Problem 54 - Elementary Algebra

Question

Set up an algebraic inequality and then solve it. A certain brand of makeup is guaranteed not to run if the temperature is less than \(35^{\circ} \mathrm{C}\). Give an equivalent statement using degrees Fahrenheit.

Step-by-Step Solution

Answer

The makeup won't run if the temperature is less than 95°F.

Step 1: Understanding the Problem

We need to convert the inequality for temperature from degrees Celsius to degrees Fahrenheit. The inequality in Celsius is given as: \[ T_{C} < 35^{\circ} \text{C} \] Our goal is to express this in terms of Fahrenheit equivalents.

Step 2: Celsius to Fahrenheit Conversion Formula

To convert temperature from Celsius to Fahrenheit, we use the formula: \[ T_{F} = \frac{9}{5}T_{C} + 32 \] where \( T_{F} \) is the temperature in Fahrenheit and \( T_{C} \) is the temperature in Celsius.

Step 3: Substituting Celsius Inequality into the Formula

Substitute \( T_{C} = 35 \) into the conversion formula to find the corresponding Fahrenheit temperature:\[ T_{F} = \frac{9}{5} \times 35 + 32 \]

Step 4: Performing the Calculation

Calculate the value:\[ T_{F} = \frac{9}{5} \times 35 + 32 = 63 + 32 = 95 \]Thus, the equivalent temperature in Fahrenheit is 95°F.

Step 5: Writing the Inequality in Fahrenheit

Now replace 35°C in the original inequality with 95°F:\[ T_{F} < 95^{\circ} \text{F} \] This is the equivalent inequality in degrees Fahrenheit.

Key Concepts

Celsius to Fahrenheit Conversion

Converting temperatures between Celsius and Fahrenheit can be simple if you know the correct formula. The relationship between these two temperature scales is represented by the formula:

• \[ T_{F} = \frac{9}{5}T_{C} + 32 \]

This means you multiply the Celsius temperature \(T_{C}\) by \(\frac{9}{5}\) and then add 32 to convert it to Fahrenheit \(T_{F}\).
Let's take an example. If we have a temperature of \(35^{\circ} \text{C}\), we simply plug it into the formula:

• \[ T_{F} = \frac{9}{5} \times 35 + 32 \]

After performing the calculation, we find that \(35^{\circ} \text{C}\) equals 95°F. This formula will always help you switch between Celsius and Fahrenheit with ease.

Temperature Inequality

In our problem, we're interested in temperatures below a certain point to ensure makeup doesn't run. Initially, we had:

• \[ T_{C} < 35^{\circ} \text{C} \]

This inequality tells us that the temperature must be less than 35°C.
When we convert this into Fahrenheit using the conversion method, we achieve:

• \[ T_{F} < 95^{\circ} \text{F} \]

This new inequality confirms that in Fahrenheit, the temperature must stay under 95°F. Understanding temperature inequalities is vital in fields like cosmetics and engineering, where specific temperature conditions must be met.
Remember, inequalities help us set boundaries and conditions that need to be maintained in different scenarios.

Inequality Solving Steps

Solving inequalities is similar to solving equations, but with special attention to the inequality sign. Here’s a step-by-step guide to solving temperature inequalities:

• **Identify the given inequality:** Start with the known inequality \(T_{C} < 35^{\circ} \text{C}\).
• **Use the conversion formula:** Substitute \(T_{C}\) into the conversion formula to switch scales. Here, \(T_{F} = \frac{9}{5}T_{C} + 32\).
• **Perform calculations:** For \(T_{C} < 35\), calculate \(T_{F}\) by plugging in 35, which yields \(T_{F} = 95\).
• **Rewrite the inequality in new terms:** Now express the inequality as \(T_{F} < 95^{\circ} \text{F}\).

By breaking down each step, solving an inequality like this ensures you handle changes between systems of measurement correctly. Always remember, inequalities require care with the direction of the inequality sign, ensuring calculations maintain their intended conditions.