To find the intersection of the intervals [tex]\( P \)[/tex] and [tex]\( Q \)[/tex], defined as [tex]\( P = \{-3 < x < 1\} \)[/tex] and [tex]\( Q = \{-1 < x < 3\} \)[/tex], we need to determine the set of all [tex]\( x \)[/tex] values that satisfy both conditions simultaneously.
1. Identify the boundaries of each interval:
- Interval [tex]\( P \)[/tex] runs from [tex]\(-3\)[/tex] to [tex]\(1\)[/tex], expressed as [tex]\(-3 < x < 1\)[/tex].
- Interval [tex]\( Q \)[/tex] runs from [tex]\(-1\)[/tex] to [tex]\(3\)[/tex], expressed as [tex]\(-1 < x < 3\)[/tex].
2. Find the start of the intersection:
- The intersection will start at the larger of the two interval starts: the maximum of [tex]\(-3\)[/tex] and [tex]\(-1\)[/tex].
- Thus, the intersection start is [tex]\(-1\)[/tex].
3. Find the end of the intersection:
- The intersection will end at the smaller of the two interval ends: the minimum of [tex]\(1\)[/tex] and [tex]\(3\)[/tex].
- Thus, the intersection end is [tex]\(1\)[/tex].
4. Combine these results:
- The intersection is between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
Therefore, the intersection [tex]\( P \cap Q \)[/tex] is given by [tex]\(-1 \leq x \leq 1\)[/tex].
So, the answer is: [tex]\(\boxed{\{-1 \leq x \leq 1\}}\)[/tex].
