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Which of the following expressions will always be negative? Select all that apply.

I. [tex]a-1[/tex]
II. [tex]1+b[/tex]
III. [tex]a+b[/tex]
IV. [tex]1-a[/tex]


Sagot :

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]