To determine which of the expressions will always be negative, we need to analyze each one based on the variable conditions that make the expression less than zero.
1. Expression I: [tex]\(a - 1\)[/tex]
- This expression will be negative if [tex]\(a < 1\)[/tex]. So, for any value of [tex]\(a\)[/tex] that is less than 1, this expression will be negative. There is a range of values of [tex]\(a\)[/tex] which make this expression negative.
2. Expression II: [tex]\(1 + b\)[/tex]
- For this to be negative, we need [tex]\(1 + b < 0\)[/tex], which simplifies to [tex]\(b < -1\)[/tex]. So, [tex]\(b\)[/tex] must be less than -1 for this expression to be negative. However, without knowing specific constraints on [tex]\(b\)[/tex], we cannot say this expression is always negative.
3. Expression III: [tex]\(a + b\)[/tex]
- This expression will be negative when the sum of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is less than 0. Without specific constraints on [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we cannot guarantee that this will always be negative; it can be positive or zero depending on the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
4. Expression IV: [tex]\(1 - a\)[/tex]
- This expression will be negative if [tex]\(a > 1\)[/tex]. That is, for any value of [tex]\(a\)[/tex] greater than 1, this expression would be negative. There is a range of values of [tex]\(a\)[/tex] which make this expression negative.
Based on these analyses:
- Expression I: [tex]\(a - 1\)[/tex] will be negative if [tex]\(a < 1\)[/tex].
- Expression IV: [tex]\(1 - a\)[/tex] will be negative if [tex]\(a > 1\)[/tex].
Thus, the expressions that will always be negative are:
[tex]\[ \boxed{1, 4} \][/tex]
