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# Integers

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

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Practice Integers
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Identify and Apply Number Properties in Integer Operations

Did you know that you can simplify an expression by using a property? Take a look at this dilemma.

Travis is working on his homework. He was doing great until he got to this problem. Take a look.

4(x+7)\begin{align*}-4(x+7)\end{align*}

Travis isn't sure how to simplify this expression since he hasn't been given a value for x\begin{align*}x\end{align*}. If Travis applies the distributive property, then he can simplify this expression. Take a look at this Concept and then you can work on this dilemma at the end of it.

### Guidance

There are many properties, the associative property, the commutative property, the additive identity property, and the additive inverse property.

You can use these properties to simplify integer expressions.

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

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

Use properties to reorganize the expression.

The commutative property states that numbers can be added in any order and this does not change the sum. When working with negative numbers, keep the negative sign with the number it belongs to. Adding parentheses around the negative number can help you keep things organized.

Apply the commutative property: (28+63)+28=(63+(28))+28\begin{align*}(-28+63)+28=(63+(-28))+28\end{align*}

The associative property states that the grouping of numbers does not change the sum.

Apply the associative property: (63+(28))+28=63+(28+28)\begin{align*}(63+(-28))+28=63+(-28+28)\end{align*}

The additive inverse property states that any number added to its opposite equals zero.

Apply the additive inverse property: 63+(28+28)=63+0\begin{align*}63+(-28+28)=63+0\end{align*}

The additive identity property states that the sum of any number and zero is that number.

Apply the additive identity property: 63+0=63\begin{align*}63+0=63\end{align*}

Exactly, and what you will find is that it is easier to perform operations with positive and negative integers when they are organized together. If you add two negative integers, then you have a negative answer. If you add two positives then you have a positive answer. Just remember to think in terms of losses and gains and it will help you keep your signs straight.

You have already learned several properties of multiplication: the associative property, the commutative property, the distributive property, the multiplicative identity, and the zero property. You can use these properties to simplify integer expressions.

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

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

I know that it may seem confusing, but start by looking at what is actually in the problem. You can see that there is a set of parentheses with values in it. There is also a small integer addition problem outside of the parentheses.

Now think about the properties that you know already. If you have these in your notebook, open up your notebook to look at the definitions of these properties. Let’s work through it.

The distributive property states that a number multiplied by a sum is equal to the sum of the products of the number multiplied by each addend.

Apply the distributive property: 3+(5)(9x+6)3=3+(5)(9x)+(5)(6)\begin{align*}3+(-5)(-9x+6)-3=3+(-5)(-9x)+(-5)(6)\end{align*}

Now simplify the multiplication: 3+(5)(9x)+(5)(6)=3+45x+(30)\begin{align*}3+(-5)(-9x)+(-5)(6)=3+45x+(-30)\end{align*}

The commutative property states that numbers can be added in any order.

Apply the commutative property: 3+45x+(30)=45x+3+(30)\begin{align*}3+45x+(-30)=45x+3+(-30)\end{align*}

Finally, simplify the addition: 45x+3+(30)=45x+(27)=45x27\begin{align*}45x+3+(-30)=45x+(-27)=45x-27\end{align*}

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

Because we don’t know the value of our variable this is as far as we can simplify this expression. Remember that simplifying doesn’t mean solve, it means make smaller.

#### Example A

Simplify4(x+6)\begin{align*}-4(x+6)\end{align*}

Solution:  4x24\begin{align*}-4x - 24\end{align*}

#### Example B

(6)(3)(4)\begin{align*}(-6)(-3)(4)\end{align*}

Solution:  72\begin{align*}72\end{align*}

#### Example C

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

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

Now let's go back to the dilemma from the beginning of the Concept.

4(x+7)\begin{align*}-4(x+7)\end{align*}

First, Travis has to distribute the negative four by multiplying it with both of the terms inside the parentheses.

4(x)+4(7)\begin{align*}-4(x)+-4(7)\end{align*}

4x+(28)\begin{align*}-4x+(-28)\end{align*}

### Vocabulary

Associative Property
states that the groupings of values can change and not affect the sum or difference.
Commutative Property
the order of the numbers being added can be changed and not affect the sum or the difference.
any number and zero is still that original number.
any number and its opposite is equal to zero.
Associative Property
the groupings of numbers being multiplied does not impact the value of the product.
Commutative Property
the order of the numbers being multiplied does not impact the value of the product.
Distributive Property
a value outside of a set of parentheses can be multiplied by the values inside the parentheses to find a product.
Multiplicative Identity
any value multiplied by 1 is still that original value.
Zero Property
any value times zero is zero.

### Guided Practice

Here is one for you to try on your own.

Name the property illustrated here.

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

Solution

Notice that there aren't any parentheses here. This expression also does not distribute any terms. The same terms are on both sides of the equals sign. The only difference is the order of the terms.

The Commutative Property is our answer.

### Practice

Directions: 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=7+5\begin{align*}-5+7+0=-7+5\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*}

Directions: 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. 12(6+4)\begin{align*}\frac{1}{2}(6 + 4)\end{align*}

### 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

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

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

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

Multiplicative Identity

The multiplicative identity for multiplication of real numbers is one.
Zero Property

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.