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Differences of Integers with the Same Sign

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Differences of Numbers with the Same Sign

Remember Jamie and the money? Take a look once again.

Jamie earned ten dollars cutting grass. He owes his brother twelve dollars from a movie that they went to. Does Jamie still owe his brother money if he gives him the whole ten dollars he made? How much does he still owe?

In the last Concept, you solved this problem by using a number line. We could also solve it by subtracting integers with the same sign.

In this Concept, you will learn how to solve this problem in a different way.

Guidance

In the last Concept, you learn how to subtract integers using a number line. You won’t always have time to draw a number line though, so this Concept will teach you how to subtract integers that have the same sign without the visual aid. Let’s begin.

First, let’s look at subtracting two positive numbers.

9 – 4 = ____

In this problem, if we use the language of losses and gains, we could say that we have a gain of nine and a loss of four. Because our loss is not greater than the gain, our answer is positive.

This is a key point. If the loss is greater than the gain, then our answer would be negative. In this example, the loss of four is not greater than the gain of nine, so our answer remains positive.

9 – 4 = 5

The answer is positive 5.

Here is a problem where we are still finding the difference between two positive numbers, but the loss is greater than the gain.

3 – 8 = _____

In this problem we start with a positive three or a gain of three. Then we have a loss of eight. The loss is greater than the gain that we started with.

3 – 8 = -5

Our answer is negative. It is a negative five.

Yes. Actually there is an easier way to think about subtracting any two integers. You can always think in terms of losses and gains, but if that is difficult, we can think of subtraction as being the opposite of addition-that is the key to making things simpler. Here is the hint.

********  MAJOR CONCEPT ********


Also:

  • This is true for any number, not just integers.
  • You can also rewrite any addition problem as a subtraction problem; adding a number is the same as subtracting its opposite.

What does this look like? How can we rewrite a subtraction problem as an addition problem?

3 – 8 = 3 + -8 = ____

The subtraction became addition.

Positive three plus a negative 8 is still a negative 5.

Our answer did not change even though our method of solving it did. The answer is still -5.

How can we find the difference of two negative numbers?

-6 – -3 = ____

We can find this difference in two ways. The first way is to think in terms of losses and gains. The second is to change subtraction to addition by adding the opposite. Let’s start with losses and gains.

If we think of this problem in terms of losses and gains, we start with a loss of 6.

-6

Next, we don’t add another loss, but we take away a loss. If you take away a loss, that is the same thing as a gain. So we have a gain of 3.

-6 combined with a gain of 3 = -3

Our answer is -3.

Now, let’s solve the problem by changing subtraction to addition by adding the opposite.

-6 – -3 = -6 + 3

We changed the subtraction to addition and added the opposite. The opposite of the given value of negative three is positive three. Now we can solve the addition problem.

-6 + 3 = -3

Notice that the answer is the same no matter which way you approach it. The answer is still -3.

Practice what you have learned by finding the differences of the following integer pairs.

Example A

5 – 10 = ____

Solution: - 5

Example B

14 – 7 = ____

Solution: 7

Example C

-4 – -8 = ____

Solution: 4

Here is the original problem once again.

Jamie earned ten dollars cutting grass. He owes his brother twelve dollars from a movie that they went to. Does Jamie still owe his brother money if he gives him the whole ten dollars he made? How much does he still owe?

First, let's work with the information that we have been given.

Jamie earned ten dollars.

10

He owes 12 dollars.

-12

We can put it together for an expression.

10 - 12 = 2

Jamie still owes his brother two dollars.

Vocabulary

Here are the vocabulary words in this Concept.

Sum
the result of an addition problem.
Difference
the result of a subtraction problem.

Guided Practice

Here is one for you to try on your own.

The bank started out the day with a loss of 25 points. Over the course of the day, a twelve point loss was taken away. What was the final statistic at the end of the day?

Answer

To work on this problem, we can write each loss as a negative number.

The bank started with a loss of 25 points.

-25

Over the course of the day, a 12 point loss was taken away.

- (-12)

We can put it together for a complete expression.

-25 - (-12)

The bank's ending statistic was still a loss of 13 points.

Video Review

Here are videos for review.

James Sousa, Subtracting Integers: The Basics

James Sousa, Subtracting Integers

James Sousa, Example of Subtracting Integers

Practice

Directions : Subtract each pair of integers.

1. 14 - 19

2. 24 - 19

3. -1 - 7

4. -4 - 12

5. -14 - 29

6. -24 - (-19)

7. 9 - 11

8. 13 - (-1)

9. 23 - 19

10. -31 - 15

11. -18 - (-19)

12. -14 - 6

13. -74 - 39

14. 54 - (-29)

15. 64 - 99

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