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7 signed numbers addition and subtraction

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The document discusses signed numbers and how they are used to represent the direction or type (increase/decrease) of measurements and transactions. Signed numbers use positive and negative signs, with positive (+) representing increases/deposits and negative (-) representing decreases/withdrawals. The combining of signed numbers into a single number is called the combining operation, and the absolute value of a number represents its magnitude without sign.

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Signed Numbers
We track the “directions” of measurements by giving them positive (+) or negative (-) signs. Signed Numbers
Signed Numbers We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases,
Signed Numbers We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies,
Signed Numbers We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on.
Signed Numbers We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers. Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account
Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account and “–” for withdraw or debit from the account, We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account and “–” for withdraw or debit from the account, then the transactions may be listed as: +400, –350. (In this section we will use red color for negative numbers for emphasis.) We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account and “–” for withdraw or debit from the account, then the transactions may be listed as: +400, –350. (In this section we will use red color for negative numbers for emphasis.) The amount left in the account is $50. We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account and “–” for withdraw or debit from the account, then the transactions may be listed as: +400, –350. (In this section we will use red color for negative numbers for emphasis.) The amount left in the account is $50. This is a surplus so its +50. We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account and “–” for withdraw or debit from the account, then the transactions may be listed as: +400, –350. (In this section we will use red color for negative numbers for emphasis.) The amount left in the account is $50. This is a surplus so its +50. We write the entire transactions as +400 – 350 = +50. We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account?
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600.
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200.
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200.
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left?
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left? The transactions are listed as +400, +200, –350, +250, –600.
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left? The transactions are listed as +400, +200, –350, +250, –600. To find the final balance in the account, we note that the total of the deposits is +850
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left? The transactions are listed as +400, +200, –350, +250, –600. To find the final balance in the account, we note that the total of the deposits is +850 and the total of the withdrawals is –950,
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left? The transactions are listed as +400, +200, –350, +250, –600. To find the final balance in the account, we note that the total of the deposits is +850 and the total of the withdrawals is –950, so the account is short of $100, or there is “–100” left in the account.
Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left? The transactions are listed as +400, +200, –350, +250, –600. To find the final balance in the account, we note that the total of the deposits is +850 and the total of the withdrawals is –950, so the account is short of $100, or there is “–100” left in the account. We write these transactions as +400 + 200 – 350 + 250 – 600 = –100.
Signed Numbers The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Example B: +100 + 200 = +300, The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
The above operation of totaling two or more signed numbers into a single signed number is called the combining operation. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number.
Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. In example A of the bank account, if we are only interested in the amount of the transactions but not the type of transactions, this amount is called the absolute value. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. In example A of the bank account, if we are only interested in the amount of the transactions but not the type of transactions, this amount is called the absolute value. The absolute value of a number x is written as |x|. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. In example A of the bank account, if we are only interested in the amount of the transactions but not the type of transactions, this amount is called the absolute value. The absolute value of a number x is written as |x|. Hence, |500| = 500, The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. In example A of the bank account, if we are only interested in the amount of the transactions but not the type of transactions, this amount is called the absolute value. The absolute value of a number x is written as |x|. Hence, |500| = 500, |-350| = 350, The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. In example A of the bank account, if we are only interested in the amount of the transactions but not the type of transactions, this amount is called the absolute value. The absolute value of a number x is written as |x|. Hence, |500| = 500, |-350| = 350, |-600| = 600, etc.. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
Signed Numbers Rules for Combining Signed Numbers
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign,
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers.
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300,
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value,
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers.
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200,
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 There are different ways to combine multiple signed numbers, we may combine them from left to right.
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 There are different ways to combine multiple signed numbers, we may combine them from left to right.
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 There are different ways to combine multiple signed numbers, we may combine them from left to right.
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 = 10 There are different ways to combine multiple signed numbers, we may combine them from left to right.
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 = 10 To combine many numbers, an alternative way is to do it as in example A of the bank transactions. There are different ways to combine multiple signed numbers, we may combine them from left to right.
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 = 10 To combine many numbers, an alternative way is to do it as in example A of the bank transactions. That is, we combined all the positive ones (deposits) first, There are different ways to combine multiple signed numbers, we may combine them from left to right.
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 = 10 To combine many numbers, an alternative way is to do it as in example A of the bank transactions. That is, we combined all the positive ones (deposits) first, then combine all the negative ones (withdrawals), There are different ways to combine multiple signed numbers, we may combine them from left to right.
Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 = 10 To combine many numbers, an alternative way is to do it as in example A of the bank transactions. That is, we combined all the positive ones (deposits) first, then combine all the negative ones (withdrawals), then combine the two results. There are different ways to combine multiple signed numbers, we may combine them from left to right.
* This way is easier if there are many numbers to combine. Signed Numbers
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s.
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4 +2
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4 +2 –4 –19
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4 +2 –4 –19 in pairs again
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4 +2 –4 –19 in pairs again = –2 – 23 = –25
* This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4 +2 –4 –19 in pairs again = –2 – 23 = –25 (Two Method Strategy) Use these two methods to cross check the + or – of multiple signed numbers.
Addition and Subtraction of Signed Numbers
Addition of Signed Numbers Addition and Subtraction of Signed Numbers
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Addition and Subtraction of Signed Numbers
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, Addition and Subtraction of Signed Numbers
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , Addition and Subtraction of Signed Numbers
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)  remove “( )” = 2 – 6
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)  remove “( )” = 2 – 6 = – 4
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)  remove “( )” = 2 – 6 = – 4 d. –4  (–8)
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)  remove “( )” = 2 – 6 = – 4 d. –4  (–8)  remove “( )” = – 4 – 8
Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)  remove “( )” = 2 – 6 = – 4 d. –4  (–8)  remove “( )” = – 4 – 8 = –12
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers.
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other.
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x.
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x.
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x. So the opposite of 6 is –6 , the opposite of –12 is –(–12) = 12.
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x. So the opposite of 6 is –6 , the opposite of –12 is –(–12) = 12. Note that the opposite of a negative number is positive.
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x. So the opposite of 6 is –6 , the opposite of –12 is –(–12) = 12. Note that the opposite of a negative number is positive. Rule for Subtraction of Signed Numbers: To subtract a signed number x, remove the parenthesis and combine with the opposite of x,
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x. So the opposite of 6 is –6 , the opposite of –12 is –(–12) = 12. Note that the opposite of a negative number is positive. Rule for Subtraction of Signed Numbers: To subtract a signed number x, remove the parenthesis and combine with the opposite of x, that is, a – (+b) = a – b opposite
Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x. So the opposite of 6 is –6 , the opposite of –12 is –(–12) = 12. Note that the opposite of a negative number is positive. Rule for Subtraction of Signed Numbers: To subtract a signed number x, remove the parenthesis and combine with the opposite of x, that is, a – (+b) = a – b a – (–b) = a + b opposite opposite
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers:
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b Example C. Remove the parentheses, then combine. a. –6 – (–8) – (–2) – (9)
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b Example C. Remove the parentheses, then combine. a. –6 – (–8) – (–2) – (9) remove “( )” = –6 + 8 + 2 – 9
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b Example C. Remove the parentheses, then combine. a. –6 – (–8) – (–2) – (9) remove “( )” = –6 + 8 + 2 – 9 = 8 + 2 – 6 – 9
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b Example C. Remove the parentheses, then combine. a. –6 – (–8) – (–2) – (9) remove “( )” = –6 + 8 + 2 – 9 = 8 + 2 – 6 – 9 = 10 – 15
Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b Example C. Remove the parentheses, then combine. a. –6 – (–8) – (–2) – (9) remove “( )” = –6 + 8 + 2 – 9 = 8 + 2 – 6 – 9 = 10 – 15 = –5
b. 2  (–4) – (–8) – (5)  (–9 ) Addition and Subtraction of Signed Numbers
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 Addition and Subtraction of Signed Numbers
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 Addition and Subtraction of Signed Numbers
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 Addition and Subtraction of Signed Numbers
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols.
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts.
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol.
