To determine the Fahrenheit temperature range at which the antifreeze protects the car, we need to solve the inequality given the formula for converting Celsius to Fahrenheit: [tex]\( C = \frac{5}{9} (F - 32) \)[/tex].
We are given the Celsius temperature range [tex]\(-40^{\circ}C\)[/tex] to [tex]\(125^{\circ}C\)[/tex]. So we need to solve the inequality:
[tex]\[ -40 < \frac{5}{9}(F - 32) < 125 \][/tex]
Let's solve this step by step.
1. Start with the inequality:
[tex]\[ -40 < \frac{5}{9}(F - 32) < 125 \][/tex]
2. To eliminate the fraction [tex]\(\frac{5}{9}\)[/tex], multiply all parts of the inequality by [tex]\(\frac{9}{5}\)[/tex]:
[tex]\[ -40 \times \frac{9}{5} < F - 32 < 125 \times \frac{9}{5} \][/tex]
3. Perform the multiplications:
[tex]\[ -72 < F - 32 < 225 \][/tex]
4. Isolate [tex]\( F \)[/tex] by adding 32 to each part of the inequality:
[tex]\[ -72 + 32 < F - 32 + 32 < 225 + 32 \][/tex]
[tex]\[ -40 < F < 257 \][/tex]
So, the Fahrenheit temperature range at which the antifreeze protects the car is:
[tex]\[ -40^{\circ}F < F < 257^{\circ}F \][/tex]
The correct inequality for the given problem is:
[tex]\[ -40 < \frac{5}{9}(F - 32) < 125 ; -40 < F < 257 \][/tex]
