To understand why Juan had to use the additive inverse before he could use the associative and commutative properties, let's break down the original expression and the properties themselves.
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Original Expression: Juan's original expression, \( 16 - 9 - 12 + 22 \), involves both positive and negative numbers represented in a way that doesn't immediately facilitate the use of properties like associative or commutative.
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Additive Inverse: Juan rewrote the expression as \( 16 + (-9) + (-12) + 22 \). The additive inverse is a concept where you express subtraction in terms of addition. For example, \( -9 \) can be seen as \( + (-9) \) and \( -12 \) as \( + (-12) \). By rewriting subtractions as the addition of their negatives, he is setting up the expression in a form that highlights all terms as sums.
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Associative Property: This property states that the way in which numbers are grouped in an addition does not change their sum. For instance, \( (a + b) + c = a + (b + c) \). When Juan rewrote his expression to \( 16 + (-9) + (-12) + 22 \), it allowed him to regroup terms as needed, such as \( (16 + 22) + (-9) + (-12) \).
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Commutative Property: This property states that the order of addition does not affect the sum, meaning \( a + b = b + a \). By rewriting the expression using the additive inverses, Juan can rearrange the terms in any order that might make it easier to calculate, such as moving the negative terms together or pairing them with positives.
In summary, Juan had to use the additive inverse in order to express all parts of the equation uniformly as additions. This allowed him to effectively apply the associative and commutative properties, simplifying the expression and making it easier to calculate his profit over the selected days. Without using the additive inverses, he would have mixed addition and subtraction, making it difficult to reorganize the terms using these properties.
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