Sure, let's solve the given set notation problem step-by-step.
We need to interpret the set notation and find the elements that satisfy the condition within the given range. The given notation represents a set:
[tex]\[
\{x \mid x \in \mathbb{Z}, -2 \leq x \leq 4\}
\][/tex]
Here's what this notation means in detail:
1. [tex]\(x\)[/tex] is an element of the set of integers ([tex]\(x \in \mathbb{Z}\)[/tex]).
2. [tex]\(x\)[/tex] must satisfy the condition [tex]\(-2 \leq x \leq 4\)[/tex].
Let's determine the integers that fall within the specified range from [tex]\(-2\)[/tex] to [tex]\(4\)[/tex].
1. Start with the lower bound:
- The smallest integer that satisfies the inequality [tex]\(-2 \leq x\)[/tex] is [tex]\(-2\)[/tex].
2. Then continue to the next integers to make sure they are within the bounds:
- The next integer is [tex]\(-1\)[/tex], which satisfies [tex]\(-2 \leq -1\)[/tex].
- The next integer is [tex]\(0\)[/tex], which satisfies [tex]\(-2 \leq 0\)[/tex].
- The next integer is [tex]\(1\)[/tex], which satisfies [tex]\(-2 \leq 1\)[/tex].
- The next integer is [tex]\(2\)[/tex], which satisfies [tex]\(-2 \leq 2\)[/tex].
- The next integer is [tex]\(3\)[/tex], which satisfies [tex]\(-2 \leq 3\)[/tex].
- The next integer is [tex]\(4\)[/tex], which satisfies [tex]\(-2 \leq 4\)[/tex].
We stop at [tex]\(4\)[/tex] since it is the upper bound.
So, the integers that satisfy the conditions [tex]\(-2 \leq x \leq 4\)[/tex] form the set:
[tex]\[
\{-2, -1, 0, 1, 2, 3, 4\}
\][/tex]
Thus, the solution for the given set notation is:
[tex]\[
\{ -2, -1, 0, 1, 2, 3, 4 \}
\][/tex]
