Let’s analyze each statement and fill in the blanks with the appropriate properties:
1. Statement Analysis:
a. The sum of an integer and its opposite is zero.
- This statement is True. It is known as the Additive Inverse Property, which states that for every integer \(a\), there exists an integer \(-a\) such that \(a + (-a) = 0\).
b. The sum of two positive integers is always less than zero.
- This statement is False. The sum of two positive integers is always a positive integer, as the addition of two positive numbers results in a larger positive number.
c. The sum of a negative integer and a positive integer is always negative.
- This statement is False. The sum of a negative integer and a positive integer can be negative, positive, or zero depending on their magnitudes. If the positive integer is larger, the result would be positive, and if the negative integer is larger, the result would be negative.
d. The sum of two negative integers is always greater than zero.
- This statement is False. The sum of two negative integers is always negative, as adding two negative numbers results in a larger negative number.
2. Fill in the blanks with the properties:
- 1. The sum of an integer and its opposite is zero.
- Property: Additive Inverse Property
- 2. The sum of two positive integers is always positive.
- Property: Closure Property of Addition for positive integers
- 3. The sum of a negative integer and a positive integer can be either positive, negative, or zero.
- Property: This depends on the magnitudes of the integers involved. (No specific name; it's based on comparison.)
- 4. The sum of two negative integers is always negative.
- Property: Closure Property of Addition for negative integers
- 5. Additive inverse of an integer is always negative.
- This statement is False. The additive inverse of a positive integer is negative, and the additive inverse of a negative integer is positive. Hence, the additive inverse is not always negative; it is the opposite sign of the original integer.
Therefore, the results are summarized as follows:
1. True - Additive Inverse Property
2. False
3. False
4. False
5. False
And the corresponding blanks are filled as above.
