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## Sums of opposite numbers and distance from zero

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On Monday, Marty borrows $50.00 from his father. On Thursday, he gives his father$28.00. Can you write an addition statement to describe Marty’s financial transactions?

When adding integers, you need to make sure you follow two rules:

1. Integers with unlike signs must be subtracted. The answer will have the same sign as that of the higher number.
2. Integers with the same sign must be added. The answer will have the same sign as that of the numbers being added.

In order to understand why these rules work, you can represent the addition of integers with manipulatives such as color counters or algebra tiles. A number line can also be used to show the addition of integers. The following problems show how to use these manipulatives to understand the rules for adding integers.

#### Let's practice adding the integers using color counters:

1. 5+(3)=?\begin{align*}5+(-3)=?\end{align*}

This problem can be represented with color counters. In this case, the red counters represent positive numbers and the yellow ones represent the negative numbers.

One positive counter and one negative counter equals zero because 1+(1)=0\begin{align*}1+(-1)=0\end{align*}. Draw a line through the counters that equal zero.

The remaining counters represent the answer. Therefore, 5+(3)=2\begin{align*}5+(-3)=2\end{align*}. The answer is the difference between 5 and 3. The answer takes on the sign of the larger number. In this case, the five has a positive value and it is greater than 3.

1. 4+(7)=?\begin{align*}4+(-7)=?\end{align*}

Draw a line through the counters that equal zero.

The remaining counters represent the answer. Therefore, 4+(7)=3\begin{align*}4+(-7)=-3\end{align*}. The answer is the difference between 7 and 4. The answer takes on the sign of the larger number, which is 7 in this case.

#### Now, let's add the integers using algebra tiles:

6x+(8x)=?\begin{align*}6x+(-8x)=?\end{align*}

This same method can be extended to adding variables. Algebra tiles can be used to represent positive and negative values.

The green algebra tiles represent positive x\begin{align*}x\end{align*} and the white tiles represent negative x\begin{align*}x\end{align*}. There are 6 positive x\begin{align*}x\end{align*} tiles and 8 negative x\begin{align*}x\end{align*} tiles.

The remaining algebra tiles represent the answer. There are two negative x\begin{align*}x\end{align*} tiles remaining. Therefore, (6x)+(8x)=2x\begin{align*}(6x)+(-8x)=-2x\end{align*}. The answer is the difference between 8x\begin{align*}8x\end{align*} and 6x\begin{align*}6x\end{align*}. The answer takes on the sign of the larger coefficient, which in this case is 8.

#### Finally, let's add the integers using a number line:

(3)+(5)=?\begin{align*}(-3)+(-5)=?\end{align*}

You can solve this problem with a number line. Indicate the starting point of -3 by using a dot. From this point, add a -5 by moving five places to the left. You will stop at -8.

The point where you stopped is the answer to the problem. Therefore, (3)+(5)=8\begin{align*}(-3)+(-5)=-8\end{align*}

### Examples

#### Example 1

Earlier, you were told that Marty borrows $50.00 from his father on Monday and gives his father$28.00 back on Thursday. How much money does Marty have on Thursday after he pays his father? How much does he owe?

First, Marty borrows $50.00 which he must repay to his father. Therefore on Monday, Marty has$50.00\begin{align*}-\50.00\end{align*}.

He then returns $28.00 to his father. Now Marty has$50.00+($28.00)=$22.00.\begin{align*}-\50.00+(\28.00)=-\22.00.\end{align*} He still owes his father \$22.00.

#### Example 2

Add the integers: (7)+(+5)=?\begin{align*}(-7)+(+5)=?\end{align*}

(7)+(+5)=57=2\begin{align*}(-7)+(+5)=5-7=-2\end{align*}

#### Example 3

Add the integers: 8+(2)=?\begin{align*}8+(-2)=?\end{align*}

8+(2)=82=6\begin{align*}8+(-2)=8-2=6\end{align*}

#### Example 4

Determine the answer to (6)+(3)=?\begin{align*}(-6)+(-3)=?\end{align*} and (2)+(5)=?\begin{align*}(2)+(-5)=?\end{align*} by using the rules for adding integers.

(6)+(3)=9\begin{align*}(-6)+(-3)=-9\end{align*}.

(2)+(5)=25=3\begin{align*}(2)+(-5)=2-5=-3\end{align*}.

### Review

Complete the following addition problems using any method.

1. (7)+(2)\begin{align*}(-7)+(-2)\end{align*}
2. (6)+(8)\begin{align*}(6)+(-8)\end{align*}
3. (5)+(4)\begin{align*}(5)+(4)\end{align*}
4. (7)+(9)\begin{align*}(-7)+(9)\end{align*}
5. \begin{align*}(-1)+(5)\end{align*}
6. \begin{align*}(8)+(-12)\end{align*}
7. \begin{align*}(-2)+(-5)\end{align*}
8. \begin{align*}(3)+(4)\end{align*}
9. \begin{align*}(-6)+(10)\end{align*}
10. \begin{align*}(-1)+(-7)\end{align*}
11. \begin{align*}(-13)+(9)\end{align*}
12. \begin{align*}(-3)+(-8)+(12)\end{align*}
13. \begin{align*}(14)+(-6)+(5)\end{align*}
14. \begin{align*}(15)+(-8)+(-9)\end{align*}
15. \begin{align*}(7)+(6)+(-9)+(-8)\end{align*}

For each of the following models, write an addition problem and answer the problem.

1. .

1. .

1. .

1. .

1. .

To see the Review answers, open this PDF file and look for section 1.1.

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### Vocabulary Language: English

Irrational Number

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

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

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

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

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

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