In an experiment, a solution of hydrochloric acid is to be kept between $30^{\circ}$ and $35^{\circ}$ Celsius. What is the range of temperature in degree Fahrenheit if conversion formula is given by $\mathrm{C}=\frac{5}{9} \quad(\mathrm{~F}-32)$, where $\mathrm{C}$ and $\mathrm{F}$ represent temperature in degree Celsius and degree Fahrenheit, respectively.

Step 1: Set up the inequality for Celsius temperature

The problem states that the temperature in Celsius, denoted by $C$, must be kept between $30^{\circ}$ and $35^{\circ}$. This can be written as a compound inequality.

Step 2: Substitute the conversion formula

We are given the conversion formula $C = \frac{5}{9}(F-32)$. We substitute this expression for $C$ into the inequality to relate the Fahrenheit temperature $F$ to the given Celsius range.

Step 3: Multiply by the reciprocal to isolate $(F-32)$

To isolate the term $(F-32)$, we multiply all parts of the inequality by the reciprocal of $\frac{5}{9}$, which is $\frac{9}{5}$. This operation maintains the direction of the inequality signs because we are multiplying by a positive number.

Step 4: Simplify the inequality

Now, we perform the multiplication on both sides of the inequality. $30 \times \frac{9}{5} = 6 \times 9 = 54$ and $35 \times \frac{9}{5} = 7 \times 9 = 63$.

Step 5: Add 32 to all parts to find the range of F

To find the range of $F$, we add $32$ to all parts of the inequality. This isolates $F$ and gives us the temperature range in Fahrenheit.

Step 6: Calculate the final Fahrenheit range

Performing the addition, we get $54 + 32 = 86$ and $63 + 32 = 95$. Thus, the temperature in Fahrenheit must be between $86^{\circ}$ and $95^{\circ}$.

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