Yes, an integer and its additive inverse always have different signs. The additive inverse of a number \( x \) is defined as the number that, when added to \( x \), results in zero. The additive inverse is \( -x \).
- If \( x \) is positive (e.g., \( x = 5 \)), then its additive inverse \( -x \) will be negative (e.g., \( -5 \)).
- Conversely, if \( x \) is negative (e.g., \( x = -3 \)), then its additive inverse \( -x \) will be positive (e.g., \( 3 \)).
- If \( x \) is zero, its additive inverse is also zero, and in this case, the two are not considered to have different signs.
In summary, for non-zero integers, an integer and its additive inverse will always have different signs. Only in the case of zero does the issue of sign not apply, as both the integer and its additive inverse are the same (zero).
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