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To determine which of the provided values could be the value of [tex]\( c \)[/tex] that satisfies the given inequalities, we need to assess each inequality step-by-step:
1. [tex]\( -2 < a \)[/tex]:
- This means [tex]\( a \)[/tex] must be an integer greater than -2. The possible integer values for [tex]\( a \)[/tex] are: [tex]\(-1, 0, 1, 2, 3, \ldots\)[/tex].
2. [tex]\( b < 8 \)[/tex]:
- This means [tex]\( b \)[/tex] must be an integer less than 8. The possible integer values for [tex]\( b \)[/tex] are: [tex]\(0, 1, 2, 3, 4, 5, 6, 7\)[/tex].
3. [tex]\( a < c < b \)[/tex]:
- This means [tex]\( c \)[/tex] must be an integer that is greater than [tex]\( a \)[/tex] and less than [tex]\( b \)[/tex].
Given these constraints, we need to check each of the provided values:
### Checking the Value [tex]\(-4\)[/tex]
To be a valid [tex]\( c \)[/tex]:
- [tex]\( -2 < a < -4 < b < 8 \)[/tex]
However:
- [tex]\( a \)[/tex] must be greater than [tex]\(-2\)[/tex], so [tex]\( a \)[/tex] can never be less than [tex]\(-4\)[/tex].
Thus, [tex]\(-4\)[/tex] cannot be a valid value for [tex]\( c \)[/tex].
### Checking the Value [tex]\(-2\)[/tex]
To be a valid [tex]\( c \)[/tex]:
- [tex]\( -2 < a < -2 < b < 8 \)[/tex]
However:
- Since [tex]\( c \)[/tex] must be strictly greater than [tex]\( a \)[/tex] ([tex]\( a < c \)[/tex]), [tex]\( c \)[/tex] cannot be equal to [tex]\(-2\)[/tex].
Thus, [tex]\(-2\)[/tex] cannot be a valid value for [tex]\( c \)[/tex].
### Checking the Value [tex]\( 3 \)[/tex]
To be a valid [tex]\( c \)[/tex]:
- [tex]\( -2 < a < 3 < b < 8 \)[/tex]
Let's explore possible values for [tex]\( a \)[/tex]:
- If [tex]\( a = -1 \)[/tex], then [tex]\( c \)[/tex] can be 3 when [tex]\( b > 3 \)[/tex]. Possible [tex]\( b \)[/tex] values are [tex]\(4, 5, 6, 7\)[/tex].
- If [tex]\( a = 0 \)[/tex], then [tex]\( c \)[/tex] can be 3 when [tex]\( b > 3 \)[/tex]. Possible [tex]\( b \)[/tex] values are [tex]\(4, 5, 6, 7\)[/tex].
- If [tex]\( a = 1 \)[/tex], then [tex]\( c \)[/tex] can be 3 when [tex]\( b > 3 \)[/tex]. Possible [tex]\( b \)[/tex] values are [tex]\(4, 5, 6, 7\)[/tex].
- If [tex]\( a = 2 \)[/tex], then [tex]\( c \)[/tex] can be 3 when [tex]\( b > 3 \)[/tex]. Possible [tex]\( b \)[/tex] values are [tex]\(4, 5, 6, 7\)[/tex].
- If [tex]\( a = 3 \)[/tex], then [tex]\( c \)[/tex] cannot be 3 because [tex]\( c \)[/tex] must be strictly greater than [tex]\( a \)[/tex].
Since the value [tex]\( c = 3 \)[/tex] fits within the specified ranges with permissible [tex]\( a \)[/tex] and [tex]\( b \)[/tex] values, it is indeed a valid value for [tex]\( c \)[/tex].
Therefore, the answer is:
[tex]\[ 3 \][/tex]
1. [tex]\( -2 < a \)[/tex]:
- This means [tex]\( a \)[/tex] must be an integer greater than -2. The possible integer values for [tex]\( a \)[/tex] are: [tex]\(-1, 0, 1, 2, 3, \ldots\)[/tex].
2. [tex]\( b < 8 \)[/tex]:
- This means [tex]\( b \)[/tex] must be an integer less than 8. The possible integer values for [tex]\( b \)[/tex] are: [tex]\(0, 1, 2, 3, 4, 5, 6, 7\)[/tex].
3. [tex]\( a < c < b \)[/tex]:
- This means [tex]\( c \)[/tex] must be an integer that is greater than [tex]\( a \)[/tex] and less than [tex]\( b \)[/tex].
Given these constraints, we need to check each of the provided values:
### Checking the Value [tex]\(-4\)[/tex]
To be a valid [tex]\( c \)[/tex]:
- [tex]\( -2 < a < -4 < b < 8 \)[/tex]
However:
- [tex]\( a \)[/tex] must be greater than [tex]\(-2\)[/tex], so [tex]\( a \)[/tex] can never be less than [tex]\(-4\)[/tex].
Thus, [tex]\(-4\)[/tex] cannot be a valid value for [tex]\( c \)[/tex].
### Checking the Value [tex]\(-2\)[/tex]
To be a valid [tex]\( c \)[/tex]:
- [tex]\( -2 < a < -2 < b < 8 \)[/tex]
However:
- Since [tex]\( c \)[/tex] must be strictly greater than [tex]\( a \)[/tex] ([tex]\( a < c \)[/tex]), [tex]\( c \)[/tex] cannot be equal to [tex]\(-2\)[/tex].
Thus, [tex]\(-2\)[/tex] cannot be a valid value for [tex]\( c \)[/tex].
### Checking the Value [tex]\( 3 \)[/tex]
To be a valid [tex]\( c \)[/tex]:
- [tex]\( -2 < a < 3 < b < 8 \)[/tex]
Let's explore possible values for [tex]\( a \)[/tex]:
- If [tex]\( a = -1 \)[/tex], then [tex]\( c \)[/tex] can be 3 when [tex]\( b > 3 \)[/tex]. Possible [tex]\( b \)[/tex] values are [tex]\(4, 5, 6, 7\)[/tex].
- If [tex]\( a = 0 \)[/tex], then [tex]\( c \)[/tex] can be 3 when [tex]\( b > 3 \)[/tex]. Possible [tex]\( b \)[/tex] values are [tex]\(4, 5, 6, 7\)[/tex].
- If [tex]\( a = 1 \)[/tex], then [tex]\( c \)[/tex] can be 3 when [tex]\( b > 3 \)[/tex]. Possible [tex]\( b \)[/tex] values are [tex]\(4, 5, 6, 7\)[/tex].
- If [tex]\( a = 2 \)[/tex], then [tex]\( c \)[/tex] can be 3 when [tex]\( b > 3 \)[/tex]. Possible [tex]\( b \)[/tex] values are [tex]\(4, 5, 6, 7\)[/tex].
- If [tex]\( a = 3 \)[/tex], then [tex]\( c \)[/tex] cannot be 3 because [tex]\( c \)[/tex] must be strictly greater than [tex]\( a \)[/tex].
Since the value [tex]\( c = 3 \)[/tex] fits within the specified ranges with permissible [tex]\( a \)[/tex] and [tex]\( b \)[/tex] values, it is indeed a valid value for [tex]\( c \)[/tex].
Therefore, the answer is:
[tex]\[ 3 \][/tex]
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