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9)
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )”
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1)
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1 = – 5
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1 = – 5 b. (–6 – 8) – 9
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1 = – 5 b. (–6 – 8) – 9 do the calculation inside the “( )” = (–14) – 9
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1 = – 5 b. (–6 – 8) – 9 do the calculation inside the “( )” = (–14) – 9 remove parentheses = – 14 – 9
b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1 = – 5 b. (–6 – 8) – 9 do the calculation inside the “( )” = (–14) – 9 remove parentheses = – 14 – 9 = – 23
Addition and Subtraction of Signed Numbers If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] do the calculation inside the “( )” = 2 – [–6 – (17)] If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] do the calculation inside the “( )” = 2 – [–6 – (17)] remove parentheses = 2 – [– 6 – 17] If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] do the calculation inside the “( )” = 2 – [–6 – (17)] remove parentheses = 2 – [– 6 – 17] do the calculation inside the “[ ]” = 2 – [– 23] If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] do the calculation inside the “( )” = 2 – [–6 – (17)] remove parentheses = 2 – [– 6 – 17] do the calculation inside the “[ ]” = 2 – [– 23] = 2 + 23 If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] do the calculation inside the “( )” = 2 – [–6 – (17)] remove parentheses = 2 – [– 6 – 17] do the calculation inside the “[ ]” = 2 – [– 23] = 2 + 23 = 25 If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
Exercise A. Combine 1. 2 + 3 2. 10 + 6 3. 34 + 21 + 4 + 17 4. –6 –2 5. –11 – 5 6. –14 –15 7. –26 –15 – 5 –14 8. –3 + 2 9. 5 –11 10. –14 + 15 11. 26 –15 12. 12 – 13 13. –23 +18 B. Combine by moving the positive numbers to the front first. Combine the positive numbers, the negative numbers separately then then combine the two results. 14. 23 – 18 +7 –12 15. –6 –2 + 10 + 6 16. –14 + 23 –15 – 3 +12 17. –26 + 15 –5 –14 + 9 18. 19 – 13 – 9 – 3 + 15 19. –6 + 19 – 15 + 5 – 9 20. – 4 + 7 – 23 + 8 + 17 – 8 + 6 + 9 – 22 – 2 21. Try to get the same answer for #20 by combining two numbers at a time without separating the positive numbers from the negative numbers. Signed Numbers

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7 signed numbers addition and subtraction

  • 2. We track the “directions” of measurements by giving them positive (+) or negative (-) signs. Signed Numbers
  • 3. Signed Numbers We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases,
  • 4. Signed Numbers We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies,
  • 5. Signed Numbers We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on.
  • 6. Signed Numbers We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
  • 7. Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
  • 8. We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers. Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account
  • 9. Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account and “–” for withdraw or debit from the account, We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
  • 10. Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account and “–” for withdraw or debit from the account, then the transactions may be listed as: +400, –350. (In this section we will use red color for negative numbers for emphasis.) We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
  • 11. Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account and “–” for withdraw or debit from the account, then the transactions may be listed as: +400, –350. (In this section we will use red color for negative numbers for emphasis.) The amount left in the account is $50. We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
  • 12. Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account and “–” for withdraw or debit from the account, then the transactions may be listed as: +400, –350. (In this section we will use red color for negative numbers for emphasis.) The amount left in the account is $50. This is a surplus so its +50. We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
  • 13. Signed Numbers Example A: a. We deposited $400 into a bank account then withdrew $350 from the account, write the transactions using signed numbers. How much is left in the account? We use “+” for deposit or having surplus in the account and “–” for withdraw or debit from the account, then the transactions may be listed as: +400, –350. (In this section we will use red color for negative numbers for emphasis.) The amount left in the account is $50. This is a surplus so its +50. We write the entire transactions as +400 – 350 = +50. We track the “directions” of measurements by giving them positive (+) or negative (-) sign. Signed measurements represent amounts of increases versus decreases, surpluses versus deficiencies, credits versus debits and so on. Numbers with signs are called signed numbers.
  • 14. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account?
  • 15. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600.
  • 16. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200.
  • 17. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200.
  • 18. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200
  • 19. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left?
  • 20. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left? The transactions are listed as +400, +200, –350, +250, –600.
  • 21. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left? The transactions are listed as +400, +200, –350, +250, –600. To find the final balance in the account, we note that the total of the deposits is +850
  • 22. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left? The transactions are listed as +400, +200, –350, +250, –600. To find the final balance in the account, we note that the total of the deposits is +850 and the total of the withdrawals is –950,
  • 23. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left? The transactions are listed as +400, +200, –350, +250, –600. To find the final balance in the account, we note that the total of the deposits is +850 and the total of the withdrawals is –950, so the account is short of $100, or there is “–100” left in the account.
  • 24. Signed Numbers b. We deposited $400 into the account then withdrew $600 from the account, write the transactions using signed numbers. How much is left in the account? The transactions may be listed as: +400, –600. The account is short by $200. This is a deficiency so its –200. We write the entire transactions as +400 – 600 = –200 c. We deposited $400 in a bank account, later deposited another $200, then withdrew $350, then deposited $250, then withdrew $600, make a list of the transactions using signed numbers. How much is left? The transactions are listed as +400, +200, –350, +250, –600. To find the final balance in the account, we note that the total of the deposits is +850 and the total of the withdrawals is –950, so the account is short of $100, or there is “–100” left in the account. We write these transactions as +400 + 200 – 350 + 250 – 600 = –100.
