To solve this problem, we need to convert the given Celsius temperature range into the corresponding Fahrenheit temperature range. We start with the given compound inequality:
[tex]\[ -40 < \frac{5}{9}(F-32) < 125 \][/tex]
This compound inequality can be broken down into two separate inequalities that we need to solve. Let's solve them step by step:
### Part 1: Solving [tex]\(-40 < \frac{5}{9}(F-32)\)[/tex]
1. Remove the fraction by multiplying all terms by [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[
\left(\frac{9}{5}\right)(-40) < F - 32
\][/tex]
2. Simplify [tex]\(\left(\frac{9}{5}\right)(-40)\)[/tex]:
[tex]\[
\left(\frac{9 \cdot -40}{5}\right) = -72
\][/tex]
So, the inequality now reads:
[tex]\[
-72 < F - 32
\][/tex]
3. Isolate [tex]\(F\)[/tex] by adding 32 to both sides:
[tex]\[
-72 + 32 < F
\][/tex]
4. Simplify:
[tex]\[
-40 < F
\][/tex]
So, the first part tells us that the Fahrenheit temperature must be greater than [tex]\(-40^\circ F\)[/tex].
### Part 2: Solving [tex]\(\frac{5}{9}(F-32) < 125\)[/tex]
1. Remove the fraction by multiplying all terms by [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[
\left(\frac{9}{5}\right) \left(\frac{5}{9}\right) (F - 32) < \left(\frac{9}{5}\right)125
\][/tex]
2. Simplify the left-hand side (the fractions cancel out):
[tex]\[
F - 32 < \left(\frac{9 \cdot 125}{5}\right)
\][/tex]
3. Simplify [tex]\(\left(\frac{9 \cdot 125}{5}\right)\)[/tex]:
[tex]\[
9 \cdot 25 = 225
\][/tex]
So, the inequality now reads:
[tex]\[
F - 32 < 225
\][/tex]
4. Isolate [tex]\(F\)[/tex] by adding 32 to both sides:
[tex]\[
F < 225 + 32
\][/tex]
5. Simplify:
[tex]\[
F < 257
\][/tex]
### Combining Both Parts
Now combining the results of both parts, we obtain the compound inequality:
[tex]\[
-40 < F < 257
\][/tex]
Therefore, the Fahrenheit temperature range at which the antifreeze protects the car is given by:
[tex]\[
-40 < F < 257
\][/tex]
In summary, the correct compound inequality corresponding to the temperature range mentioned is:
[tex]\[
-40 < \frac{5}{9}(F-32) < 125
\][/tex]
