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## Sums of opposite numbers and distance from zero

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Smith Middle School has started a school store. The first day that the school store was open, the students had many, many customers. They sold pencils, erasers and pens to sixth, seventh and eighth graders. At the end of the first day, Mallory calculated the total amount of sales and found that the students had raised $87.00. Not bad for a first day. On the second day, Trevor worked in the store. He realized that the school store still owed Mr. Janus$20.00 from the day before and so he took that out of the total sales. Then he sold $45.00 more dollars in supplies to students. On the third day, Kelly worked in the store and there were three returns for a total of$3.00. Then Kelly sold \$25.00 in supplies.

When the students met with Mr. Janus on Wednesday afternoon, here were the numbers that they wrote on the board.

Monday +87.00

Tuesday -20.00

+45.00

Wednesday -3.00

+ 25.00

The students began to figure out their total earnings for the week.

Use this Concept to learn how to add and subtract integers. Then you will understand how to calculate the total earnings of the school store.

### Guidance

You are already familiar with whole numbers such as 0, 1, 2, 3 and so on. Well, in this lesson you will learn how to work with integers. First, let’s look at defining the term integer.

What is an integer?

Integers include the set of whole numbers and their opposites: {... -3, -2, -1, 0, 1, 2, 3, ...}. Notice that as we work with integers we are working with both positive and negative numbers. Sometimes, you will hear people talk about integers as only being negative numbers. The real thing to remember is that integers include the set of whole numbers this includes both positive and negative numbers.

We can work with integers when adding, subtracting, multiplying and dividing. In this lesson, you will learn to add and subtract integers. When working with integers and operations, it’s important to keep track of the sign of the number.

Let’s look at adding and subtracting integers.

When we think about adding and subtracting integers, it is helpful to think about money and this will help you to think in terms of losses and gains. We can think of a gain as positive and a loss as negative.

If you have two losses, then you have more loss and that is negative.

If you have two gains, then you have a gain and that is positive.

When you have a loss and then a gain, you will have to look at how much of a loss or how much of a gain is occurring to figure out whether your answer is positive or negative.

You can think of these statements as some of the basic rules of working with integers.

Take a look at this situation.

\begin{align*}-5+ -6= -11\end{align*}

Here we have a loss of five and another loss of six so we end up with a total loss of 11.

This was is a little harder. Think through it carefully.

Add: \begin{align*}-623+215\end{align*}

Here we have a loss of 623 and then a gain of 215. If you think about this logically, the loss is greater than the gain. Therefore the answer will still be negative. We can find the difference between these two values to figure out how much of a loss we still have after a gain of 215.

When we subtract integers, we take away a loss or a gain. If you can keep thinking in terms of losses and gains then it will be a lot simpler when you work with the negative integers.

Subtract: \begin{align*}-412 - 244\end{align*}

Here we have a loss of 412 and then we are going to take away a positive 244. If we take a positive 244 from a negative 412, then we will have a negative answer.

You can think of this as an addition problem in disguise.

\begin{align*}-412+(-244)\end{align*}

A negative number plus a negative number equals a negative number. So, add the two numbers and write a negative sign.

\begin{align*}-412+(-244)=-656\end{align*}

Here is another one.

\begin{align*}54- -789\end{align*}

Let’s look at this problem. We start with a gain of 54. Then we are going to take away a loss of 789. If we take a loss away, then we are moving into the positive zone. We can think of this problem in terms of addition.

\begin{align*}54+789\end{align*}

#### Example A

\begin{align*}-89 + 11\end{align*}

Solution:  \begin{align*}-78\end{align*}

#### Example B

\begin{align*}45 + -19\end{align*}

Solution:  \begin{align*}26\end{align*}

#### Example C

\begin{align*}23 - -12\end{align*}

Solution:  \begin{align*}35\end{align*}

Now let's go back to the dilemma from the beginning of the Concept. First, we need to write an equation and then we can find a sum.

For the equation, we use \begin{align*}x\end{align*} as the sum of the earnings. This will include both losses and gains.

Now we write in the losses and gains. We use a + for a gain and a – for a loss.

\begin{align*}x=87.00+(-20.00)+45.00+(-3.00)+25.00\end{align*}

Next we can find the value of \begin{align*}x\end{align*} .

\begin{align*}x=\134.00\end{align*}

Not bad for the first three days of sales.

### Guided Practice

Here is one for you to try on your own.

Add: \begin{align*}-229+563\end{align*}

Solution

Here we have a loss of 229 and a gain of 563. In this problem, the gain is greater than the loss so we know that our answer is going to be positive. We can find the difference between the two values to find out how much of a gain we have.

### Explore More

1. \begin{align*}6 + 7 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}5 + -8 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}8 + -8 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}6 + -10 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}8 + -2 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}9 + -4 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}-14 + -7 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}-12 + -14 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}-13 + -10 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}-18 + -30 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Directions: Subtract the following integers.

1. \begin{align*}-9 - 5 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}-8 - 7 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}-12 - 8 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}6 - 9 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}10 - 15 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}18 - - 5 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}12 - - 4 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}23 - - 9 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}-5 - - 2 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}-8 - - 5 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

### Vocabulary Language: English

Integer

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...
Irrational Number

Irrational Number

An irrational number is a number that can not be expressed exactly as the quotient of two integers.
Natural Numbers

Natural Numbers

The natural numbers are the counting numbers and consist of all positive, whole numbers. The natural numbers are the numbers in the list 1, 2, 3... and are often referred to as positive integers.
number line

number line

A number line is a line on which numbers are marked at intervals. Number lines are often used in mathematics to show mathematical computations.
operation

operation

Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.
rational number

rational number

A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.
Real Number

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.