To solve the problem given [tex]\(\frac{1}{5} < \frac{1}{y} < \frac{1}{3}\)[/tex], we need to find the corresponding range for the expression [tex]\(5 - 2y\)[/tex].
Step 1: Understand the inequality [tex]\(\frac{1}{5} < \frac{1}{y} < \frac{1}{3}\)[/tex]
First, let's rewrite the inequality to isolate [tex]\(y\)[/tex]:
- Since [tex]\(\frac{1}{y}\)[/tex] is between [tex]\(\frac{1}{5}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex], we need to invert these fractions. Remember that when you invert an inequality with fractions, the inequality signs flip because the function [tex]\(f(y) = \frac{1}{y}\)[/tex] is a decreasing function.
Inverted, the inequality becomes:
[tex]\[ 5 > y > 3 \][/tex]
So, the range of [tex]\(y\)[/tex] is:
[tex]\[ 3 < y < 5 \][/tex]
Step 2: Finding the range for [tex]\(5 - 2y\)[/tex]
Now, we need to substitute the bounds of [tex]\(y\)[/tex] into the expression [tex]\(5 - 2y\)[/tex] to find its range.
1. Evaluate the expression at the lower bound [tex]\(y = 3\)[/tex]:
[tex]\[
5 - 2 \times 3 = 5 - 6 = -1
\][/tex]
2. Evaluate the expression at the upper bound [tex]\(y = 5\)[/tex]:
[tex]\[
5 - 2 \times 5 = 5 - 10 = -5
\][/tex]
Step 3: Combine the results
The expression [tex]\(5 - 2y\)[/tex] will vary between these two results, as [tex]\(y\)[/tex] varies from 3 to 5. Therefore:
[tex]\[ -5 < 5 - 2y < -1 \][/tex]
Summary
The possible values of the expression [tex]\(5 - 2y\)[/tex] are in the range:
[tex]\[
\boxed{-5 < 5 - 2y < -1}
\][/tex]
