Name the property: [tex]$\frac{-11}{16} + \frac{11}{16} = 0$[/tex]
i) commutative
ii) distributive
iii) closure
iv) additive inv
Answer (2)
The property demonstrated by the equation − 16 11 + 16 11 = 0 is the additive inverse property, which states that a number added to its opposite equals zero. The chosen option is iv) additive inv .
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The equation shows the sum of a number and its negative equals zero.
The additive inverse property states that a + ( − a ) = 0 .
In this case, 16 11 + ( − 16 11 ) = 0 .
Therefore, the property demonstrated is the additive inverse. a dd i t i v e in v erse
Explanation
Understanding the Problem The given equation is − 16 11 + 16 11 = 0 . We need to identify the property that this equation demonstrates from the given options: commutative, distributive, closure, and additive inverse.
Recalling the Additive Inverse Property The equation shows that the sum of a number and its negative (or additive inverse) equals zero. The additive inverse property states that for any number a , there exists a number − a such that a + ( − a ) = 0 .
Applying the Definition In our case, a = 16 11 , and its additive inverse is − 16 11 . When we add these two numbers, we get zero: 16 11 + ( − 16 11 ) = 0 . This matches the definition of the additive inverse property.
Identifying the Correct Property Therefore, the property demonstrated by the equation − 16 11 + 16 11 = 0 is the additive inverse property.
Examples
The additive inverse property is useful in balancing equations and simplifying expressions in algebra. For example, if you have the equation x + 5 = 0 , you can add the additive inverse of 5 (which is -5) to both sides of the equation to solve for x : x + 5 + ( − 5 ) = 0 + ( − 5 ) , which simplifies to x = − 5 . This concept is also used in physics to understand forces that cancel each other out, resulting in equilibrium.