  • 25. Signed Numbers The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 26. Signed Numbers Example B: +100 + 200 = +300, The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 27. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 28. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 29. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 30. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number.
  • 31. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 32. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 33. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. In example A of the bank account, if we are only interested in the amount of the transactions but not the type of transactions, this amount is called the absolute value. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 34. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. In example A of the bank account, if we are only interested in the amount of the transactions but not the type of transactions, this amount is called the absolute value. The absolute value of a number x is written as |x|. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 35. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. In example A of the bank account, if we are only interested in the amount of the transactions but not the type of transactions, this amount is called the absolute value. The absolute value of a number x is written as |x|. Hence, |500| = 500, The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 36. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. In example A of the bank account, if we are only interested in the amount of the transactions but not the type of transactions, this amount is called the absolute value. The absolute value of a number x is written as |x|. Hence, |500| = 500, |-350| = 350, The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 37. Signed Numbers Example B: +100 + 200 = +300, –100 – 200 = –300 +500 – 300 = +200, –500 + 300 = –200 A number written without a sign is treated as a positive number. Therefore, 100 + 200 is the same as +100 + 200 and both combined to 300. In order to state precisely the rules for combining signed numbers, we introduce the notion of absolute values. In example A of the bank account, if we are only interested in the amount of the transactions but not the type of transactions, this amount is called the absolute value. The absolute value of a number x is written as |x|. Hence, |500| = 500, |-350| = 350, |-600| = 600, etc.. The above operation of totaling two or more signed numbers into a single signed number is called the combining operation.
  • 38. Signed Numbers Rules for Combining Signed Numbers
  • 39. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign,
  • 40. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers.
  • 41. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300,
  • 42. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300
  • 43. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value,
  • 44. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers.
  • 45. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200,
  • 46. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200
  • 47. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 There are different ways to combine multiple signed numbers, we may combine them from left to right.
  • 48. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 There are different ways to combine multiple signed numbers, we may combine them from left to right.
  • 49. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 There are different ways to combine multiple signed numbers, we may combine them from left to right.
  • 50. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 = 10 There are different ways to combine multiple signed numbers, we may combine them from left to right.
  • 51. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 = 10 To combine many numbers, an alternative way is to do it as in example A of the bank transactions. There are different ways to combine multiple signed numbers, we may combine them from left to right.
  • 52. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 = 10 To combine many numbers, an alternative way is to do it as in example A of the bank transactions. That is, we combined all the positive ones (deposits) first, There are different ways to combine multiple signed numbers, we may combine them from left to right.
  • 53. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 = 10 To combine many numbers, an alternative way is to do it as in example A of the bank transactions. That is, we combined all the positive ones (deposits) first, then combine all the negative ones (withdrawals), There are different ways to combine multiple signed numbers, we may combine them from left to right.
  • 54. Signed Numbers Rules for Combining Signed Numbers I. To combine two or more numbers of the same signs, keep the sign, take the sum of the absolute values of the numbers. +100 + 200 = +300, –100 – 200 = –300 II. To combine two numbers of different signs, keep the sign of the number with larger absolute value, take the difference of the absolute values of the numbers. +500 – 300 = +200, –500 + 300 = –200 Example C: 8 – 9 + 11 = –1 + 11 = 10 To combine many numbers, an alternative way is to do it as in example A of the bank transactions. That is, we combined all the positive ones (deposits) first, then combine all the negative ones (withdrawals), then combine the two results. There are different ways to combine multiple signed numbers, we may combine them from left to right.
  • 55. * This way is easier if there are many numbers to combine. Signed Numbers
  • 56. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers
  • 57. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11
  • 58. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front
  • 59. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11
  • 60. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11
  • 61. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36
  • 62. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25
  • 63. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s.
  • 64. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs
  • 65. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4
  • 66. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4 +2
  • 67. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4 +2 –4 –19
  • 68. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4 +2 –4 –19 in pairs again
  • 69. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4 +2 –4 –19 in pairs again = –2 – 23 = –25
  • 70. * This way is easier if there are many numbers to combine. * When doing this, it helps to move all the positive ones to the front and the negative ones to the back. Signed Numbers Example D: 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 positive ones to the front = 7 + 14 + 15 – 11 – 12 – 19 – 8 – 11 = 36 – 61 = –25 Another method for combining many signed numbers is to do two in groups of two’s. Hence 7 – 11 + 14 – 12 + 15 – 19 – 8 – 11 group them in pairs = –4 +2 –4 –19 in pairs again = –2 – 23 = –25 (Two Method Strategy) Use these two methods to cross check the + or – of multiple signed numbers.
