Let's analyze the given equation [tex]\( -6 + 6 = 0 \)[/tex] and determine which property of real numbers it demonstrates.
The equation shows that when you add -6 and 6, the result is 0. This means that the two numbers essentially cancel each other out. Now, we need to identify the property of real numbers that describes this relationship.
### Possible properties:
1. Associative Property of Addition:
- This property states that the way in which numbers are grouped in addition does not change the result. For example, for any numbers [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], we have:
[tex]\[
(a + b) + c = a + (b + c)
\][/tex]
- This property is not about the numbers canceling each other out, so it's not the property shown by the given equation.
2. Commutative Property of Addition:
- This property states that the order in which two numbers are added does not change the result. For example, for any numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we have:
[tex]\[
a + b = b + a
\][/tex]
- Although this property is true, it does not explain why [tex]\( -6 + 6 \)[/tex] equals 0.
3. Identity Property of Addition:
- This property states that there is an additive identity (0) such that any number added to 0 will result in the original number. For any number [tex]\( a \)[/tex], we have:
[tex]\[
a + 0 = a
\][/tex]
- The given equation doesn't involve adding 0 to a number, so this is not the relevant property here.
4. Inverse Property of Addition:
- This property states that for any real number [tex]\( a \)[/tex], there exists a number [tex]\( -a \)[/tex] such that [tex]\( a + (-a) = 0 \)[/tex]. Essentially, it shows that every number has an additive inverse that cancels it out to get 0.
- For the given equation, we have [tex]\( a = -6 \)[/tex] and [tex]\( -a = 6 \)[/tex]. It fits this property perfectly:
[tex]\[
-6 + 6 = 0
\][/tex]
- This clearly demonstrates the inverse property of addition, where [tex]\( -6 \)[/tex] and [tex]\( 6 \)[/tex] are additive inverses.
### Conclusion:
The property of real numbers shown by the equation [tex]\( -6 + 6 = 0 \)[/tex] is the inverse property of addition.
