing Integers In determining the sign of the quotient when dividing with integers, division follows the same rules as does multiplication. Procedure: To divide two integers: 1. Determine sign of answer: 2. Divide the integers 3. Attach sign from 1 to the answer Example 9. Divide the following: Answers: Your Turn Problem #9 Divide the following: Answers: a) –3 b) 18 c) –17 d) 7 Important Note: When dividing with zero, if zero is in the dividend (numerator), the answer is zero. If zero is in the divisor (denominator), the answer is undefined. Writing zero would be wrong. –5 12 –13 2

13. Averages Procedure: To calculate an average. 1. Find the sum of all of the numbers 2. Divide the sum by the amount of numbers given. Example 10. During a period of 5 days Anchorage, Alaska the temperature was as follows: 7C, 2C, 1C, 4, and 5C. Find the average temperature in Anchorage, Alaska for the 5 days. Solution: Step 2: 15  5 = 3 Answer: The average temperature for the 5 days was 3C. Your Turn Problem #10 The speed of the first five cars in the preliminaries in the Dayton 500 were as follows: 210mph, 180mph, 200mph , 195mph and 170mph. Calculate the average speed of the first five cars. Step 1: 7 + 2 + (1) + (4) + (5) = 14 Answer: 191 mph The End B.R. 4-23-07