Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 5 Subtracting Integers

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1 Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 5 of this DVD before working these problems. The DVD is located at: Page 1

2 Part 1: Subtracting Positive Integers 1) Find the difference of the following positive integers. 2) Find the difference of the following positive integers. 3) Find the difference of the following positive integers. 4) Find the difference of the following positive integers. 5) Find the difference of the following positive integers. 6) Find the difference of the following positive integers. Page 2

3 Part 2: Subtracting Positive Integers from Negative Integers 7) Find the difference of the following integers. 8) Find the difference of the following integers. 9) Find the difference of the following integers. 10) Find the difference of the following integers. 11) Find the difference of the following integers. 12) Find the difference of the following integers. Page 3

4 Part 3: Subtracting Negative Integers from Positive Integers 13) Find the difference of the following integers. 14) Find the difference of the following integers. 15) Find the difference of the following integers. 16) Find the difference of the following integers. 17) Find the difference of the following integers. 18) Find the difference of the following integers. Page 4

5 Part 4: Subtracting Negative Integers from Negative Integers 19) Find the difference of the following negative integers. 20) Find the difference of the following negative integers. 21) Find the difference of the following negative integers. 22) Find the difference of the following negative integers. 23) Find the difference of the following negative integers. 24) Find the difference of the following negative integers. Page 5

6 Part 5: Evaluate and solve for the following expressions 28) 29) 30) 31) 32) 33) 34) 35) Page 6

7 Question 1) Find the difference of the following positive integers. Answer (bigger #) - (smaller #) = positive First we notice that we are subtracting a smaller number from a bigger number, which tells us we will end up with a positive answer. What we have is bigger in quantity than what will be taken away, which means we will still have some left over. We then subtract 6 from 12 as we are used to doing in regular subtraction. If we also picture a number line and start at 12, we can move 6 units to the left since we are taking away something and we see we end up at 6. Ans: 6 Page 7

8 2) Find the difference of the following positive integers. (bigger #) - (smaller #) = positive First we notice that we are subtracting a smaller number from a bigger number, which tells us we will end up with a positive answer. What we have is bigger in quantity than what will be taken away, which means we will still have some left over. We then subtract 5 from 25 as we are used to doing in regular subtraction. If we also picture a number line and start at 25, we can move 5 units to the left since we are taking away something and we see we end up at 20. Ans: 20 Page 8

9 3) Find the difference of the following positive integers. (smaller #) - (bigger #) = negative First we notice we are subtracting a bigger number from a smaller number. This means we will end up with a negative number or owing some value since we don t have enough to continue taking more of what we have away. Next we simply subtract the numbers as a regular subtraction (smaller number from bigger number) and put the negative sign in front of the answer. Another way of solving for this is we can rewrite the expression to something we are familiar to and solve that way. Recall that we will take the sign of the larger absolute value of the two integers which in this case is negative (-32). We then simply subtract the two numbers in a way we are used to doing and place the negative sign in front. Ans: -10 Page 9

10 4) Find the difference of the following positive integers. (smaller #) - (bigger #) = negative First we notice we are subtracting a bigger number from a smaller number. This means we will end up with a negative number or owing some value since we don t have enough to continue taking more of what we have away. Next we simply subtract the numbers as a regular subtraction (smaller number from bigger number) and put the negative sign in front of the answer. Another way of solving for this is we can rewrite the expression to something we are familiar to and solve that way. Recall that we will take the sign of the larger absolute value of the two integers which in this case is negative (-14). We then simply subtract the two numbers in a way we are used to doing and place the negative sign in front. Ans: -7 Page 10

11 5) Find the difference of the following positive integers. (bigger #) - (smaller #) = positive First we notice that we are subtracting a smaller number from a bigger number, which tells us we will end up with a positive answer. What we have is bigger in quantity than what will be taken away, which means we will still have some left over. We then subtract 8 from 16 as we are used to doing in regular subtraction. If we also picture a number line and start at 16, we can move 8 units to the left since we are taking away something and we see we end up at 8. Ans: 8 Page 11

12 6) Find the difference of the following positive integers. (smaller #) - (bigger #) = negative First we notice we are subtracting a bigger number from a smaller number. This means we will end up with a negative number or owing some value since we don t have enough to continue taking more of what we have away. Next we simply subtract the numbers as a regular subtraction (smaller number from bigger number) and put the negative sign in front of the answer. Another way of solving for this is we can rewrite the expression to something we are familiar to and solve that way. Recall that we will take the sign of the larger absolute value of the two integers which in this case is negative (-21). We then simply subtract the two numbers in a way we are used to doing and place the negative sign in front. Ans: -11 Page 12

