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Answer :
To solve the inequality [tex]\( 36.4 \leq \frac{5}{9}(F - 32) \leq 37.6 \)[/tex] and transform it step-by-step, follow these instructions:
1. Identify the inequality:
[tex]\[
36.4 \leq \frac{5}{9}(F - 32) \leq 37.6
\][/tex]
2. Multiply each part of the inequality by [tex]\(\frac{9}{5}\)[/tex] to eliminate the fraction:
- Multiply the left side by [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[
\frac{9}{5} \cdot 36.4
\][/tex]
- Simplify the left side:
[tex]\[
\frac{9 \times 36.4}{5} = \frac{327.6}{5} = 65.52
\][/tex]
- The middle part (which initially has the fraction [tex]\(\frac{5}{9}\)[/tex]) will just become [tex]\(F - 32\)[/tex]:
[tex]\[
\frac{9}{5} \cdot \frac{5}{9}(F - 32) = 1 \cdot (F - 32) = F - 32
\][/tex]
- Multiply the right side by [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[
\frac{9}{5} \cdot 37.6
\][/tex]
- Simplify the right side:
[tex]\[
\frac{9 \times 37.6}{5} = \frac{338.4}{5} = 67.68
\][/tex]
3. Combine all the parts back into the inequality:
[tex]\[
65.52 \leq F - 32 \leq 67.68
\][/tex]
4. Isolate [tex]\(F\)[/tex] by adding 32 to each part of the inequality:
[tex]\[
65.52 + 32 \leq F \leq 67.68 + 32
\][/tex]
Simplifying this we get:
[tex]\[
97.52 \leq F \leq 99.68
\][/tex]
So the final answer is:
[tex]\[
97.52 \leq F \leq 99.68
\][/tex]
This means that the temperature in Fahrenheit (F) will range from 97.52°F to 99.68°F.
1. Identify the inequality:
[tex]\[
36.4 \leq \frac{5}{9}(F - 32) \leq 37.6
\][/tex]
2. Multiply each part of the inequality by [tex]\(\frac{9}{5}\)[/tex] to eliminate the fraction:
- Multiply the left side by [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[
\frac{9}{5} \cdot 36.4
\][/tex]
- Simplify the left side:
[tex]\[
\frac{9 \times 36.4}{5} = \frac{327.6}{5} = 65.52
\][/tex]
- The middle part (which initially has the fraction [tex]\(\frac{5}{9}\)[/tex]) will just become [tex]\(F - 32\)[/tex]:
[tex]\[
\frac{9}{5} \cdot \frac{5}{9}(F - 32) = 1 \cdot (F - 32) = F - 32
\][/tex]
- Multiply the right side by [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[
\frac{9}{5} \cdot 37.6
\][/tex]
- Simplify the right side:
[tex]\[
\frac{9 \times 37.6}{5} = \frac{338.4}{5} = 67.68
\][/tex]
3. Combine all the parts back into the inequality:
[tex]\[
65.52 \leq F - 32 \leq 67.68
\][/tex]
4. Isolate [tex]\(F\)[/tex] by adding 32 to each part of the inequality:
[tex]\[
65.52 + 32 \leq F \leq 67.68 + 32
\][/tex]
Simplifying this we get:
[tex]\[
97.52 \leq F \leq 99.68
\][/tex]
So the final answer is:
[tex]\[
97.52 \leq F \leq 99.68
\][/tex]
This means that the temperature in Fahrenheit (F) will range from 97.52°F to 99.68°F.
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