We are tasked with finding the maximum value of [tex]\(|y - 2x|\)[/tex] where [tex]\(2 \leq x \leq 5\)[/tex] and [tex]\(-4 < y \leq -3\)[/tex].
Let's go through the ranges for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
1. Given [tex]\(2 \leq x \leq 5\)[/tex], the possible integer values for [tex]\(x\)[/tex] are [tex]\(2\)[/tex], [tex]\(3\)[/tex], [tex]\(4\)[/tex], and [tex]\(5\)[/tex].
2. Given [tex]\(-4 < y \leq -3\)[/tex], the possible integer values for [tex]\(y\)[/tex] are [tex]\(-4\)[/tex] (but since it's not included, we skip [tex]\(-4\)[/tex]) and [tex]\(-3\)[/tex].
To find [tex]\(|y - 2x|\)[/tex], we need to calculate this expression for all combinations of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] within the given ranges.
Let's compute it step-by-step for each combination:
1. For [tex]\(x = 2\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[
|y - 2x| = |-4 - 2(2)| = |-4 - 4| = |-8| = 8
\][/tex]
2. For [tex]\(x = 5\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[
|y - 2x| = |-4 - 2(5)| = |-4 - 10| = |-14| = 14
\][/tex]
3. For [tex]\(x = 2\)[/tex] and [tex]\(y = -3\)[/tex]:
[tex]\[
|y - 2x| = |-3 - 2(2)| = |-3 - 4| = |-7| = 7
\][/tex]
4. For [tex]\(x = 5\)[/tex] and [tex]\(y = -3\)[/tex]:
[tex]\[
|y - 2x| = |-3 - 2(5)| = |-3 - 10| = |-13| = 13
\][/tex]
Now, we compare these computed values:
[tex]\[
8, 14, 7, 13
\][/tex]
The maximum value from this set is [tex]\(14\)[/tex].
Thus, the maximum value of [tex]\(|y - 2x|\)[/tex] is [tex]\(\boxed{14}\)[/tex].