13 7) Find the difference of the following integers. (negative #) - (positive #) = negative First thing we notice is we are subtracting from a negative number, which means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -14, then go 9 places to the left since we are subtracting, we notice we land on -23. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 14 already, now I am going to owe 9 more, which means in total, I owe 23, hence the -23 of something we owe. Another way of looking at this problem is to use what we learned in adding integers. We can rewrite the expression into an addition. This is the exact same thing; we are adding the total number of things we owe. So we simply add the two values and place a negative sign in front to get the total of things we owe. Ans: -23 Page 13

14 8) Find the difference of the following integers. (negative #) - (positive #) = negative First thing we notice is we are subtracting from a negative number, which means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -2, then go 10 places to the left since we are subtracting, we notice we land on -12. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 2 already, now I am going to owe 10 more, which means in total, I owe 12, hence the -12 of something we owe. Another way of looking at this problem is to use what we learned in adding integers. We can rewrite the expression into an addition. This is the exact same thing; we are adding the total number of things we owe. So we simply add the two values and place a negative sign in front to get the total of things we owe. Ans: -12 Page 14

15 9) Find the difference of the following integers. (negative #) - (positive #) = negative First thing we notice is we are subtracting from a negative number, which means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -18, then go 2 places to the left since we are subtracting, we notice we land on -20. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 18 already, now I am going to owe 2 more, which means in total, I owe 20, hence the -20 of something we owe. Another way of looking at this problem is to use what we learned in adding integers. We can rewrite the expression into an addition. This is the exact same thing; we are adding the total number of things we owe. So we simply add the two values and place a negative sign in front to get the total of things we owe. Ans: -20 Page 15

16 10) Find the difference of the following integers. (negative #) - (positive #) = negative First thing we notice is we are subtracting from a negative number, which means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -1, then go 1 place to the left since we are subtracting, we notice we land on -2. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 1 already, now I am going to owe 1 more, which means in total, I owe 2, hence the -2 of something we owe. Another way of looking at this problem is to use what we learned in adding integers. We can rewrite the expression into an addition. This is the exact same thing; we are adding the total number of things we owe. So we simply add the two values and place a negative sign in front to get the total of things we owe. Ans: -2 Page 16

17 11) Find the difference of the following integers. (negative #) - (positive #) = negative First thing we notice is we are subtracting from a negative number, which means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -7, then go 12 places to the left since we are subtracting, we notice we land on -19. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 7 already, now I am going to owe 12 more, which means in total, I owe 19, hence the -19 of something we owe. Another way of looking at this problem is to use what we learned in adding integers. We can rewrite the expression into an addition. This is the exact same thing; we are adding the total number of things we owe. So we simply add the two values and place a negative sign in front to get the total of things we owe. Ans: -19 Page 17

18 12) Find the difference of the following integers. (negative #) - (positive #) = negative First thing we notice is we are subtracting from a negative number, which means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -9, then go 11 places to the left since we are subtracting, we notice we land on -20. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 9 already, now I am going to owe 11 more, which means in total, I owe 20, hence the -20 of something we owe. Another way of looking at this problem is to use what we learned in adding integers. We can rewrite the expression into an addition. This is the exact same thing; we are adding the total number of things we owe. So we simply add the two values and place a negative sign in front to get the total of things we owe. Ans: -20 Page 18

19 13) Find the difference of the following integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number First, we notice we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. We then simply add the two integers and the result is our final answer. Ans: 15 Page 19

20 14) Find the difference of the following integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number First, we notice we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. We then simply add the two integers and the result is our final answer. Ans: 9 Page 20

21 15) Find the difference of the following integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number First, we notice we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. We then simply add the two integers and the result is our final answer. Ans: 30 Page 21

22 16) Find the difference of the following integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number First, we notice we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. We then simply add the two integers and the result is our final answer. Ans: 47 Page 22

23 17) Find the difference of the following integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number First, we notice we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. We then simply add the two integers and the result is our final answer. Ans: 20 Page 23

24 18) Find the difference of the following integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number First, we notice we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. We then simply add the two integers and the result is our final answer. Ans: 8 Page 24

25 19) Find the difference of the following negative integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number The first thing we should notice right away is that we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. Now we are left with an addition problem where we are adding two integers with opposite signs. Recall from the previous section (Adding Integers) that it doesn t matter if we are adding a negative to a positive or a positive to a negative. The sign of your result will always be carried by the sign of the largest absolute value in the expression. In this case, the absolute value of 15 is greater so the final result will be negative. Remember that when adding two integers with opposite signs, we are really in fact subtracting. So next we subtract the absolute values from each other as we would with a regular subtraction. Once you get your result don t forget to indicate the sign of the result you found earlier, which is negative. Ans: -10 Page 25