  • 71. Addition and Subtraction of Signed Numbers
  • 72. Addition of Signed Numbers Addition and Subtraction of Signed Numbers
  • 73. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Addition and Subtraction of Signed Numbers
  • 74. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, Addition and Subtraction of Signed Numbers
  • 75. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , Addition and Subtraction of Signed Numbers
  • 76. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers
  • 77. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)
  • 78. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4
  • 79. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9
  • 80. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)
  • 81. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3
  • 82. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4
  • 83. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)
  • 84. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)  remove “( )” = 2 – 6
  • 85. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)  remove “( )” = 2 – 6 = – 4
  • 86. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)  remove “( )” = 2 – 6 = – 4 d. –4  (–8)
  • 87. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)  remove “( )” = 2 – 6 = – 4 d. –4  (–8)  remove “( )” = – 4 – 8
  • 88. Addition of Signed Numbers Adding signed numbers is the same as combining the numbers. Rule for Addition of Signed Numbers: To add two signed numbers, remove the parenthesis and combine the numbers, that is, a  (+b) = a + b , a + (–b) = a – b Addition and Subtraction of Signed Numbers Example A. Remove parentheses then combine. a. 5  (+4)  remove “( )” = 5 + 4 = 9 b. –7  (3)  remove “( )” = –7 + 3 = – 4 c. 2  (–6)  remove “( )” = 2 – 6 = – 4 d. –4  (–8)  remove “( )” = – 4 – 8 = –12
  • 89. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6
  • 90. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6
  • 91. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6
  • 92. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17
  • 93. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4
  • 94. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers
  • 95. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers.
  • 96. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other.
  • 97. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x.
  • 98. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x.
  • 99. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x. So the opposite of 6 is –6 , the opposite of –12 is –(–12) = 12.
  • 100. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x. So the opposite of 6 is –6 , the opposite of –12 is –(–12) = 12. Note that the opposite of a negative number is positive.
  • 101. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x. So the opposite of 6 is –6 , the opposite of –12 is –(–12) = 12. Note that the opposite of a negative number is positive. Rule for Subtraction of Signed Numbers: To subtract a signed number x, remove the parenthesis and combine with the opposite of x,
  • 102. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x. So the opposite of 6 is –6 , the opposite of –12 is –(–12) = 12. Note that the opposite of a negative number is positive. Rule for Subtraction of Signed Numbers: To subtract a signed number x, remove the parenthesis and combine with the opposite of x, that is, a – (+b) = a – b opposite
  • 103. Addition and Subtraction of Signed Numbers e. 5  (4) + (–3) + (–8) + 4 – 6 remove ( )’s = 5 + 4 – 3 – 8 + 4 – 6 = 5 + 4 + 4 – 3 – 8 – 6 = 13 – 17 = –4 Subtraction of Signed Numbers For subtraction of signed numbers, we need the notion of “opposite” numbers. The numbers x and –x are said to be the opposite or the negative of each other. The opposite of x is –x. The opposite of –x is –(–x) = x. So the opposite of 6 is –6 , the opposite of –12 is –(–12) = 12. Note that the opposite of a negative number is positive. Rule for Subtraction of Signed Numbers: To subtract a signed number x, remove the parenthesis and combine with the opposite of x, that is, a – (+b) = a – b a – (–b) = a + b opposite opposite
  • 104. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)
  • 105. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4
  • 106. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1
  • 107. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)
  • 108. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7
  • 109. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10
  • 110. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)
  • 111. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5
  • 112. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7
  • 113. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers:
  • 114. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b
  • 115. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b
  • 116. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b Example C. Remove the parentheses, then combine. a. –6 – (–8) – (–2) – (9)
  • 117. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b Example C. Remove the parentheses, then combine. a. –6 – (–8) – (–2) – (9) remove “( )” = –6 + 8 + 2 – 9
  • 118. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b Example C. Remove the parentheses, then combine. a. –6 – (–8) – (–2) – (9) remove “( )” = –6 + 8 + 2 – 9 = 8 + 2 – 6 – 9
  • 119. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b Example C. Remove the parentheses, then combine. a. –6 – (–8) – (–2) – (9) remove “( )” = –6 + 8 + 2 – 9 = 8 + 2 – 6 – 9 = 10 – 15
  • 120. Addition and Subtraction of Signed Numbers Example B. Remove parentheses then combine. a. 5 – (+4)  remove “( )”, change to opposite = 5 – 4 = 1 b. 3 – (–7)  remove “( )”, change to opposite = 3 + 7 = 10 c. –12 – (–5)  remove “( )”, change to opposite = –12 + 5 = –7 Summary for removing parentheses for addition and subtraction of signed numbers: For addition: a  (+b) = a + b a + (–b) = a – b For subtraction: a – (+b) = a – b a – (–b) = a + b Example C. Remove the parentheses, then combine. a. –6 – (–8) – (–2) – (9) remove “( )” = –6 + 8 + 2 – 9 = 8 + 2 – 6 – 9 = 10 – 15 = –5
  • 121. b. 2  (–4) – (–8) – (5)  (–9 ) Addition and Subtraction of Signed Numbers
  • 122. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 Addition and Subtraction of Signed Numbers
  • 123. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 Addition and Subtraction of Signed Numbers
  • 124. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 Addition and Subtraction of Signed Numbers
  • 125. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers
  • 126. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols.
