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Sums of Integers with Different Signs

Add positive and negative numbers together to find a new sum.

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Sums of Integers with Different Signs
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Collette has just moved to New York City for school and can't quite believe how expensive everything is. She had $1000 in her bank account when she got to the city. She makes $520 at the coffee shop where she works her first two weeks in the city. That's supposed to be automatically deposited into her account on the 14th of the month. Her rent is $1250. It is supposed to be taken out on the 16th of the month. Will she have enough money in her account to cover the withdrawal?

In this concept, you will learn to add integers with different signs.

Adding Integers with Different Signs

positive number is a number greater than zero. It can be written with or without a + symbol in front of it. A gain in something is written with a positive number. 

negative number is a number that is less than zero. It is always written with a - symbol in front of it. A loss is written with a negative number. 

An integer is any positive whole number or its opposite. Here, opposite means sign. So a positive integer has a negative opposite and vice versa.

One way to add integers with different signs is to ignore the sign and find the difference between the two integers. Then give the answer the sign of the bigger integer.

Here is an example:

8 + -3 = ____

First, ignore the sign and find the difference between the two numbers. Difference means to subtract.

8 - 3 = 5.

Next, determine which of the original integers was bigger.

8 is bigger than 5.

Then, check the sign of the integer that is bigger.

The sign of 8 is "+". 

Finally, assign the answer that sign.

The answer is positive 5.

You can check your work by using a number line.

Start with positive 8.

Add a negative three to that by moving three units to the left.

The answer is 5.

Here is another example:

-9 + 4 = ____

First, ignore the signs and find the difference between the two values.

9 - 4 = 5

Next, determine which integer is bigger.

9 is bigger than 4.

Then, check the sign of that original integer.

9 was negative.

Finally, assign that sign to the answer.

The answer is -5.

Examples

Example 1

Earlier, you were given a problem about Collette and her money troubles.

She has to figure out if she is going to have enough money in her bank account on the 16th to make her rent of $1,250. Her bank account started with $1000 and she made $520 at work. Will she have enough?

First, she writes all the relevant quantities as positive and negative integers.

Her bank account starts positive, so that is: +1000. Her earnings are added to that, so they are positive, as well: +520. Her rent will be taken from her account, so that is a negative integer: -1250.

Next, she checks that her wages will be in the bank before her rent comes out. Since they come in the 14th, they should be. Then she adds what she starts with to her wages. 

\begin{align*}1000 + 520 = 1520\end{align*}

Then, she adds the positive amount in her bank account to the negative amount to be taken out to see if she has enough.

\begin{align*}1520+ -1250=\end{align*}

To do this problem, first she takes the difference of the two numbers, ignoring the signs:

\begin{align*}1520-1250=270\end{align*}

 Next, she checks the sign of the original integers.

The positive was bigger. That means the final answer will be positive. 

She concludes, that her final bank account will be a positive number, which means that there will still be money in her account. This means, she will have enough money in her account to cover her rent.

Example 2

Find the sum.

-2 + 8 = _____

First, ignore the signs and find the difference between the two values.

8 - 2 = 6

Next, decide which integer is bigger.

8 is bigger than 2.

Then, check the original sign of that integer.

8 is positive.

Finally, assign the answer that sign.

The answer is 6.

Example 3

Find the sum.

7 + -13 = ____

First, ignore the signs and find the difference between the two values.

13 - 7 = 6

Next, determine which integer is bigger.

13 is bigger than 7.

Then, check the sign of that original integer.

13 was negative.

Finally, assign that sign to the answer.

The answer is -6.

Example 4

Find the sum.

-22 + 10 = ____

First, ignore the signs and find the difference between the two values.

22 - 10 = 12

Next, determine which integer is bigger.

22 is bigger than 10.

Then, check the sign of that original integer.

22 was negative.

Finally, assign that sign to the answer.

The answer is -12.

Example 5

Find the sum.

-1 + 16 = ____

First, ignore the signs and find the difference between the two values.

16 - 1 = 15

Next, determine which integer is bigger.

16 is bigger than 1.

Then, check the sign of that original integer.

16 was positive.

Finally, assign that sign to the answer.

The answer is +15.

Review

Add the following pairs of integers.

  1. \begin{align*}12 + -4\end{align*}
  2. \begin{align*}-24 + 14\end{align*}
  3. \begin{align*}32 + -4\end{align*}
  4. \begin{align*}-52 + 14\end{align*}
  5. \begin{align*}67 + -64\end{align*}
  6. \begin{align*}55 + -64\end{align*}
  7. \begin{align*}78 + -84\end{align*}
  8. \begin{align*}99 + -104\end{align*}
  9. \begin{align*}-112 + 114\end{align*}
  10. \begin{align*}-19 + -4 + 12\end{align*}
  11. \begin{align*}-32 + -24 + 65\end{align*}
  12. \begin{align*}98 + -12 + -34\end{align*}
  13. \begin{align*}70 + -34 + 23\end{align*}
  14. \begin{align*}82 + -54 + 27\end{align*}
  15. \begin{align*}98 + -34 + -18\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.8. 

Resources

 

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Vocabulary

Absolute Value

The absolute value of a number is the distance the number is from zero. Absolute values are never negative.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

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