26 20) Find the difference of the following negative integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number The first thing we should notice right away is that we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. Now we are left with an addition problem where we are adding two integers with opposite signs. Recall from the previous section (Adding Integers) that it doesn t matter if we are adding a negative to a positive or a positive to a negative. The sign of your result will always be carried by the sign of the largest absolute value in the expression. In this case, the absolute value of 8 is greater so the final result will be positive. Remember that when adding two integers with opposite signs, we are really in fact subtracting. So next we subtract the absolute values from each other as we would with a regular subtraction. Once you get your result don t forget to indicate the sign of the result you found earlier, which is positive. Ans: 6 Page 26

27 21) Find the difference of the following negative integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number The first thing we should notice right away is that we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. Now we are left with an addition problem where we are adding two integers with opposite signs. Recall from the previous section (Adding Integers) that it doesn t matter if we are adding a negative to a positive or a positive to a negative. The sign of your result will always be carried by the sign of the largest absolute value in the expression. In this case, the absolute value of 50 is greater so the final result will be negative. Remember that when adding two integers with opposite signs, we are really in fact subtracting. So next we subtract the absolute values from each other as we would with a regular subtraction. Once you get your result don t forget to indicate the sign of the result you found earlier, which is negative. Ans: -30 Page 27

28 22) Find the difference of the following negative integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number The first thing we should notice right away is that we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. Now we are left with an addition problem where we are adding two integers with opposite signs. Recall from the previous section (Adding Integers) that it doesn t matter if we are adding a negative to a positive or a positive to a negative. The sign of your result will always be carried by the sign of the largest absolute value in the expression. In this case, the absolute value of 14 is greater so the final result will be positive. Remember that when adding two integers with opposite signs, we are really in fact subtracting. So next we subtract the absolute values from each other as we would with a regular subtraction. Once you get your result don t forget to indicate the sign of the result you found earlier, which is positive. Ans: 11 Page 28

29 23) Find the difference of the following negative integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number The first thing we should notice right away is that we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. Now we are left with an addition problem where we are adding two integers with opposite signs. Recall from the previous section (Adding Integers) that it doesn t matter if we are adding a negative to a positive or a positive to a negative. The sign of your result will always be carried by the sign of the largest absolute value in the expression. In this case, the absolute value of 9 is greater so the final result will be negative. Remember that when adding two integers with opposite signs, we are really in fact subtracting. So next we subtract the absolute values from each other as we would with a regular subtraction. Once you get your result don t forget to indicate the sign of the result you found earlier, which is negative. Ans: -1 Page 29

30 24) Find the difference of the following negative integers. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number The first thing we should notice right away is that we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. Now we are left with an addition problem where we are adding two integers with opposite signs. Recall from the previous section (Adding Integers) that it doesn t matter if we are adding a negative to a positive or a positive to a negative. The sign of your result will always be carried by the sign of the largest absolute value in the expression. In this case, the absolute value of 22 is greater so the final result will be negative. Remember that when adding two integers with opposite signs, we are really in fact subtracting. So next we subtract the absolute values from each other as we would with a regular subtraction. Once you get your result don t forget to indicate the sign of the result you found earlier, which is negative. Ans: -2 Page 30

31 28) First, let s evaluate the expression by substituting the values expressed by the letters. We also notice there is an absolute value expression. We substitute for that as well. Now we can use what we have learned previously to find the difference. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number First, we notice we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. We then simply add the two integers and the result is our final answer. Ans: 17 Page 31

32 29) First, let s evaluate the expression by substituting the values expressed by the letters. Now we can use what we have learned previously to find the difference. (bigger #) - (smaller #) = positive First we notice that we are subtracting a smaller number from a bigger number, which tells us we will end up with a positive answer. What we have is bigger in quantity than what will be taken away, which means we will still have some left over. We then subtract 4 from 10 as we are used to doing in regular subtraction. If we also picture a number line and start at 10, we can move 4 units to the left since we are taking away something and we see we end up at 6. Ans: 6 Page 32

33 30) First, let s evaluate the expression by substituting the values expressed by the letters. Now we can use what we have learned previously to find the difference. (negative #) - (positive #) = negative First thing we notice is we are subtracting from a negative number, which means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -12, then go 4 places to the left since we are subtracting, we notice we land on -16. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 12 already, now I am going to owe 4 more, which means in total, I owe 16, hence the -16 of something we owe. Another way of looking at this problem is to use what we learned in adding integers. We can rewrite the expression into an addition. This is the exact same thing; we are adding the total number of things we owe. So we simply add the two values and place a negative sign in front to get the total of things we owe. Ans: -16 Page 33