  • 127. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts.
  • 128. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol.
  • 129. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9)
  • 130. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )”
  • 131. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1)
  • 132. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1
  • 133. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1 = – 5
  • 134. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1 = – 5 b. (–6 – 8) – 9
  • 135. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1 = – 5 b. (–6 – 8) – 9 do the calculation inside the “( )” = (–14) – 9
  • 136. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1 = – 5 b. (–6 – 8) – 9 do the calculation inside the “( )” = (–14) – 9 remove parentheses = – 14 – 9
  • 137. b. 2  (–4) – (–8) – (5)  (–9 ) remove the ( )’s = 2 – 4 + 8 – 5 – 9 = 2 + 8 – 4 – 5 – 9 = 10 – 18 = –8 Addition and Subtraction of Signed Numbers The ( ), [ ], or { } are grouping symbols. Each set of symbol must contain both the left-hand and right-hand parts. Each set encloses calculations that are to be done within the symbol. Example D. a. –6 – (8 – 9) do the calculation inside the “( )” = – 6 – (– 1) remove parentheses = – 6 + 1 = – 5 b. (–6 – 8) – 9 do the calculation inside the “( )” = (–14) – 9 remove parentheses = – 14 – 9 = – 23
  • 138. Addition and Subtraction of Signed Numbers If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
  • 139. Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
  • 140. Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] do the calculation inside the “( )” = 2 – [–6 – (17)] If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
  • 141. Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] do the calculation inside the “( )” = 2 – [–6 – (17)] remove parentheses = 2 – [– 6 – 17] If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
  • 142. Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] do the calculation inside the “( )” = 2 – [–6 – (17)] remove parentheses = 2 – [– 6 – 17] do the calculation inside the “[ ]” = 2 – [– 23] If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
  • 143. Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] do the calculation inside the “( )” = 2 – [–6 – (17)] remove parentheses = 2 – [– 6 – 17] do the calculation inside the “[ ]” = 2 – [– 23] = 2 + 23 If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
  • 144. Addition and Subtraction of Signed Numbers Example E. 2 – [–6 – (8 + 9)] do the calculation inside the “( )” = 2 – [–6 – (17)] remove parentheses = 2 – [– 6 – 17] do the calculation inside the “[ ]” = 2 – [– 23] = 2 + 23 = 25 If there is a set of grouping symbol inside another set of grouping symbol, the inner set is to be calculated first.
  • 145. Exercise A. Combine 1. 2 + 3 2. 10 + 6 3. 34 + 21 + 4 + 17 4. –6 –2 5. –11 – 5 6. –14 –15 7. –26 –15 – 5 –14 8. –3 + 2 9. 5 –11 10. –14 + 15 11. 26 –15 12. 12 – 13 13. –23 +18 B. Combine by moving the positive numbers to the front first. Combine the positive numbers, the negative numbers separately then then combine the two results. 14. 23 – 18 +7 –12 15. –6 –2 + 10 + 6 16. –14 + 23 –15 – 3 +12 17. –26 + 15 –5 –14 + 9 18. 19 – 13 – 9 – 3 + 15 19. –6 + 19 – 15 + 5 – 9 20. – 4 + 7 – 23 + 8 + 17 – 8 + 6 + 9 – 22 – 2 21. Try to get the same answer for #20 by combining two numbers at a time without separating the positive numbers from the negative numbers. Signed Numbers