Let's examine each step in the given expression and identify which properties are used:
1. Starting expression:
[tex]\[
(x+2) \cdot 5 - 7
\][/tex]
2. First step:
[tex]\[
(x+2) \cdot 5 - 7 = 5(x+2) - 7
\][/tex]
Here, the commutative property of multiplication is used. This property states that the order in which two numbers are multiplied does not change the product, allowing us to rewrite [tex]\((x+2) \cdot 5\)[/tex] as [tex]\(5(x+2)\)[/tex].
3. Second step:
[tex]\[
5(x+2) - 7 = (5x + 10) - 7
\][/tex]
In this step, the distributive property is used. This property allows us to distribute the [tex]\(5\)[/tex] across the terms inside the parentheses [tex]\((x+2)\)[/tex], resulting in [tex]\(5x + 10\)[/tex].
4. Third step:
[tex]\[
(5x + 10) - 7 = 5x + (10 - 7)
\][/tex]
The associative property of addition is applied here. This property allows us to change the grouping of the numbers being added, rearranging [tex]\( (5x + 10) - 7 \)[/tex] to [tex]\( 5x + (10 - 7) \)[/tex].
5. Final step:
[tex]\[
5x + (10 - 7) = 5x + 3
\][/tex]
In this step, we simply perform the arithmetic operation of subtracting 7 from 10, which simplifies to 3.
### Summary of Used Properties:
1. Commutative property of multiplication: [tex]\((x+2) \cdot 5 = 5(x+2)\)[/tex]
2. Distributive property: [tex]\(5(x+2) = 5x + 10\)[/tex]
3. Associative property of addition: [tex]\((5x + 10) - 7 = 5x + (10 - 7)\)[/tex]
### Property Not Used:
- Commutative property of addition: This property states that the order in which two numbers are added does not change the sum. In this sequence of simplification steps, we did not use the commutative property of addition at any point.
Therefore, the property that is not used in simplifying the given expression is the commutative property of addition.
