To find the possible values of [tex]\( x \)[/tex] that satisfy both conditions, let's analyze each inequality separately and then find the intersection of the solutions.
### Analyzing the First Inequality
The first inequality is:
[tex]\[ -5 < x \leq 1 \][/tex]
Since [tex]\( x \)[/tex] is an integer:
- [tex]\( x \)[/tex] must be greater than -5.
- [tex]\( x \)[/tex] must be less than or equal to 1.
Thus, the possible integer values for [tex]\( x \)[/tex] are:
[tex]\[ -4, -3, -2, -1, 0, 1 \][/tex]
### Analyzing the Second Inequality
The second inequality is:
[tex]\[ 1 \leq x + 3 < 8 \][/tex]
To isolate [tex]\( x \)[/tex], we need to subtract 3 from all parts of the inequality:
[tex]\[ 1 - 3 \leq x + 3 - 3 < 8 - 3 \][/tex]
[tex]\[ -2 \leq x < 5 \][/tex]
Since [tex]\( x \)[/tex] is an integer:
- [tex]\( x \)[/tex] must be greater than or equal to -2.
- [tex]\( x \)[/tex] must be less than 5.
Thus, the possible integer values for [tex]\( x \)[/tex] are:
[tex]\[ -2, -1, 0, 1, 2, 3, 4 \][/tex]
### Finding the Intersection
To find the values of [tex]\( x \)[/tex] that satisfy both inequalities, we need to find the common values in both sets:
- From the first inequality, we have [tex]\( x \in \{-4, -3, -2, -1, 0, 1\} \)[/tex].
- From the second inequality, we have [tex]\( x \in \{-2, -1, 0, 1, 2, 3, 4\} \)[/tex].
The common values are:
[tex]\[ -2, -1, 0, 1 \][/tex]
### Final Answer
Thus, the possible values of [tex]\( x \)[/tex] that satisfy both conditions are:
[tex]\[ -2, -1, 0, 1 \][/tex]
