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# Integers

## Associative, commutative, identity, and inverse properties with positive and negative numbers

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Identify and Apply Number Properties in Integer Operations

### [Figure1] License: CC BY-NC 3.0

You need to determine the size of a flower garden so you will know how many rows to plant. The area of your entire back yard is 59625 ft2\begin{align*}59625 \ ft^2\end{align*}. If x\begin{align*}x\end{align*} is the width of the flower garden and your entire back yard is 225 feet long, use the diagram to determine the unknown width.

In this concept, you will learn to identify and apply number properties in integer operations.

### Integer Properties

There are many number properties that you can use to simplify integer expressions.

• The associative property states that the grouping of numbers does not change the sum.
• The commutative property states that numbers can be added in any order and this does not change the sum.
• The additive inverse property states that any number added to its opposite equals zero.
• The additive identity property states that the sum of any number and zero is that number.

Let’s look at an example.

Simplify the following expression. Justify each step by identifying the property used.

(28+63)+28\begin{align*}(-28+63)+28\end{align*}

First, use commutative property to reorganize the expression.

(28+63)+28=(63+(28))+28\begin{align*}(-28 + 63) + 28 = (63 + (-28)) + 28\end{align*}

Next, use the associative property to reorganize the expression.

(63+(28))+28=63+(28+28)\begin{align*}(63 + (-28)) + 28 = 63 + (-28+28)\end{align*}

Then, apply the additive inverse property.

63+(28+28)=63+0\begin{align*}63+(-28+28)=63 + 0\end{align*}

Then, apply the additive identity property.

63+0=63\begin{align*}63+0=63\end{align*}

Let’s try another example.

Simplify the following expression. Justify each step by writing the property used.

3+(5)(9x+6)\begin{align*}3+(-5)(-9x+6)\end{align*}

First, use the distributive property.

3+(5)(9x+6)=3+(5×9x)+(5×6)\begin{align*}3 + (-5)(-9x +6) = 3 + (-5 \times -9x) + (-5 \times 6)\end{align*}

Next, simplify the expression.

3+(5×9x)+(5×6)=3+45x30\begin{align*}3 + (-5 \times - 9x) + (-5 \times 6) = 3+45 x - 30\end{align*}

Then, combine like terms and then the numbers together. Like terms have the same literal coefficients. For example, 2x\begin{align*}2x\end{align*} and 3x\begin{align*}3x\end{align*} are like terms, while 2a\begin{align*}2a\end{align*} and 3b\begin{align*}3b\end{align*} are unlike terms.

3+45x30=45x27\begin{align*}3+45 x - 30 = 45x - 27\end{align*}

The answer is 45x27\begin{align*}45x - 27\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about the flower garden problem.

You need to figure out the width of the flower garden.

First, consider what you know.

Area of the land is 59625 ft2\begin{align*}59625\ ft^2\end{align*}.

AreaLengthWidth===Length×Width200 ft225+x\begin{align*}\begin{array}{rcl} \text{Area} &=& \text{Length}\times \text{Width} \\ \text{Length} &=& 200 \ ft \\ \text{Width} &=& 225 + x \end{array}\end{align*}

Next, use what you know to set up your equation.

59625=(225+x)×225\begin{align*}59625 = (225 + x) \times 225\end{align*}

Next, use the distributive property.

59625=45000+200x\begin{align*}59625 = 45000 + 200 x\end{align*}

Next, combine like terms.

59625506259000==225x225x\begin{align*}\begin{array}{rcl} 59625 - 50625 &=& 225 x \\ 9000 &=& 225 x \end{array}\end{align*}

Then, solve for x\begin{align*}x\end{align*} by dividing by 225.

9000225=225x225\begin{align*}\frac{9000}{225} = \frac{225 x}{225}\end{align*}

The unknown width is 40 feet.

#### Example 2

Name the property illustrated here.

4x+(5x)+8y=(5x)+8y+4x\begin{align*}4x + (-5x) +8 y = (-5x) + 8y + 4x\end{align*}

The only difference is the order of the terms.

The answer is the commutative property.

#### Example 3

Simplify:

4(x+6)\begin{align*}-4(x+6)\end{align*}

First, use the distributive property.

4(x+6)=4x24\begin{align*}-4(x+6)=-4x-24\end{align*}

The answer is 4x24\begin{align*}-4x-24\end{align*}.

#### Example 4

Simplify:

(63)+3\begin{align*}(-6-3)+3\end{align*}

First, reorganize the expression so that you are only involving addition.

(63)+3=(6+(3))+3\begin{align*}(-6-3)+3 = (-6+(-3)) +3\end{align*}

Next, use the associative property to reorganize the expression.

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

Then, apply the additive inverse property.

6+(3+3)=6+0\begin{align*}-6 + (-3 +3) = -6+0\end{align*}

Then, apply the additive identity property.

6+0=6\begin{align*}-6 + 0 = -6\end{align*}

#### Example 5

Simplify:

8x+3y+4x\begin{align*}-8x + 3y +4x\end{align*}

First, use commutative property to reorganize the expression.

8x+3y+4x=8x+4x+3y\begin{align*}-8x +3y +4x = -8x +4x +3y\end{align*}

Next, combine like terms.

8x+4x+3y=4x+3y\begin{align*}-8x +4x +3y =-4x +3y\end{align*}

The answer is 4x+3y\begin{align*}-4x +3y\end{align*}.

### Review

Identify each property illustrated.

1. 3x+4x+7y=7y+3x+4x\begin{align*}3x+4x+7y =7y+3x+4x\end{align*}

2. 5+7+0=5+7\begin{align*}-5+7+0=-5+7\end{align*}

3. (6+5)+9=6+(5+9)\begin{align*}(-6+5)+9 = -6+(5+9)\end{align*}

4. 5+x+8y=x+5+8y\begin{align*}-5+-x+8y = -x+5 +8y\end{align*}

5. 6(x+y)=6x+6y\begin{align*}6(x+y) = 6x+6y\end{align*}

6. 7y(1)=7y\begin{align*}-7y(1) = -7y\end{align*}

7. x(8+y)=8x+xy\begin{align*}x(8+y) = 8x+xy\end{align*}

Simplify each expression.

8. 4(y5)+3y\begin{align*}4(y-5) + -3y\end{align*}

9. 5(x4)\begin{align*}-5(x-4)\end{align*}

10. 4x+7x+73y\begin{align*}-4x +7x +7 -3y\end{align*}

11. 6(y+4)\begin{align*}-6(y+4)\end{align*}

12. 3(y2)+2(y+6)\begin{align*}-3(y-2) + 2(y+6)\end{align*}

13. 8(x+4)3(x+2)\begin{align*}8(x+4) - 3(x+2)\end{align*}

14. 9y(3+2)\begin{align*}-9y(3+2)\end{align*}

15. \begin{align*}\frac{1}{2} (6+4)\end{align*}

To see the Review answers, open this PDF file and look for section 2.13.

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Color Highlighted Text Notes

### Vocabulary Language: English

The sum of any number and zero is the number itself.

The additive inverse or opposite of a number x is -1(x). A number and its additive inverse always sum to zero.

Associative Property

The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).

Commutative Property

The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example $a+b=b+a \text{ and\,} (a)(b)=(b)(a)$.

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.

Multiplicative Identity

The multiplicative identity for multiplication of real numbers is one.

Zero Property

The zero property of multiplication says that the product of any number and zero is zero. The zero property of addition states that the sum of any number and zero is the number.