To understand the transition from step 3 to step 4 in Simonne's simplification, let’s carefully review the steps involved:
1. Initial Expression:
[tex]\[
12 - 3(-2x + 4)
\][/tex]
2. Step 1: Distribution
[tex]\[
12 + (-3)(-2x) + (-3)(4)
\][/tex]
This step involves distributing the [tex]$-3$[/tex] across both terms inside the parentheses.
3. Step 2: Simplification
[tex]\[
12 + 6x + (-12)
\][/tex]
Here, [tex]$(-3) \times (-2x) = 6x$[/tex] and [tex]$(-3) \times 4 = -12$[/tex].
4. Step 3: Combine Like Terms
[tex]\[
12 + (-12) + 6x
\][/tex]
The constants [tex]$12$[/tex] and [tex]$-12$[/tex] are written together.
5. Step 4: Simplification
[tex]\[
0 + 6x
\][/tex]
In this step, [tex]$12 + (-12) = 0$[/tex] because they are additive inverses of each other.
6. Step 5: Final Simplification
[tex]\[
6x
\][/tex]
Any number added to [tex]$0$[/tex] remains unchanged, applying the identity property of addition.
Now, let us identify the mathematical property used to transition from step 3 to step 4:
[tex]\[
12 + (-12) + 6x \rightarrow 0 + 6x
\][/tex]
Here, the property used to simplify [tex]$12 + (-12)$[/tex] is the inverse property of addition. This property states that the sum of any number and its additive inverse (opposite) is zero. Since [tex]$-12$[/tex] is the additive inverse of [tex]$12$[/tex], their sum is [tex]$0$[/tex].
Therefore, the property of real numbers used in the transition from step 3 to step 4 is the:
[tex]\[
\boxed{\text{inverse property of addition}}
\][/tex]