34 31) First, let s evaluate the expression by substituting the values expressed by the letters. We also notice there is an absolute value expression. We substitute for that as well. Now we can use what we have learned previously to find the difference. We now have three integers that we need to find the difference for. We can make it easier by tackling two integers at a time from left to right and then finding the difference between that result and the remaining integer in the expression. (smaller #) - (bigger #) = negative result: First we notice we are subtracting a bigger number from a smaller number. This means we will end up with a negative number or owing some value since we don t have enough to continue taking more of what we have away. Next we simply subtract the numbers as a regular subtraction (smaller number from bigger number) and put the negative sign in front of the answer. Re-evaluate original expression with new result. (negative #) - (positive #) = negative First thing we notice is we are subtracting from a negative number, which means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -5, then go 10 places to the left since we are Page 34

35 subtracting, we notice we land on -15. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 5 already, now I am going to owe 10 more, which means in total, I owe 15, hence the -15 of something we owe. Ans: -15 Page 35

36 32) First, let s evaluate the expression by substituting the values expressed by the letters. Now we can use what we have learned previously to find the difference. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number The first thing we should notice right away is that we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. Now we are left with an addition problem where we are adding two integers with opposite signs. Recall from the previous section (Adding Integers) that it doesn t matter if we are adding a negative to a positive or a positive to a negative. The sign of your result will always be carried by the sign of the largest absolute value in the expression. In this case both absolute values are equal. This should give us a clue to what the final answer will be. Remember that when adding two integers with opposite signs, we are really in fact subtracting. So next we subtract the absolute values from each other as we would with a regular subtraction. We see that we end up with 0, which is why there was no clear sign to pick from since zero is neutral (neither negative or positive). Page 36

37 Ans: 0 33) First, let s evaluate the expression by substituting the values expressed by the letters. We also notice there is an absolute value expression. We substitute for that as well. Now we can use what we have learned previously to find the difference. We now have three integers that we need to find the difference for. We can make it easier by tackling two integers at a time from left to right and then finding the difference between that result and the remaining integer in the expression. (negative #) - (positive #) = negative result: Next thing we notice is we are subtracting from a negative number, which means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -8, then go 9 places to the left since we are subtracting, we notice we land on -17. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 8 already, now I am going to owe 9 more, which means in total, I owe 17, hence the -17 of something we owe. Re-evaluate original expression with new result. Page 37

38 (negative #) - (positive #) = negative Next thing we notice is we are subtracting from a negative number. We just finished doing a similar problem. This means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -17, then go 4 places to the left since we are subtracting, we notice we land on -21. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 17 already, now I am going to owe 4 more, which means in total, I owe 21, hence the -21 of something we owe. Ans: -21 Page 38

39 34) First, let s evaluate the expression by substituting the values expressed by the letters. Now we can use what we have learned previously to find the difference. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number We now have three integers that we need to find the difference for. We can make it easier by tackling two integers at a time from left to right and then finding the difference between that result and the remaining integer in the expression. First, we notice we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next to each other cancel out and you end up with a plus sign. We then simply add the two integers and the result is our final answer. result: Re-evaluate original expression with new result. (a #) - (negative #) = (a #) + (positive #) We are adding the opposite of the negative number Next, we notice we are attempting to subtract with a negative number. This means we have a minus sign in front of a negative and as you recall, this is where we are going to add the opposite. This means the two negative signs next Page 39

40 to each other cancel out and you end up with a plus sign. We then simply add the two integers and the result is our final answer. Ans: 30 Page 40

41 35) First, let s evaluate the expression by substituting the values expressed by the letters. Now we can use what we have learned previously to find the difference. (negative #) - (positive #) = negative First thing we notice is we are subtracting from a negative number, which means we are going to increase the number of things we owe by the number we are subtracting with. If we visualize a number line and start at -12, then go 10 places to the left since we are subtracting, we notice we land on -22. So we notice the result is simply the sum of the two numbers with a negative sign in front. This is because we are taking away from something that is already negative. I owed 12 already, now I am going to owe 10 more, which means in total, I owe 22, hence the -22 of something we owe. Another way of looking at this problem is to use what we learned in adding integers. We can rewrite the expression into an addition. This is the exact same thing; we are adding the total number of things we owe. So we simply add the two values and place a negative sign in front to get the total of things we owe. Ans: -22 Page 41

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