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 \begin{align*}59625 \ ft^2\end{align*}. If \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.

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

First, use commutative property to reorganize the expression.

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

Next, use the associative property to reorganize the expression.

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

Then, apply the additive inverse property.

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

Then, apply the additive identity property.

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

The answer is 63.

Let’s try another example.

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

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

First, use the distributive property.

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

Next, simplify the expression.

\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, \begin{align*}2x\end{align*} and \begin{align*}3x\end{align*} are like terms, while \begin{align*}2a\end{align*} and \begin{align*}3b\end{align*} are unlike terms.

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

The answer is \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 \begin{align*}59625\ ft^2\end{align*}.

\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.

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

Next, use the distributive property.

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

Next, combine like terms.

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

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

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

The answer is 40.

The unknown width is 40 feet.

#### Example 2

Name the property illustrated here.

\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:

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

First, use the distributive property.

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

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

#### Example 4

Simplify:

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

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

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

Next, use the associative property to reorganize the expression.

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

Then, apply the additive inverse property.

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

Then, apply the additive identity property.

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

The answer is -6.

#### Example 5

Simplify:

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

First, use commutative property to reorganize the expression.

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

Next, combine like terms.

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

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

### Review

Identify each property illustrated.

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

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

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

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

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

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

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

Simplify each expression.

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

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

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

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

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

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

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

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

### Review (Answers)

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