Sure, let's work through the given questions step-by-step.
### Part (a)
Describe the values of [tex]\( b \)[/tex] that are solutions of the inequality [tex]\( b > -2 \)[/tex].
To determine which values of [tex]\( b \)[/tex] satisfy the inequality [tex]\( b > -2 \)[/tex]:
- We need [tex]\( b \)[/tex] to be any number that is greater than [tex]\(-2\)[/tex].
- This means that [tex]\( b \)[/tex] can take on any value that is larger than [tex]\(-2\)[/tex].
So, the set of values for [tex]\( b \)[/tex] that satisfy this inequality are all numbers greater than [tex]\(-2\)[/tex].
In interval notation, this is represented as:
[tex]\[ (-2, \infty) \][/tex]
In set notation:
[tex]\[ \{ b \in \mathbb{R} : b > -2 \} \][/tex]
### Part (b)
Describe the values of [tex]\( b \)[/tex] that are not solutions of the inequality. Write an inequality for these values.
To determine the set of values of [tex]\( b \)[/tex] that do not satisfy the inequality [tex]\( b > -2 \)[/tex]:
- We need [tex]\( b \)[/tex] to be any number that is less than or equal to [tex]\(-2\)[/tex].
- This means [tex]\( b \)[/tex] can be [tex]\(-2\)[/tex] or any number smaller than [tex]\(-2\)[/tex].
So, the set of values for [tex]\( b \)[/tex] that do not satisfy the inequality [tex]\( b > -2 \)[/tex] are all numbers less than or equal to [tex]\(-2\)[/tex].
In interval notation, this is represented as:
[tex]\[ (-\infty, -2] \][/tex]
In set notation:
[tex]\[ \{ b \in \mathbb{R} : b \leq -2 \} \][/tex]
### Part (c)
What do all the values in parts (a) and (b) represent? Is this true for any inequality?
The values in part (a) represent all the solutions to the inequality [tex]\( b > -2 \)[/tex]. These are the values for which the inequality holds true.
The values in part (b) represent all the values that are not solutions to the inequality [tex]\( b > -2 \)[/tex]. These are the values for which the inequality does not hold true.
Together, the values in parts (a) and (b) partition the number line into two distinct intervals:
- One interval where the inequality [tex]\( b > -2 \)[/tex] is satisfied (solutions).
- Another interval where the inequality [tex]\( b > -2 \)[/tex] is not satisfied (non-solutions).
This is true for any inequality: an inequality partitions the number line (or space) into regions where the inequality is satisfied and regions where it is not satisfied. For example, for any inequality of the form [tex]\( b < c \)[/tex], the solution set would be [tex]\( b < c \)[/tex] and the non-solution set would be [tex]\( b \geq c \)[/tex].
