To solve for the possible values of the expression [tex]\(5 - 2y\)[/tex] given the inequality [tex]\(\frac{1}{5} < \frac{1}{y} < \frac{1}{3}\)[/tex], follow these steps:
1. Analyze and Invert the Inequality:
Given:
[tex]\[
\frac{1}{5} < \frac{1}{y} < \frac{1}{3}
\][/tex]
We need to find the bounds for [tex]\(y\)[/tex]. Invert the inequality:
[tex]\[
\frac{1}{5} < \frac{1}{y} < \frac{1}{3}
\][/tex]
Inverting the inequality means reversing it and flipping the inequality signs:
[tex]\[
5 > y > 3
\][/tex]
So, [tex]\(y\)[/tex] lies between 3 and 5.
2. Find the Minimum and Maximum Values of [tex]\(y\)[/tex]:
From the inequality [tex]\(5 > y > 3\)[/tex]:
- The minimum value of [tex]\(y\)[/tex] is 3.
- The maximum value of [tex]\(y\)[/tex] is 5.
3. Substitute and Solve for the Expression [tex]\(5 - 2y\)[/tex]:
We need to find the range of values for [tex]\(5 - 2y\)[/tex] within the bounds [tex]\(3 < y < 5\)[/tex].
- Substitute [tex]\(y = 3\)[/tex] into [tex]\(5 - 2y\)[/tex]:
[tex]\[
5 - 2(3) = 5 - 6 = -1
\][/tex]
- Substitute [tex]\(y = 5\)[/tex] into [tex]\(5 - 2y\)[/tex]:
[tex]\[
5 - 2(5) = 5 - 10 = -5
\][/tex]
4. Determine the Range of the Expression:
From the calculated values:
[tex]\[
5 - 2(3) = -1 \quad \text{and} \quad 5 - 2(5) = -5
\][/tex]
Since 3 < [tex]\(y\)[/tex] < 5, [tex]\(5 - 2y\)[/tex] ranges from [tex]\(-1\)[/tex] to [tex]\(-5\)[/tex].
Therefore, the inequality for the expression [tex]\(5 - 2y\)[/tex] is:
[tex]\[
-5 < 5 - 2y < -1
\][/tex]
Summarizing:
[tex]\[
\begin{array}{l}
5-2 y \\
-5 < 5-2 y < -1
\end{array}
\][/tex]
